The Characteristic Initial Value Problem for the ...

M. Dossa and C. Tadmon (2010) “The Characteristic Initial Value Problem,” Applied Mathematics Research eXpress, Vol. 2010, No. 2, pp. 154–231 doi:10.1093/amrx/abq014

The Characteristic Initial Value Problem for the Einstein–Yang–Mills–Higgs System in Weighted Sobolev Spaces Marcel Dossa1 and Calvin Tadmon2 1 Department

Correspondence to be sent to: [email protected]

We revisit and complete existence and uniqueness results stated and partially ¨ established by Muller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two intersecting smooth null hypersurfaces. The new ingredient of this investigation consists of some Moser estimates expressed in the same weighted Sobolev spaces as those used ¨ by Muller zum Hagen. These estimates, combined with energy inequalities obtained ¨ by Muller zum Hagen for the linearized Goursat problem, permit us to develop a fixed point method which leads clearly to an existence and uniqueness result for the quasilinear Goursat problem. As an application we locally solve, under finite differentiability conditions, the characteristic initial value problem for the Einstein–Yang–Mills–Higgs system using harmonic gauge for the gravitational potentials and Lorentz gauge for the Yang–Mills potentials. 1 Introduction This work is devoted to the characteristic initial value problem for the Einstein–Yang– Mills–Higgs (EYMH) system with initial data prescribed on two intersecting smooth null hypersurfaces. The interests and physical motivations for studying characteristic initial value problems have been widely mentioned in [34, 37]. These problems, for example, play a fundamental role in the recent theory of black holes formation made Received February 28, 2010; Revised June 10, 2010; Accepted July 22, 2010 c The Author(s) 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected].

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of Mathematics, Faculty of Science, University of Yaounde I, Po. Box 812, Yaounde, Cameroon and 2 Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, Po. Box 67, Dschang, Cameroon

The Characteristic Initial Value Problem

155

up by Christodoulou [11]. It is well known that the EYMH system is not an evolution system as it stands. In order to reduce it to a hyperbolic system, one needs to impose to the unknown functions some supplementary conditions called gauge conditions which, due to the deep structure of the system, must satisfy the following properties: (i) whenever these gauge conditions are fulfilled in the entire space–time, the EYMH system reduces to a quasilinear hyperbolic system called the evolution system or the reduced EYMH system. (ii) whenever the associated evolution system is satisfied in the entire space– time and the gauge conditions are satisfied on the null hypersurfaces that system are satisfied in the entire space–time. It therefore follows that when the choice of gauge conditions is made, the initial value problem for the EYMH system is decomposed into two parts called the evolution problem and the constraints problem. The resolution of the evolution problem is equivalent to the resolution of the reduced EYMH system thanks to the choice of the gauge conditions. Due to the gauge conditions, the initial data for the reduced EYMH system cannot be given freely. It is necessary to construct, from an arbitrary choice of some components of the gravitational potentials and Yang–Mills potentials (called free data) on the initial null hypersurfaces, all the initial data such that the solution of the reduced EYMH system with these initial data satisfies the gauge conditions on the initial null hypersurfaces. The construction of such data is referred to as the resolution of the constraints problem. Throughout the work, we will use harmonic gauge for the Einstein system and Lorentz gauge for the Yang–Mills system. The reduced EYMH system stands as a hyperbolic quasilinear system of secondorder differential equations when harmonic and Lorentz gauges are used. The Goursat problem (i.e., the characteristic initial value problem with initial data prescribed on two intersecting smooth null hypersurfaces) for such a system has been studied by ¨ Rendall [37] and Muller zum Hagen [35] in 1990. Rendall [37] established a C ∞ existence and uniqueness result for quasilinear hyperbolic systems of second order with C ∞ data prescribed on two intersecting smooth null hypersurfaces, by transforming this problem to an ordinary spatial Cauchy problem with zero initial data. Then, by using the harmonic gauge condition, he applied this C ∞ existence and uniqueness result to solve the characteristic initial value problem (evolution and constraints problems) for the Einstein equations in vacuum and with


carry the initial data, then these gauge conditions and the complete EYMH

156

M. Dossa and C. Tadmon

relativistic perfect fluid source. For the sake of more physical applications, it is known that, solutions of partial differential equations (PDE) of finite differentiability order are more important and realistic than those of infinite differentiability order. Rendall [37, Section 7] mentioned briefly how results of finite differentiability order can be obtained ¨ for data of finite differentiability order although proofs were not given. Muller zum Hagen [35] (see also [12, 13]) used Sobolev-type inequalities to derive energy inequalities that enable him solve, in some weighted Sobolev spaces (results of finite differentiability order), the characteristic initial value problem for linear hyperbolic systems of second ¨ order. In the abstract of his paper, Muller zum Hagen [35] proposed to extend, in a but to our knowledge this has never been done. ¨ In the present paper we propose to fill the gap left by Muller zum Hagen [35] by giving a detailed proof of the local existence and uniqueness result announced in [12, 13, 35] for the quasilinear Goursat problem. We achieve this goal by implementing a fixed point method. The main tools consist basically of certain Sobolev inequalities and Moser estimates combined with the energy inequalities established in [35] for the linearized Goursat problem. These tools are similar to those used by Dionne [17] for the spatial ordinary Cauchy problem. Analogous tools were made up by Dossa [19, 20] and they were used to treat the characteristic initial value problem with initial data prescribed on a null cone. Apart from the fundamental papers [13, 35, 37], some other works on Goursat problems can be found in [3–5, 16, 26, 27, 34, 36, 38–41]. As an application we establish, under finite differentiability assumptions, a local existence and uniqueness result for the Goursat problem associated to EYMH system in harmonic and Lorentz gauges, generalizing thereby some preceding results such as those obtained by Rendall [37] and Damour and Schmidt [15] under C ∞ assumptions. The existence results on the spatial ordinary Cauchy problem associated to YMH or YM systems have been achieved by several authors (see e.g., [7–9, 14, 17, 22, 23, 28, 30]). In a forthcoming work we intend to discuss the global resolution of the Goursat problem for the EYMH system in harmonic coordinates and Lorentz gauge by adapting recent methods developed by Lindblad and Rodnianski [29] to globally treat ordinary spatial Cauchy problems for Einstein equations in vacuum. The paper is organized as follows. In Section 2, we recall basic definitions and ¨ assumptions introduced by Muller zum Hagen [35]. In Section 3, we begin by recalling the definition of the weighted Sobolev spaces with their corresponding norms as used in [35]. We then present the results of [35] concerning an initial value problem for second-order hyperbolic linear systems with data assigned to two intersecting null


forthcoming work, the existence and uniqueness result obtained to the quasilinear case,


157

hypersurfaces. Our main result concerning the initial value problem for second-order hyperbolic quasilinear systems with data prescribed on two intersecting null hypersurfaces is stated at the end of Section 3. Section 4 is devoted to the statement of Sobolevtype inequalities while Section 5 deals with the establishment of Moser-type estimates. In Section 6, we provide the detailed proof of our main result stated in Section 3. Section 7 is concerned properly with the resolution of the Goursat problem associated to the EYMH system. Both the evolution and the constraints problems are treated.

2 Geometric Background and Notation

be studied. Throughout the paper, the notations of [35] are used. Let n be an integer such that n≥ 2, and let L denote a compact domain of Rn+1 with a piecewise smooth boundary ∂ L, G 1 and G 2 are two n-dimensional surfaces such that G ω ⊂ L for ω = 1, 2. We assume that G ω are defined by G ω = {x ∈ L : xω = 0},

ω = 1, 2,

(2.1)

where x = (xa) = (x1 , . . . , xn+1 ) is the global canonical coordinates system of Rn+1 . In addition, we assume that G 1 ∪ G 2 ⊂ ∂ L. Set τ (x) = x1 + x2

and

T0 = sup τ (x).

(2.2)

x∈L

For t ∈ [0, T0 ], define the following point sets: L t = {x ∈ L : 0 ≤ τ (x) ≤ t},

Λt = {x ∈ L : τ (x) = t},

G ωt = {x ∈ G ω : 0 ≤ τ (x) ≤ t},

Γtω = {x ∈ G ω : τ (x) = t}.

(2.3)

It is clear that

Lt =

0≤σ ≤t

Λσ ,

G ωt =

Γσω ,

Γ ≡ G 1 ∩ G 2 = Γ01 = Γ02 .

(2.4)

0≤σ ≤t

If S is a surface (submanifold) of Rn+1 , then D S denotes a basis of derivatives tangent to S. E.g., DΓtω = (D3 , . . . , Dn+1 ),

DG 1 = (D2 , . . . , Dn+1 ),

DG 2 = (D1 , D3 , . . . , Dn+1 ), where Da =

∂ ∂ xa

DΛt = (Dε , D3 , . . . , Dn+1 ),

for a = 1, . . . , n + 1

and

1 Dε = (D1 − D2 ). 2


In this section we introduce the geometric framework where the Goursat problem will

158


For k ∈ N, define D Sk = D S . . . D S . k-times

(2.5)

3 Recalling the Result of [35] and Statement of the Main Result for the Quasilinear Problem 3.1 Spaces of functions used

spaces has been governed by the following phenomena encountered in the study of the Goursat problem in L and described in [12, 13, 18, 34, 35]: • the blowup behavior of Sobolev constants on subdomains L t , G t , and Λt of L, as t → 0, • the fact that, unlike the ordinary spatial Cauchy problem, the restrictions to the initial surfaces G ω of the derivatives of the solution of the Goursat problem on L, associated to a second-order hyperbolic system (S), are determined by integrating the restrictions to G ω of the system (S) and its derivatives. ¨ We recall the definition of the weighted Sobolev spaces introduced by Muller Zum Hagen [35]. Let S be some surface such that S ∈ {L, L t , G t , Λt } and let v be a tensor field on S. Set

|v|2S =

|v|2 dS,

(3.1)

S

where dS is the volume element induced on S by dx1 . . . dxn+1 , |v| is the norm of v with respect to the Kronecker metric δ ab. For a vector field v = (v I ) we have |v| = [ I (v I )2 ]1/2 . Define

|St | = tα

⎧ 1 ⎪ ⎪ ⎪ ⎨2 where α = 1 ⎪ ⎪ ⎪ ⎩ 0

if St = Λt , G ωt , if St = L t ,

(3.2)

if St = Γtω , Γ .

For s ∈ N, set |v|sSt ,R = |St |−1

s

1/2 |D kRv|2St

,

(3.3)

k=0

|v|sSt = |v|sSt ,St ,

(3.4)


We shall use the same spaces of functions as in [35]. The choice of these functional


159

where St is any of the points sets defined in (2.3) and that appear in (3.2), R is a surface of Rn+1 such that St ⊂ R ⊂ L, L is defined in Section 2. It is easy to see that |v|sSt = |St |−1 v H s (St ) , where H s (St ) is the usual Sobolev space endowed with the norm

s 1/2 k 2 v H s (St ) = |D St v| St , k=0

D Skt v is understood in the distributional sense. When H s (St ) is endowed with the equivFor convenience, we set

v H s (St ,R) =

s

1/2 |D kRv|2St

.

k=0

Then |v|sSt ,R = |St |−1 v H s (St ,R) . As in [35], let V(St ) denote the volume of St , i.e., V(St ) =

St

dSt , then V(Λt ) and V(G ωt )

are (up to a multiplicative constant) equal to t, V(L t ) is (up to a multiplicative constant) equal to t2 , and V(Γtω ) is a constant with respect to t. Hence 0 < lim |St |−1 (V(St ))1/2 < +∞.

(3.5)

t→0

We also define the following modified Sobolev norm (see [35]) σ ,St |v| St ,s = ess sup |v|Σ , s

(3.6)

0≤σ ≤t

where St is decomposed into a congruence of surfaces Σσ , i.e., St =

Σσ ,

0≤σ ≤t

where Σσ = St ∩ Λσ . We define further norms as follows: s−1 s−1 1/2 1/2 ω ω ω G ωt G G G t t v s = (|Dωk v|2(s−k)−1 )2 , v s,1t = (|Dωk v|2(s−k)−1 )2 , v sL t

=

k=0

(|v|sL t )2

+

2

Gω ( v s t )2

ω=1

k=1

1/2 ,

s−1 1/2 k 2 (|Dω v|G ωt ,2(s−k)−1 ) , v G ωt ,s = k=0

Lt v s,1

=

(|v|sL t )2

2 Gω + ( v s,1t )2 ω=1

1/2 ,

s−1 1/2 k 2 v G ωt ,s,1 = (|Dω v|G ωt ,2(s−k)−1 ) , k=1


alent norm |v|sSt , it is denoted Hs (St ). The factor |St |−1 of (3.3) is called the weight factor.

160


v L t ,s = (|v| L t ,s ) + 2

Γ ,G ω ,1 v s t

=

s−1

2

1/2 ( v G ωt ,s )

ω=1

2

,

2 2 v G ωt ,s,1 v L t ,s,1 = (|v| L t ,s ) +

,

ω=1

1/2

Γ ,G ωt (|Dωk v|2(s−k)−2 )2

1/2

2

.

(3.7)

k=0

C ∞ (L t ) denotes the space of restrictions to L t of functions which are C ∞ on Rn+1 . From the Hilbertian structure of the usual Sobolev space H s (L t ), it is easy to see that Hs (L t ) is a Hilbert space and C ∞ (L t ) is dense in Hs (L t ). Hs (L t ) denotes the completion of C ∞ (L t ) with respect to the norm v sL t . Define for Mt = L t or G ωt , Es (L t ) = {v ∈ Hs (L t ) : v L t ,s < +∞}.

E s (Mt ) is endowed with the norm |v| Mt ,s and Es (L t ) is endowed with the norm v L t ,s to be Banach spaces. Let W be an open subset of Rl , 0 < l ∈ N. C bk(L t × W) denotes the space of functions f : L t × W → R such that D i f exists (in the usual sense) for all i = 0, . . . , k and are continuous and bounded on L t × W. C bk(L t × W) is endowed with the norm f C bk(L t ×W) =

sup

(x,w)∈L t ×W, |α+β|≤k

|Dxα Dwβ f(x, w)|.

3.2 The linear Goursat problem: statement of the existence and uniqueness result of [35]

¨ We recall the result obtained by Muller zum Hagen [35] for the second-order linear hyperbolic systems. In order to reduce the length of the paper we do not go into details concerning energy inequalities and C ∞ existence and uniqueness result combined to obtain the main result of [35]. The references [21, 27, 37] where C ∞ existence and uniqueness results are clearly stated and proved they can be consulted. The energy inequalities are established in [35] (see also [19, 20]). Here and throughout the remainder of the paper, Einstein summation convention is used, i.e., v aua = a v aua. The following initial value problem is considered: gab Dabu + B a Dau + Cu = f

in L T ,

u= u on G ωT ,

(3.8)

ω

where u= (u ),

∂uA , Dau= (Dau ) = ∂ xa

f = ( f A),

u = (uA), ω = 1, 2, a, b, . . . = 1, . . . , n + 1, A, I , J, . . . = 1, . . . , N.

A

A

ω

ω

∂ 2 uA Dabu= , ∂ xa∂ xb

B a = (B JaI ),

C = (C JI ), (3.9)


E s (Mt ) = {v ∈ Hs (Mt ) : |v| Mt ,s < +∞},


161

In addition to the assumptions of Section 2, the following supplementary assumptions are made.

Assumptions (Lin). ⎫ gab(x) is a Lorentz metric of signature (−, +, . . . , +), ⎪ ⎪ ⎬ G 1 and G 2 are null hypersurfaces w.r.t. gab, ⎪ ⎪ ⎭ Γ ≡ G 1 ∩ G 2 is a (n − 1)-smooth spatial surface.

(3.10)

gab Daτ Dbτ < 0 ab

g

in L,

(3.11)

is regularly hyperbolic in L with hyperbolicity constant h ∈ (0, ∞),

(3.12)

i.e., any past directed null curve in L can be extended to a null curve, which hits G 1 ∪ G 2 and lies completely in L; moreover, R = ∂ L\(G 1 ∪ G 2 ) is a non-timelike piecewise smooth surface. There exists constants h, h1 , h2 , h3 ∈ (0, ∞) such that ⎫ −gabtatb > h1 where ta = Daτ , ⎪ ⎪ ⎬ ab 2 ab 2 ab (g katb = 0) ⇒ (h2 |k| < g kakb) where |k| = kakbδ , ⎪ ⎪ ⎭ |g| < h where |g|2 = gac gbdδ δ , −1 h = max(h−1 1 , h2 , h3 ),

3

(3.13)

ab cd

where δab is the Kronecker metric in L defined by δab =

⎧ ⎨1

if a = b,

⎩0

if a = b.

We are now in a position to state the main theorem of [35]. Before doing this we adopt the following notation: [w]Γ denotes the restriction of w to Γ , for any function w defined on G ω . Theorem 3.1. Let n and s be two integers such that n/2 < s − 2, n≥ 2. Let gab be a regularly hyperbolic metric with hyperbolicity constant h ∈ (0, ∞) (see (3.12) and (3.13)). We assume that g, B, C ∈ Hs (L T ), u ∈ E 2s−1 (G ωT ), ω

f ∈ Hs−1 (L T ), [u]Γ ∈ H2s−1 (Γ ), ω

u = u on Γ . 1

2

(3.14)


The function τ (x) = x1 + x2 is timelike, i.e.,

162


Then the linear characteristic initial value problem (3.8) has in L T a unique solution u∈ Es (L T ). Furthermore u satisfies the following energy inequality for 0 < t ≤ T: u L t ,s ≤ c(h, γ sL t , γ Γs , t)

(|u|G ωt ,2s−1 +

ω=1,2

ω

Γ ,G ω ,1 f s−1 t )

+ t

Lt f s−1

,

(3.15)

where c is a non-decreasing continuous function of each of its arguments h, γ sL t , γ Γs , t, with

= t( C ,

B, g sL t ),

γ Γs

=

2

Γ ,G C , B, g s−1

ω

,1

,

(3.16)

ω=1

C , B, g sL T = C sL T + B sL T + g sL T , C ,

Γ ,G ω ,1 B, g s−1

Γ ,G ω ,1 = C s−1

+

Γ ,G ω ,1 B s−1

+

(3.17)

Γ ,G ω ,1 g s−1 .

(3.18)

See [35]. One mainly uses C ∞ existence and uniqueness results combined appro-

Proof.

priately with energy inequalities and density arguments.

3.3 The quasilinear Goursat problem: statement of the main local existence and uniqueness result of this paper

We state the local existence and uniqueness result for characteristic second-order quasilinear hyperbolic problems. The proof will be given later in Section 6. The functional framework used here consists of the weighted Sobolev spaces defined above. The following quasilinear initial value problem is considered with unknown u gab(x, u)Dabu= f(x, u, Du) u= u on ω

in L T ,

(3.19)

G ωT ,

where ∂uA , Du= (Dau ) = ∂ xa

T ∈ (0, T0 ], f = ( f A),

u= (u ), A

u = (uA), ω

ω

A

ω = 1, 2,

∂ 2 uA Dabu= , ∂ xa∂ xb

a, b, . . . = 1, . . . , n + 1,

A, I , J, . . . = 1, . . . , N.


γ sL t


163

The following assumptions on the coefficients and the initial data of (3.19) will be needed. ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

s is an integer such that s > n/2 + 2, gab(x, u) ∈ C 2s−1 (U × V) where U is an open domain of Rn+1 and V is an open domain of R N , such that L ⊂ U , R = ∂ L\(G 1 ∪ G 2 ) is a piecewise smooth surface that admits, at each point x ∈ R, a future-directed normal na(x) such that

ω

ω

u∈ ω

(3.20)

ω

E 2s−1 (G ωT ),

[u]Γ ∈ H2s−1 (Γ ),

u = u on Γ .

ω

1

2

(3.21)

The local existence and uniqueness result for the characteristic initial value problem for the second-order quasilinear hyperbolic systems is stated as follows:

Theorem 3.2. (i) Assume (3.20) and (3.21). Then ∃T1 ∈ (0, T] such that the quasilinear characteristic initial value problem (3.19) has in L T1 a unique solution u∈ Es (L T1 ). (ii) Assume in addition to (3.20) and (3.21) that 0 ∈ V, f0 ∈ Hs−1 (L T )

and

∂2 f ∂2 f ∂2 f , , ∈ C 2s−3 (U × V × R N(n+1) ), (∂u)2 ∂u∂ Du (∂ Du)2

where f0 is the function defined on U by f0 (x) = f(x, 0, 0). There exists a positive real number d such that if f0 L T ,s−1 < d,

2 ω=1

|u|G ωT ,2s−1 < d, ω

then the solution in (i) is global, i.e., T1 = T.

Remark 3.3. Theorem 3.2 was announced in [12, 13, 35] under slightly more restricted assumptions, but to our knowledge the rigorous details of its proof are not yet available in the literature. It is our aim, in this paper, to provide a detailed proof of Theorem 3.2. In order to do this, we implement a fixed point method. The fundamental tools consist


⎪ ⎪ ⎪ ⎪ ⎪ ab ⎪ −g (x, u)tatb > 0, ∀(x, u) ∈ U × V where ta = Daτ , ⎪ ⎪ ⎪ ⎪ ab ab ⎪ (g (x, u)katb = 0) ⇒ (0 < g (x, u)kakb) ∀(x, u) ∈ U × V, ⎪ ⎪ ⎪ ⎪ ⎪ 2s−3 N(n+1) ⎪ (U × V × R ), f(x, u, Du) ∈ C ⎪ ⎪ ⎪ ω ω ω ab ⎭ u is continuous on G , u(G T ) ⊂ V and G is characteristic w.r.t. g (x, u(x)), ⎪ gab(x, u)na(x)nb(x) ≤ 0 ∀u∈ V,

164


essentially of Theorem 3.1 (namely the energy inequality (3.15)), Sobolev-type inequali

ties, and original Moser-type estimates.

4 Sobolev-type Inequalities In this section, we establish Sobolev-type inequalities in weighted Sobolev spaces defined in Section 3. Those Sobolev inequalities play a key role in the resolution of the linear characteristic initial value problem, essentially when energy inequalities are concerned (see [35]). It will also be seen that the establishment of the Moser-type of the long list of Sobolev-type inequalities which are provided in this section. The role of the Moser-type estimates will be explained in Section 5. We begin by a generic lemma. Before doing this, we recall the definition of some usual Sobolev spaces (see [1]) which will be needed. If Ω is an open subset of R N and (m, p) ∈ N × (0, ∞), then Wm, p(Ω) denotes the usual Sobolev space of functions u: Ω → R such that D α u exist in the distributional sense for all α = (α1 , . . . , α N ) ∈ N N with |α| ≤ m and ⎡ u Wm, p(Ω) ≡ ⎣

|α|≤m

⎤1/ p D α u L p(Ω) ⎦ p

< +∞,

where D α u L p(Ω) = p

Ω

|D α u(x)| p dx,

Dα =

∂ |α| , (∂ x1 )α1 . . . (∂ xN )α N

|α| = α1 + · · · + α N .

C bm (Ω) denotes the space of functions u: Ω → R such that D i u(x) exist in the usual sense for all i = 0, . . . , m and D i u are continuous and bounded on Ω. C bm (Ω) is endowed with the norm u C bm (Ω) =

sup x∈Ω,|α|≤m

|D α u(x)|.

Wm, p(Kt ) is the space of restrictions to Kt of functions which are in Wm, p(R N ), C bm (Kt ) is the space of restrictions to Kt of functions which are in C bm (R N ). Lemma 4.1. We assume that ∀t ∈ [0, T], T ∈ (0, ∞), Kt is a domain of R N such that: Kt = i∈I τi (Ptk), where Ptk can be written as follows: Ptk = {(xi ) ∈ R N : 0 ≤ xi ≤ t for i = 1, . . . , k and 0 ≤ xi ≤ 1 for i = k + 1, . . . , N},


estimates requires a great number of those Sobolev inequalities. These are the merits


165

(under an affine and regular mapping independent of t), I being a finite set of indices and the τi are translations of R N , i.e., ∃b ∈ R N , ∀x ∈ R N , τi (x) = b + x. The following Sobolevtype inequalities hold (with precise constants) provided that all the quantities of the right-hand side exist and are finite. v W j,q (Kt ) ≤ ctk(1/q−1/ p) v Wm+ j, p(Kt ) ,

(4.1)

if Np N − mp

or

(mp ≥ N and p ≤ q < +∞),

where c > 0 is a constant which depends only on N, p, m, q, j, and T and which is a nondecreasing function of T: v C j (Kt ) ≤ ct−k/ p v Wm+ j, p(Kt ) b

if (mp > N) or ( p = 1 and m = N),

(4.2)

c > 0 being a constant, which depends only on N, p, m, j, and T and which is a nondecreasing function of T : uv Ws, p(Kt ) ≤ ct−k/ p u Ws1 , p(Kt ) v Ws2 , p(Kt )

if s < s1 + s2 −

N p

and

s ≤ min(s1 , s2 ), (4.3)

where c > 0 is a constant which depends only on N, p, s, s1 , s2 , and T and which is a non-decreasing function of T.

Proof.

The crucial point in this proof is to give the precise dependence with respect to

t of the Sobolev constants on the domains Kt . To reach this target, one uses appropriate scale transformations to pass from the domains Kt to the domain K1 where the Sobolev constants are independent of t. See [18–20] for details.

Applying Lemma 4.1 to Kt ∈ {Λt , Γtω , G ωt , L t } and using the definition of the weighted norms (see (3.4), (3.6), (3.7)) and their corresponding spaces, we derive further inequalities listed in the following Theorem 4.2.

Theorem 4.2. Let St ∈ {Λt , Γtω , G ωt , L t } and Mt ∈ {G ωt , L t }. There are constants c > 0 independent of t, u, and v such that the following inequalities hold (it is assumed that all the


mp < N and p ≤ q ≤

166


quantities of the right-hand side exist and are finite). |u|sMt ≤ c|u| Mt ,s ,

(4.4)

u sMt ≤ c u Mt ,s ,

(4.5)

Mt u s,1

(4.6)

≤ c u Mt ,s,1 ,

|uv|sSt ≤ c|u|sS1t |v|sS2t

uv sMt ≤ c u sM1 t v sM2 t Mt t t ≤ c u sM1 ,1 v sM2 ,1 uv s,1

|uv| Mt ,s ≤ c|u| Mt ,s1 |v| Mt ,s2 −

−1 + dim Mt 2 −1 + dim Mt 2

and

−1 + dim Mt 2

uv sMt ≤ c u sMt v sMt

s ≤ min(s1 , s2 ),

and

s ≤ min(s1 , s2 ),

and

s ≤ min(s1 , s2 ),

dim Mt , 2 −1 + dim Mt , if s > 2

n if s > , 2 n− 1 Γtω u C 0 (Γtω ) ≤ c|u|s , if s > 2 t t Gω Γ ω ,G ω σ ,L σ 1/2 |u|Λ dσ ≤ ct3/2 |u|sL t |u|s σ dσ ≤ ct|u|s t . s t u C 0 (Λt ) ≤ c|u|Λ s

0

Proof.

(4.10)

(4.11)

if s < s1 + s2

if s >

uv Mt ,s ≤ c u Mt ,s v Mt ,s

(4.9)

if s < s1 + s2

uv Mt ,s,1 ≤ c u Mt ,s1 ,1 v Mt ,s2 ,1 −

(4.8)

if s < s1 + s2

uv Mt ,s ≤ c u Mt ,s1 v Mt ,s2 −

(4.7)

0

(4.12) (4.13) (4.14) (4.15) (4.16) (4.17)

[Proof of (4.4), (4.5), and (4.6)] By a simple computation we have t σ ,St 2 |Σσ |2 [|u|Σ ] dσ . [|u|sSt ]2 = |St |−2 s 0

(i) For St = L t we have Σσ = Λσ and |Σσ | = σ 1/2 . Therefore 2 t t2 σ ,L t [|u|sL t ]2 ≤ |L t |−2 ess sup |u|Λ σ dσ = t−2 × (|u| L t ,s )2 . s 2 0≤σ ≤t 0 √ Hence |u|sL t ≤ 2/2(|u| L t ,s ).


dim St and s ≤ min(s1 , s2 ), 2 dim Mt and s ≤ min(s1 , s2 ), if s < s1 + s2 − 2 dim Mt and s ≤ min(s1 , s2 ), if s < s1 + s2 − 2

if s < s1 + s2 −


167

(ii) For St = G ωt we have Σσ = Γσω and |Σσ | = 1. Therefore 2 t Gω Γ ω ,G ω [|u|s t ]2 ≤ |G ωt |−2 ess sup |u|s σ t dσ = t−1 × t(|u|G ωt ,s )2 . 0≤σ ≤t

0

Gω

Thus, |u|s t ≤ |u|G ωt ,s . Hence the first relation (4.4) follows. Equations (4.5) and (4.6) follow directly from (4.4) by definitions of the norms due to (3.7).

[Proof of (4.7), (4.8), and (4.9)] (i) (4.7) is obtained directly by applying inequality (4.3) of Lemma 4.1 for Kt = Λt , Γtω , G ωt , L t , and (N, p) = (n, 2). (ii) It holds that Gω uv s t

s−1 1/2 G ωt k 2 = (|Dω uv|2(s−k)−1 )

and

dim G ωt = n.

k=0

For 0 ≤ k ≤ s − 1, Leibniz formula gives Dωk uv =

C ki Dωi uDωj v.

(4.18)

i+ j=k

Therefore Gω

t ≤ ck |Dωk uv|2(s−k)−1

Gω

t |Dωi uDωj v|2(s−k)−1 .

(4.19)

i+ j=k

For i + j = k, (4.7) implies for St = G ωt that Gω

Gω

Gω

t |Dωi uDωj v|2(s−k)−1 ≤ κ|Dωi u|2(st 1 −i)−1 |Dωj v|2(st 2 − j)−1 ,

(4.20)

if 2(s − k) − 1 < 2(s1 − i) − 1 + 2(s2 − j) − 1 −

n , 2

2(s − k) − 1 ≤ min{2(s1 − i) − 1, 2(s2 − j) − 1}.

(4.21)

Inequalities (4.21) are equivalent to s − (k − i) ≤ s1 , s − i ≤ s2

and

s < s1 + s2 −

1 n + . 2 4

(4.22)


Proof.

168


Our assumption s < s1 + s2 − n/2 and s ≤ min(s1 , s2 ) implies (4.21) since n≥ 2. Therefore (4.20) is satisfied under our assumption. Hence G ωt Gω Gω |Dωi uDωj v|2(s−k)−1 ≤κ |Dωi u|2(st 1 −i)−1 |Dωj v|2(st 2 − j)−1 . i+ j=k

(4.23)

i+ j=k

Using Schwarz inequality, we have

⎤1/2 k 1/2 ⎡ k ω ω G 2⎦ ⎣ (|Dωj u|G t |Dωi uDωj v|2(s−k)−1 ≤ κ (|Dωi u|2(st 1 −i)−1 )2 . 2(s2 − j)−1 ) G ωt

i+ j=k

i=0

j=0

(4.24)

(4.25)

i+ j=k

Using (4.19) and (4.25) we gain (4.8) for Mt = G ωt . Equation (4.8) for Mt = L t follows directly from (4.7) and the definitions of the norms. (iii) The proof of (4.9) is similar to that of (4.8). Proof.

[Proof of (4.10), (4.11), and (4.12)] (i) By definition we have σ ,Mt σ ,Mt |uv| Mt ,s = ess sup |uv|Σ , |uv|Σ = |Σσ |−1 uv H s (Σσ ,Mt ) with Mt = s s

0≤σ ≤t

Σσ .

0≤σ ≤t

Applying inequality (4.3) of Lemma 4.1 for Kt = Λt , Γtω , G ωt , L t , and (N, p) = (n, 2) we get uv H s (Σσ ,Mt ) ≤ c|Σσ |−1 u H s1 (Σσ ,Mt ) v H s2 (Σσ ,Mt ) , if

dim Σσ and 2 As dim Σσ = −1 + dim Mt , it holds that s < s1 + s2 −

s ≤ min(s1 , s2 ).

σ ,Mt |uv|Σ ≤ c|Σσ |−2 u H s1 (Σσ ,Mt ) v H s2 (Σσ ,Mt ) , s

i.e., σ ,Mt σ ,Mt σ ,Mt |uv|Σ ≤ c|u|Σ |v|Σ . s s1 s2

We apply ess sup to obtain (4.10). (ii) The proofs of (4.11) and (4.12) are similar to those of (4.8) and (4.9), respectively.


As 0 ≤ k ≤ s − 1, we obtain, since s ≤ min(s1 , s2 ) G ωt Gω Gω |Dωi uDωj v|2(s−k)−1 ≤ κ u s1t v s2t .


Proof.

[Proof of (4.13) and (4.14)] It is sufficient to write (4.8) for s = s1 = s2 .

Proof.

[Proof of (4.15) and (4.16)] We apply (4.2) to Λt and Γtω for j = 0 and p = 2.

169

(i) For Kt = Λt , we have k = 1 and N = n. Therefore u C 0 (Λt ) ≤ ct−1/2 u H s (Λt ) if t s > n/2, i.e., u C 0 (Λt ) ≤ c|u|Λ s if s > n/2.

(ii) For Kt = Γtω , we have k = 0 and N = n − 1. Therefore u C 0 (Γtω ) ≤ c u H s (Γtω )

if s >

n− 1 , 2

Γω

u C 0 (Γtω ) ≤ c|u|s t Proof.

n− 1 . 2

if s >

[Proof of (4.17)] For σ ∈ [0, t] we have σ ,L |u|Λ s

=σ

−1/2

s

1/2 |D Lk u|2Λσ

.

k=0

¨ Using Holder inequality we gain after a simple calculation t σ 1/2 |u|sΛσ ,L dσ ≤ t3/2 |u|sL t . 0

By the same tools and computations we get t Gω Γ ω ,G ω |u|s σ dσ ≤ t|u|s t . 0

Hence (4.17) is proved and we are done with Theorem 4.2.

5 Moser-type Estimates Moser-type estimates (see [32, 33]) are the supplementary tools that are going to be combined with Sobolev and energy inequalities together with the result obtained for the linear Goursat problem to derive the existence and uniqueness result for the quasilinear Goursat problem. Here, we adapt the work of Dossa [19, 20] who used similar estimates to handle the characteristic initial value problem on the cone. We first give some supplementary notation and definitions that generalize those used so far. We are conscious of the abundance of technicalities and complicated notation that are used, but they seem to be unavoidable for the rigorous analysis of the characteristic initial value problem in the weighted Sobolev spaces defined in Section 3.


i.e.,

170


Definition 5.1. Let M be a domain of R N+1 such that M ⊂ R N × [0, T] and M = We assume that Kt is as in Lemma 4.1, i.e., Kt = i∈I τi (Ptk), where

0≤t≤T

Kt .

Ptk = {(xi ) ∈ R N : 0 ≤ xi ≤ t for i = 1, . . . , k and 0 ≤ xi ≤ 1 for i = k + 1, . . . , N}. I is a set of indices and τi are translations of R N . Let (m, p) ∈ N × (0, ∞); u a tensor field on M. We set Kt ,M −k/ p u Wm, p(Kt ,M) , |u|m, p =t

with

where |D i u(x)| = [

I ,J

|α|=i

|D i u(x)| p dKt

,

(5.2)

Kt

i=0

1/ p

m

|D α uIJ | p]1/ p, α ∈ N N+1 , and uIJ are the components of u. We

also set M −k/ p |u|m, p=T

T

0

1/ p p

u Wm, p(Kt ,M) dt

,

(5.3a)

and define the space Wm, p(M) by Wm, p(M) = {u∈ L p(M) : D i u∈ L p(M) for i = 1, . . . , m}. M p Wm, p(M) endowed with the norm |u|m, p is a Banach space. Here L (M) stands for the

usual Lebesgue space. Furthermore, we set Kt ,M |u| M,m, p = ess sup |u|m, p ,

E m, p(M) = {u∈ Wm, p(M) : |u| M,m, p < +∞}.

(5.3b)

0≤t≤T

E m, p(M) endowed with the norm |u| M,m, p is a Banach space. Let Y be an open subset of Rl , 0 < l ∈ N. C bm (M × Y) denotes the space of funcβ

tions f : M × Y → R such that all usual derivatives Dxα D y f, |α| + |β| ≤ m, exist and are bounded. C bm (M × Y) is endowed with its natural norm defined by f C bm (M×Y) = sup |Dxα D βy f(x, y)|. (x,y)∈M×Y |α|+|β|≤m

Remark 5.2. (i) It is easy to see through a simple computation that the following embedding holds: M 1/ p E m, p(M) → Wm, p(M) with |u|m, |u| M,m, p. p≤T

(ii) Lemma 4.1 still holds if Wm, p(Kt ) is replaced by Wm, p(Kt , M).

(5.4)


u Wm, p(Kt ,M) =

(5.1)


171

(iii) For p = 2 and M = Mt ∈ {L t , G ωt }, the relations between the norms of Definition 5.1 (see (5.3a) and (5.3b)) and those defined in Section 3 (see (3.4) and (3.6)) are the following: Lt Lt |u|m,2 = t1/2 |u|m ,

Gω

Gω

t |u|m,2 = |u|mt ,

|u| Mt ,m,2 = |u| Mt ,m .

We begin the series of Moser-type estimates by the following generic

theorem.

u: M → Y a mapping u(x) = (u1 (x), . . . , ul (x)); We assume that 1 < m, p ≤ s; N < (s − 1) p. Then, replacing y by u(x) in f(x, y), we obtain a function f(x, u(x)) of x which satisfies the following inequalities: f(x, u(x)) Wm, p(Kt ,M) ≤ cT (m, s, p)tk/ p f C bm (M×Y) [1 + t−k/ p u Ws, p(Kt ,M) ]m , |

M f(x, u(x))|m, p ≤ cT (m, s,

p)T

f C bm (M×Y) [1 + |u| M,s, p] ,

1/ p

m

(5.5) (5.6)

| f(x, u(x))| M,m, p ≤ cT (m, s, p) f C bm (M×Y) [1 + |u| M,s, p] , m

(5.7)

where cT (m, s, p) is a non-decreasing function of T, which also depends on m, s, and p.

Proof.

For the sake of simplicity, let u Ws, p(Kt ,M) denote the norm of u in the cartesian

product space (Ws, p(Kt , M))l . The same convention will be used for other similar norms. By definition, we have ⎡ f(x, u(x)) Wm, p(Kt ,M) = ⎣

⎤1/ p

|α|≤m

|D α { f(x, u(x))}| p dKt ⎦

.

(5.8)

Kt

Let α ∈ N N+1 with |α| ≤ m. D α { f(x, u(x))} is a linear combination with constant coefficients of terms of the form gβ,γ ,λ (x) = Dxβ Duγ f(x, u)|u=u(x) ×

!

D λ(q,ρ) uq ,

(5.9)

q,ρ

where q takes the possible values 1, . . . , l for which γq > 0. ρ takes the possible values 1, . . . , γq . λ(q, ρ) = (λ1 (q, ρ), . . . , λ N+1 (q, ρ)); λa(q, ρ) > 0 are integers for a = 1, . . . , N + 1. λ satisfies the relation β+

q,ρ

λ(q, ρ) = α.

(5.10)


Theorem 5.3. Let uj ∈ E s, p(M), j = 1, . . . , l; Y an open subset of Rl ; f(x, y) ∈ C bm (M × Y).

172


We set i = |β|

and

j = |γ | = number of possible values taken by (q, ρ).

(5.11)

We evaluate gβ,γ ,λ L p(Kt ) by discussing according to the possible values of j. First case: j = 0. The set of all (q, ρ) is therefore empty and then gβ,γ ,λ (x) = gα,0,0 (x) = Dxα f(x, u)|u=u(x) .

gβ,γ ,λ L p(Kt ) ≤ f C bm (M×Y)

1/ p

dKt

.

Kt

As the volume of Kt is (up to a multiplicative constant) equal to tk we gain gα,0,0 L p(Kt ) ≤ ctk/ p f C bm (M×Y) .

(5.12)

Now k/ p ≥ 0 and t ∈ [0, T] imply that gα,0,0 L p(Kt ) ≤ cT k/ p f C bm (M×Y) .

(5.13)

Second case: j ≥ 1. Let a be the greatest integer such that (s − a) p > N. Due to the assumption (s − 1) p > N, we have a ≥ 1. We set J1 = {(q, ρ) : |λ(q, ρ)| ≤ a},

j1 = |J1 |,

J2 = {(q, ρ) : |λ(q, ρ)| > a},

j2 = |J2 |.

(5.14)

It is clear that j = j1 + j2 . Equation (5.9) implies that " " "! " " " gβ,γ ,λ L p(Kt ) ≤ f C bm (M×Y) " D λ(q,ρ) uq " " q,ρ "

.

(5.15)

L p(Kt )

We use the following obvious decomposition " " "! " " " λ(q,ρ) D uq " " " q,ρ "

" " " " ! " ! " λ(q,ρ) λ(q,ρ) " =" D u × D u q q" " " " (q,ρ)∈J (q,ρ)∈J p 1 2 L (K ) t

. L p(Kt )

(5.16)


We get


173

For (q, ρ) ∈ J1 , by definition of J1 it holds that (s − |λ(q, ρ)|) p > N. Applying (4.2), we get " " " " ! ! " " λ(q,ρ) λ(q,ρ) " D uq × D uq " " " " p "(q,ρ)∈J1 (q,ρ)∈J2 L (Kt ) " " " ! " ! " " −k/ p λ(q,ρ) λ(q,ρ) " ≤ (ct D uq Ws−|λ(q,ρ)|, p(Kt ) ) × " D uq " . (5.17) " "(q,ρ)∈J2 " p (q,ρ)∈J1 L (Kt )

For (q, ρ) ∈ J2 , it holds that |λ(q, ρ)| ≥ a + 1. As a is the greatest integer such that

2

Since (s − 1) p > N and |α| ≤ m ≤ s, we get j2 −

p(sj2 − (|α| − i)) ps( j2 − 1) ≤ j2 − < 1. N N

Therefore, we can find positive real numbers rq,ρ , (q, ρ) ∈ J2 , such that 1 = 1, rq,ρ (q,ρ)∈J

p ≤ prq,ρ
N, we have (s − |λ(q, ρ)|) p ≤ N. Furthermore, due to (5.10), it holds that N − (s − |λ(q, ρ)|) p p(sj2 − (q,ρ)∈J2 |λ(q, ρ)|) p(sj2 − (|α| − i)) = j2 − ≤ j2 − . N N N (q,ρ)∈J

174


Thus, T −k

T 0

p

p

f(x, u(x)) Wm, p(Kt ,M) dt ≤ cT f C m (M×Y) [1 + |u| M,s, p]mp. b

Hence relation (5.6) follows. As shown above (see (5.12) and (5.21)) it holds that t−k/ p f(x, u(x)) Wm, p(Kt ,M) ≤ c f C bm (M×Y) [1 + t−k/ p u Ws, p(Kt ,M) ]m . Applying ess sup we obtain the relation (5.7).

(u1 (x), . . . , ul (x)) and v(x) = (v1 (x), . . . , vl (x)); Y be an open subset of Rl such that u(x) + θ (v(x) − u(x)) ∈ Y, ∀θ ∈ [0, 1]; f(x, y) ∈ C bm (M × Y). We assume that 1 < m, p ≤ s; N < (s − 1) p. Then the following inequalities hold f(x, v(x)) − f(x, u(x)) Wm−1, p(Kt ,M) ≤ ct v − u Ws−1, p(Kt ,M) ,

(5.22)

with ct = cT (m, s, p, N) f C bm (M×Y) [1 + t−k/ p( u Ws, p(Kt ,M) + v Ws, p(Kt ,M) )]m−1 , M M | f(x, v(x)) − f(x, u(x))|m−1, p ≤ c |v − u|s−1, p,

(5.23a)

M 1/ p | f(x, v(x)) − f(x, u(x))|m−1, |v − u| M,s−1, p, p≤cT

(5.23b)

with c = cT (m, s, p, N) f C bm (M×Y) [1 + |u| M,s, p + |v| M,s, p]m−1 , | f(x, v(x)) − f(x, u(x))| M,m−1, p ≤ c |v − u| M,s−1, p.

(5.24)

Here cT (m, s, p, N) is a non-decreasing function of T, which depends also on m, s, p,

and N.

Proof.

Thanks to the mean value formula, it holds that l f(x, v(x)) − f(x, u(x)) = (v j − uj ) j=1

1

Duj f(x, u(x) + θ (v(x) − u(x))) dθ .

(5.25)

0

Applying (4.3) to (5.25) with s = s1 = m − 1,

s2 = s − 1,

(5.26)


Corollary 5.4. Let uj , v j ∈ E s, p(M), j = 1, . . . , l; u, v : M → Y be the two mappings: u(x) =


175

we obtain f(x, v(x)) − f(x, u(x)) Wm−1, p(Kt ,M) l 1 −k/ p ≤ ct v − u Ws−1, p(Kt ,M) Duj f(x, u + θ (v − u)) Wm−1, p(Kt ,M) dθ . j=1

(5.27)

0

From (5.5), we have Duj f(x, u + θ (v − u)) Wm−1, p(Kt ,M) ≤ cT (m, s, p, N)tk/ p f C bm (M×Y) [1 + t−k/ p( u Ws, p(Kt ,M) + v Ws, p(Kt ,M) )]m−1 ,

(5.28)

and N. Equations (5.27) and (5.28) imply (5.22). Equations (5.23a), (5.23b), and (5.4) proceed easily from (5.22) by the definition of the norms.

Corollary 5.5. The notation and assumptions are those of Corollary 5.4. If w ∈ Ws−1, p(M) and f(x, y) ∈ C bm (M × Y), then the following inequalities are satisfied: [ f(x, v(x)) − f(x, u(x))]w Wm−1, p(Kt ,M) ≤ ct v − u Ws−1, p(Kt ,M) w Ws−1, p(Kt ,M) ,

(5.29)

with ct = cT (m, s, p, N)t−k/ p f C bm (M×Y) [1 + t−k/ p( u Ws, p(Kt ,M) + v Ws, p(Kt ,M) )]m−1 . If in addition w ∈ E s−1, p(M), then it holds that M M |[ f(x, v(x)) − f(x, u(x))]w|m−1, p ≤ c |v − u|s−1, p|w| M,s−1, p,

(5.30a)

M 1/ p f(x, u(x))]w|m−1, |v p≤cT

(5.30b)

|[ f(x, v(x)) −

− u| M,s−1, p|w| M,s−1, p,

with c = cT (m, s, p, N) f C bm (M×Y) [1 + |u| M,s, p + |v| M,s, p]m−1 , |[ f(x, v(x)) − f(x, u(x))]w| M,m−1, p ≤ c |v − u| M,s−1, p|w| M,s−1, p.

(5.31)

Here cT (m, s, p, N) is a non-decreasing function of T, which depends also on m, s, p,

and N. Proof.

We apply (4.3) with s = s1 = m − 1 and s2 = s − 1 to get [ f(x, v(x)) − f(x, u(x))]w Wm−1, p(Kt ,M) ≤ ct−k/ p f(x, v(x)) − f(x, u(x)) Wm−1, p(Kt ,M) w Ws−1, p(Kt ,M) .

(5.32)


where cT (m, s, p, N) is a non-decreasing function of T, which depends also on m, s, p,

176


Inserting (5.22) in (5.32), we obtain (5.29). Equations (5.30a), (5.30b), and (5.31) follow directly from (5.29) by the definition of the norms.

Similar to Theorem 5.3, we have the following theorem. Theorem 5.6. Let uj ∈ E s (L t ), j = 1, . . . , l; W be an open subset of Rl , f : L t × W → R such that f ∈ C bm (L t × W), u: L t → W a mapping u(x) = (u1 (x), . . . , ul (x)). We assume that 0 ≤ m ≤ s, n< 2s. Then the function f(x, u(x)) of x satisfies the following inequality: (5.33)

ct (m, s) being a non-decreasing function of t, depending also on m and s. If in addition, f(x, 0) = 0 and Duj f ∈ C bm (L t × W) then it holds that +

| f(x, u(x))| L t ,m ≤ ct (m, s)| f|m,t |u| L t ,s [1 + |u| L t ,s ](m−1) ,

(5.34)

where

$ # | f|m,t = max f C bm (L t ×W) , max Duj f C bm (L t ×W) , 1≤ j≤l

(m − 1)+ = max(m − 1, 0).

As E s (L t ) is embedded continuously in Hs (L t ), (5.33) and (5.34) also hold if the norm Lt . | f(x, u(x))| L t ,m is replaced by the norm | f(x, u(x))|m

Proof.

The main steps are the same as in the proof of Theorem 5.3. We just have to

make some judicious modifications. We replace M by L t , Kt by Λσ , 0 ≤ σ ≤ t since L t = 0≤σ ≤t Λσ . Then using the notation of the proof of Theorem 5.3, we have gα,0,0 L 2 (Λσ ) ≤ cσ 1/2 f C bm (Λσ ×W) , and gβ,γ ,λ L 2 (Λσ ) ≤ cσ 1/2 f C bm (Λσ ×W) [σ −1/2 u H s (Λσ ) ] j , where c = ct (m, s) is a non-decreasing function of t, depending also on m and s. Thus ess sup σ −1/2 gα,0,0 L 2 (Λσ ) ≤ c f C bm (L t ×W) . 0≤σ ≤t

(5.35)

Multiplying the inequality gβ,γ ,λ L 2 (Λσ ) ≤ cσ 1/2 f C bm (Λσ ×W) [σ −1/2 u H s (Λσ ) ] j by σ −1/2 and applying ess sup we obtain ess sup σ −1/2 gβ,γ ,λ L 2 (Λσ ) ≤ c f C bm (L t ×W) [|u| L t ,s ] j . 0≤σ ≤t

Hence the inequality (5.33) follows.

(5.36)


| f(x, u(x))| L t ,m ≤ ct (m, s) f C bm (L t ×W) [1 + |u| L t ,s ]m ,


177

To prove (5.34), we follow almost the same steps as in the proof of (5.33), apart from the estimate of gα,0,0 L 2 (Λσ ) . In this case, the assumption f(x, 0) = 0 allows us to write, using the mean value formula, gα,0,0 (x) = Dxα f(x, u)|u=u(x) =

l j=1

1

uj 0

Duj Dxα f(x, θ u(x)) dθ .

Thus, gα,0,0 L 2 (Λσ ) ≤ max Duj f C bm (L t ×W) u H s (Λσ ) . 1≤ j≤l

Hence (5.37)

0≤σ ≤t

Combining (5.37) and (5.36) we gain (5.34).

Theorem 5.7. (i) Let uj , v j ∈ E s (L t ),

j = 1, . . . , l; u, v : L t → W be two mappings u(x) =

(u1 (x), . . . , ul (x)),

v(x) = (v1 (x), . . . , vl (x));

C bm (L t l

∈ C bm−1 (L t

× W); Duj f

f : Lt × W → R

such

f∈

that

× W), for j = 1, . . . , l; W be an open subset of

R such that u(x) + θ (v(x) − u(x)) ∈ W, ∀θ ∈ [0, 1]. We assume that 1 ≤ m ≤ s; n< 2(s − 1). Then the following inequality is satisfied | f(x, v(x)) − f(x, u(x))| L t ,m−1 ≤ ct (m, s) max Duj f C bm−1 (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m−1 |v − u| L t ,s−1 , 1≤ j≤l

(5.38)

ct (m, s) being a non-decreasing function of t, depending also on m and s. (ii) If in addition w ∈ E s−1 (L t ), then it holds that |[ f(x, v(x)) − f(x, u(x))]w| L t ,m−1 ≤ ct (m, s) max Duj f C bm−1 (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m−1 |v − u| L t ,s−1 |w| L t ,s−1 . 1≤ j≤l

(5.39) (iii) If in addition to the assumptions in (i) we assume that m ≤ s − 1, Duj f ∈ C bm (L t × W) and w ∈ E s−2 (L t ), then it holds that |[ f(x, v(x)) − f(x, u(x))]w| L t ,m−1 ≤ ct (m, s) max Duj f C bm (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m |v − u| L t ,s−1 |w| L t ,s−2 . 1≤ j≤l

(5.40)


ess sup σ −1/2 gα,0,0 L 2 (Λσ ) ≤ c| f|m,t |u| L t ,s .

178


Proof.

Equations (5.38) and (5.39) follow directly from (5.4) of Corollary 5.4 and (5.31) of

Corollary 5.5. To prove (5.40), since m − 1 < m + s − 2 − n/2 and m − 1 ≤ min(m, s − 2), we use (4.10) of Theorem 4.2 to gain |[ f(x, v(x)) − f(x, u(x))]w| L t ,m−1 ≤ c| f(x, v(x)) − f(x, u(x))| L t ,m |w| L t ,s−2 ,

(5.41)

where the positive constant c is independent of t. From (5.38), we get | f(x, v(x)) − f(x, u(x))| L t ,m 1≤ j≤l

(5.42)

where dt (m, s) is a non-decreasing function of t, depending also on m and s. Inserting (5.42) in (5.41), we get (5.40).

Theorem 5.8. (i) Let uj , v j ∈ E s (L t ),

j = 1, . . . , l; u, v : L t → W be two mappings u(x) =

(u1 (x), . . . , ul (x)), and v(x) = (v1 (x), . . . , vl (x)); f : L t × W → R such that f ∈ C bm (L t × W); W be an open subset of Rl such that u(x) + θ (v(x) − u(x)) ∈ W, ∀θ ∈ [0, 1]. We assume that 1 ≤ m ≤ s; n/2 < s ≤ n/2 + 1. Then the following inequality is satisfied | f(x, v(x)) − f(x, u(x))| L t ,m−1 ≤ ct (m, s) f C bm (L t ×W) × [1 + |u| L t ,s + |v| L t ,s ]m−1 |v − u| L t ,s ,

(5.43)

ct (m, s) being a non-decreasing function of t, depending also on m and s. (ii) If in addition Duj f ∈ C bm (L t × W)∀ j = 1, . . . , l, then it holds that | f(x, v(x)) − f(x, u(x))| L t ,m−1 ≤ ct (m, s) max Duj f C bm (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m |v − u| L t ,s . 1≤ j≤l

Proof.

(5.44)

By the mean value formula it holds that f(x, v(x)) − f(x, u(x)) =

l (v j − uj ) j=1

1 0

Duj f(x, u(x) + θ (v(x) − u(x))) dθ .

(5.45)


≤ dt (m, s) max Duj f C bm (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m |v − u| L t ,s−1 ,


179

Due to (4.10) of Theorem 4.2, since m − 1 < s + m − 1 − n/2 and m − 1 ≤ min(s, m − 1), it holds that | f(x, v(x)) − f(x, u(x))| L t ,m−1 1 ≤ c|v − u| L t ,s max |Duj f(x, u(x) + θ (v(x) − u(x)))| L t ,m−1 dθ , 1≤ j≤l

(5.46)

0

where the positive constant c is independent of t. From (5.33) of Theorem 5.6, we get |Duj f(x, u(x) + θ (v(x) − u(x)))| L t ,m−1 (5.47)

where dt (m, s) is a non-decreasing function of t, depending also on m and s. We insert (5.47) into (5.46) to obtain (5.43). The proof of (5.44) is similar to that of (5.43) provided that one replaces, respectively, the inequalities (5.46) and (5.47) by the following inequalities (5.48) and (5.49) | f(x, v(x)) − f(x, u(x))| L t ,m−1 1 ≤ c|v − u| L t ,s−1 max |Duj f(x, u(x) + θ (v(x) − u(x)))| L t ,m dθ , 1≤ j≤l

(5.48)

0

|Duj f(x, u(x) + θ (v(x) − u(x)))| L t ,m ≤ dt (m, s) Duj f C bm (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m .

(5.49)

We close this section on Moser-type estimates by the following two theorems which play a crucial role in the resolution of the quasilinear Goursat problem. Theorem 5.9. Let ui ∈ Es (L t ), i = 1, . . . , l; W be an open subset of Rl ; f : L t × W → R such that f ∈ C b2m−1 (L t × W); u: L t → W be a mapping u(x) = (u1 (x), . . . , ul (x)). We assume that 1 ≤ m ≤ s, n< 2s. (i) Then the function f(x, u(x)) of x satisfies the following inequality: f(x, u(x)) L t ,m ≤ ct (m, s) f C b2m−1 (L t ×W) [1 + u L t ,s ]2m−1 ,

(5.50)

ct (m, s) being a non-decreasing function of t, depending also on m and s. As Es (L t ) is embedded continuously in Hs (L t ), it also holds that Lt ≤ dt (m, s) f C b2m−1 (L t ×W) [1 + u L t ,s ]2m−1 , f(x, u(x)) m

(5.51)

dt (m, s) being a non-decreasing function of t, depending also on m and s.


≤ dt (m, s) Duj f C bm−1 (L t ×W) [1 + |u| L t ,s + |v| L t ,s ]m−1 ,

180


(ii) If L t is replaced in (5.51) by G ωt , ω = 1, 2, then we instead get the following inequality: Gω

f(x, u(x)) mt ≤ dt (m, s)t1/2 f C b2m−1 (G ωt ×W) [1 + u G ωt ,s ]2m−1 .

Proof.

(5.51a)

One uses similar tools, with some adequate modifications, as in the proof of

Theorem 5.3. See Appendix for details.

(i) Let ui , vi ∈ Es (L t ), i = 1, . . . , l; u, v : L t → W be the two mappings, u(x) = (u1 (x), . . . , ul (x)) and v(x) = (v1 (x), . . . , vl (x)); f : L t × W → R such that f ∈ C b2m−3 (L t × W), f(x, 0) = 0 and Dui f ∈ C b2m−3 (L t × W) ∀i = 1, . . . , l; W be an open subset of Rl such that u(x) + θ (v(x) − u(x)) ∈ W, ∀θ ∈ [0, 1]. We assume that 2 ≤ m ≤ s; n + 1 < 2(s − 1). Then the following inequality is satisfied: Lt f(x, v(x)) − f(x, u(x)) m−1 Lt ≤ ct (m, s) max Dui f C b2m−3 (L t ×W) [1 + u L t ,s + v L t ,s ]2m−3 v − u s−1 , (5.52) 1≤i≤l

ct (m, s) being a non-decreasing function of t, depending also on m and s. (ii) If in addition w ∈ Es−1 (L t ), then it holds that Lt [ f(x, v(x)) − f(x, u(x))]w m−1

≤ ct (m, s) max Dui f C b2m−3 (L t ×W) [1 + u L t ,s + v L t ,s ]2m−3 v − u sL t w L t ,s−1 . 1≤i≤l

(5.53) (iii) If in addition to the assumptions in (i) we assume that m ≤ s − 1, f ∈ C b2m−3 (L t × W), Dui f ∈ C b2m−1 (L t × W) ∀i = 1, . . . , l and w ∈ Es−2 (L t ), then it holds that Lt [ f(x, v(x)) − f(x, u(x))]w m−1 Lt ≤ ct (m, s) max Dui f C b2m−1 (L t ×W) [1 + u L t ,s + v L t ,s ]2m−1 v − u s−1 w L t ,s−2 . 1≤i≤l

(5.54) Lt (iv) The inequalities (5.52), (5.53), and (5.54) still hold if the norms . m−1 and Lt are replaced, respectively, by the norms . L t ,m−1 and . L t ,s−1 . . s−1


Theorem 5.10.


Proof.

181

Thanks to the mean value formula, we have f(x, v(x)) − f(x, u(x)) =

l (vi − ui )

1

Duj f(x, u(x) + θ (v(x) − u(x))) dθ .

0

i=1

(i) Due to (4.8) and (4.5) of Theorem 4.2, since m − 1 < s − 1 + m − 1 − (n + 1)/2 and m − 1 ≤ min(s − 1, m − 1), we gain

1≤i≤l

(5.55)

0

where the positive constant c is independent of t. From (5.50), we get Duj f(x, u(x) + θ (v(x) − u(x))) L t ,m−1 ≤ ct (m, s) Dui f C b2m−3 (L t ×W) [1 + u L t ,s + v L t ,s ]2m−3 .

(5.56)

Inserting (5.56) into (5.55), we gain (5.52). (ii) From (4.8) of Theorem 4.2 we have Lt Lt ≤ c f(x, v(x)) − f(x, u(x)) m−1 w L t ,s−1 . (5.57) [ f(x, v(x)) − f(x, u(x))]w m−1

We insert (5.52) into (5.57) and thus prove (5.53). (iii) From (4.8) of Theorem 4.2, we also obtain Lt Lt ≤ c f(x, v(x)) − f(x, u(x)) m w L t ,s . [ f(x, v(x)) − f(x, u(x))]w m−1

(5.58)

Applying (5.52) where we replace m by m + 1, we get Lt f(x, v(x)) − f(x, u(x)) m Lt ≤ ct (m, s) max Dui f C b2m−1 (L t ×W) [1 + u L t ,s + v L t ,s ]2m−1 v − u s−1 . (5.59) 1≤i≤l

Inserting (5.59) into (5.58) we get (5.54). Finally the proof of (iv) is obvious. It suffices to replace, in all the steps above, the inequality (4.8) of Theorem 4.2 by the inequality (4.11) of the same theorem.

6 Proof of Theorem 3.2 Proof of item (i) of Theorem 3.2. We propose to construct, by a contraction argument, a fixed point of the mapping κ : w → κ(w) = u, from a ball B of Es (L T ) into itself, where u


Lt f(x, v(x)) − f(x, u(x)) m−1 1 Lt ≤ c v − u s−1 max Duj f(x, u(x) + θ (v(x) − u(x))) L t ,m−1 dθ ,

182


solves the following linear Goursat problem gab(x, w(x))Dabu= f(x, w(x), Dw(x))

in L T ,

u= u on G ωT .

(6.1)

ω

We proceed in three main steps. Later on, to show that the ball B is not empty, it is important to know how to construct on L T a function v1 ∈ Es (L T ), such that v1 = u on G ωT . ω

6.1 First step: Construction of an element v1 of Es (L T ) such that v1 = u on G ωT ω

mab Dabv = 0 in L T , where

⎧ ⎪ −1 ⎪ ⎪ ⎨

v = u on G ωT ,

(6.2)

ω

if (a, b) ∈ {(1, 2), (2, 1)},

mab = 1 ⎪ ⎪ ⎪ ⎩ 0

if a = b ∈ {3, . . . , n + 1}, elsewhere.

It is easy to check that the coefficients and the initial data of (6.2) satisfy all the hypotheses of Theorem 3.1. Therefore, the Goursat problem (6.2) has a unique solution v1 ∈ Es (L T ). 6.2 Second step: Properties of the mapping κ defined by the linear Goursat problem (6.1)

We begin by noting from assumptions (3.20) and (3.21) that u is continuous on G ω , ω

u = u ≡ u on Γ ≡ G 1T ∩ G 2T , 1

2

u(G ωT ) ⊂ V. ω

Since Γ is a compact subset of G ωT , it follows that u(Γ ) is a compact subset of V. Now, as V is an open subset of R N , there exists a positive real number ε1 such that Vε1 ≡ u(Γ ) + Bε1 ⊂ V, with Bε1 = {v ∈ R N : |v| < ε1 },

Bε1 = {v ∈ R N : |v| ≤ ε1 },

where |v| stands for the euclidian norm of v in R N . The following subsets of R(n+1)N will also be used: Wε∗2 ≡ U (Γ ) + Bε∗2 ,

Bε∗2 = {v ∗ ∈ R(n+1)N : |v ∗ | < ε2 },

Bε∗2 = {v ∗ ∈ R(n+1)N : |v ∗ | ≤ ε2 },


Consider the following linear Goursat problem:


183

where U = (D1 u, D2 u, D3 u, . . . , Dn+1 u), |v ∗ | denotes the euclidian norm of v ∗ in R(n+1)N and 2

1

ε2 is a positive real number which can be chosen as large as necessary. The following lemmas are relevant in order to show that the mapping κ contracts. Before stating them we need to define the regular hyperbolicity constant that will be used throughout the proof. Set for each (x, u) ∈ U × V, hab(x, u) = gab(x, u) − 2

ta(x, u)tb(x, u) tctc(x, u)

where ta = Daτ ,

ta(x, u) = gab(x, u)tb.

metric for each (x, u) ∈ U × V. Now set h1 = h2 = h3 =

inf

(x,u)∈L T ×Vε1

inf

(x,u)∈L T ×Vε1 Xa ∈Sn

(−gab(x, u)tatb),

hab(x, u)Xa Xb,

sup

gac(x, u)gbd(x, u)δabδcd,

(x,u)∈L T ×Vε1

where Sn is the n-dimensional unit sphere. Since L T × Vε1 and Sn are compact, h1 , h2 , and h3 are positive real numbers, thanks to assumption (3.20). Let now (x, u) ∈ U × V and take a non-zero covector (ka) such that gab(x, u)takb = 0. Then gab(x, u)kakb = hab(x, u)kakb ≥ h2 |k|2

where |k|2 =

n+1

|ka|2 .

a=1

Finally set −1 h = max(h−1 1 , h2 , h3 ),

where h is a hyperbolicity constant of the regularly hyperbolic metric gab(x, w(x)), for every continuous function w : L T → Vε1 . Lemma 6.1. Consider on L T a function w such that w ∈ Es (L T ) and (w(x), Dw(x)) ∈ Vε1 × Wε∗2 , ∀x ∈ L T , where Vε1 ≡ u(Γ ) + Bε1 and Wε∗2 ≡ U (Γ ) + Bε∗2 . Then the linear Goursat problem (6.1) has in L T a unique solution u∈ Es (L T ). In addition, u satisfies the following


Then in view of the hyperbolicity assumptions in (3.20), hab(x, u) is a positive definite

184


energy inequality for t ∈ (0, T]: u L t ,s ≤ C 1

2 ω=1

|u|G ωt ,2s−1 + t

1/2

ω

f C b2s−3 (L T ×Vε

∗ 1 ×Wε2 )

[1 + w L t ,s ]

2s−3

,

(6.3)

where the constant C 1 > 0 is a non-decreasing function depending continuously on each of the following arguments h, T, g C b2s−3 (L T ×Vε ) , w L T ,s , with h denoting regular hyper1

bolicity constant just defined above.

Proof.

need to show that gab(x, w(x)) and f(x, w(x), Dw(x)) satisfy the remaining assumptions of Theorem 3.1, i.e., gab(x, w(x)) ∈ Hs (L T ),

f(x, w(x), Dw(x)) ∈ Hs−1 (L T ).

For m = s, Moser estimate (5.50) of Theorem 5.9 gives g(x, w(x)) L T ,s ≤ cT (s) g C b2s−1 (L T ×Vε ) [1 + w L T ,s ]2s−1 . 1

Thus, gab(x, w(x)) ∈ Es (L T ). Since Es (L T ) is embedded in Hs (L T ), we gain gab(x, w(x)) ∈ Hs (L T ). Similarly, we estimate f(x, w(x), Dw(x)) using Moser inequality (5.50) of Theorem 5.9 for m = s − 1 to have f(x, w(x), Dw(x)) L T ,s−1 ≤ cT (s) f C b2s−3 (L T ×Vε

∗ 1 ×Wε2 )

[1 + w L T ,s ]2s−3 .

(6.4)

From this we deduce as above that f(x, w(x), Dw(x)) ∈ Hs−1 (L T ). Now Theorem 3.1 implies that the linear Goursat problem (6.1) has a unique solution u∈ Es (L T ), which satisfies the following energy inequality for 0 < t ≤ T: 2 ω Lt Γ ,G ,1 u L t ,s ≤ c (|u|G ωt ,2s−1 + f(x, w(x), Dw(x)) s−1 ) + t f(x, w(x), Dw(x)) s−1 , ω=1

(6.5)

ω

where c is a non-decreasing continuous function of its arguments h, γ sL T , γ Γs , and T, with γ sL T = T g(x, w(x)) sL T ,

γ Γs =

2

,G g(x, w(x)) Γs−1

ω

,1

.

ω=1

We will derive the inequality (6.3) by estimating the right-hand side terms of inequality ,G (6.5). The estimates of the quantities g(x, w(x)) Γs−1

ω

,1

,G and f(x, w(x), Dw(x)) Γs−1 Γ ,G ωt

done by using the definition of the norms and the inequality |w|l

G ωt

ω

,1

are

≤ c0 |w|l+1 , where c0


As gab(x, w(x)) is regularly hyperbolic with hyperbolicity constant h, We just


185

is independent of t and w (see inequality (4.60) of [35, p. 189]). Doing so we gain Γ ,G ω ,1

g(x, w(x)) s−1 t

Γ ,G ω ,1

f(x, w(x), Dw(x)) s−1 t

Gω

t ≤ c0 g(x, w(x)) s−1 ,

Gω

t ≤ c0 f(x, w(x), Dw(x)) s−1 .

(6.6a)

Using Moser inequality (5.51a) of Theorem 5.9 for m = s − 1 we get Gω

t ≤ cT (s)t1/2 g C b2s−3 (L T ×Vε ) [1 + w L t ,s ]2s−3 , g(x, w) s−1 1

f(x, w,

G ωt Dw) s−1

≤ cT (s)t

1/2

f C b2s−3 (L T ×Vε

∗ 1 ×Wε2 )

[1 + w L t ,s ]2s−3 ,

(6.6b)

where cT (s) is a non-decreasing function of T depending also on s. To estimate the Theorem 5.9, for m = s and s − 1, respectively, to have g(x, w(x)) sL T ≤ cT (s) g C b2s−1 (L T ×Vε ) [1 + w L T ,s ]2s−1 , 1

f(x, w(x),

Lt Dw(x)) s−1

≤ cT (s) f C b2s−3 (L T ×Vε

∗ [1 1 ×Wε2 )

+ w L t ,s ]2s−3 .

(6.7)

From (6.5), (6.6a), (6.6b), and (6.7) we get the desired inequality (6.3).

Lemma 6.2. Assume that s > n/2 + 2. Then there exists a real constant r1 = r1 (ε1 , ε2 ) > 0 depending only on ε1 and ε2 such that (w ∈ Es (L t ), w = u on G ωt , and t w L t ,s < r1 ) ⇒ ((w(x), Dw(x)) ∈ Vε1 × Wε∗2 , ∀x ∈ L t ). ω

Proof.

Assume s > n/2 + 2 and let x = (x1 , . . . , xn+1 ) ∈ L t , t ∈ (0, T]. Setting xΓ =

(0, 0, x3 , . . . , xn+1 ), it holds that xΓ ∈ Γ and, thanks to the mean value formula, w(x) − u(xΓ ) =

2

Dw(x) − U (xΓ ) =

∂w (σ x1 , σ x2 , x3 , . . . , xn+1 ) dσ , ∂ xi

1

∂(Dw) (σ x1 , σ x2 , x3 , . . . , xn+1 ) dσ . ∂ xi

x

0

i=1 2

1

i

xi

i=1

0

(6.8)

As Es (L T ) is embedded in C 0 (L T ) for s > (n+ 1)/2, it follows from (6.8) that there exists two positive real numbers c1 and c2 (Sobolev constants) such that sup |w(x) − u(xΓ )| ≤ c1 t w L t ,s , x∈L t

sup |Dw(x) − U (xΓ )| ≤ c2 t w L t ,s .

(6.9)

x∈L t

We now take r1 = min(ε1 /c1 , ε2 /c2 ) to gain, for t w L t ,s < r1 , w(x) ∈ Vε1 ,

Dw(x) ∈ Wε∗2

∀x ∈ L t .


Lt , we use Moser inequality (5.51) of quantities g(x, w(x)) sL T and f(x, w(x), Dw(x)) s−1

186


Lemma 6.3. The assumptions are those of Lemma 6.1. Let w1 and w2 ∈ Es (L T ) be two choices of w of Lemma 6.1 such that wi = u on G ωT , i = 1, 2. Let ui = κ(wi ) be the corresω

ponding solutions for the linear Goursat problem (6.1). Then the following inequality holds for t ∈ (0, T] : u1 − u2 L t ,s−1 ≤ C t1/2 w1 − w2 L t ,s−1 ,

(6.10)

where C is a non-decreasing function depending continuously on h, T, w1 L T ,s + w2 L T ,s , g s,T , and f s,T , with A

1

1

(6.11)

f s,T = max DW1λ f C b2s−5 (L T ×Vε

∗ , 1 ×Wε2 )

λ

where w1 = (w1A) = (w11 , . . . , w1N ),

Dw1A = D(w11 ,...,w1N ) ,

W1 = (w1 , Dw1 ) = (W1λ ) = (w11 , . . . , w1N , (Dw1 )1 , . . . , (Dw1 ) N(n+1) ), DW1λ = D(w11 ,...,w1N ,(Dw1 )1 ,...,(Dw1 ) N(n+1) ) .

Proof.

By definition of κ, it holds that gab(x, w1 (x))Dabu1 = f(x, w1 (x), Dw1 (x)) in L T ,

u1 = u on G ωT ,

(6.12)

gab(x, w2 (x))Dabu2 = f(x, w2 (x), Dw2 (x)) in L T ,

u2 = u on G ωT .

(6.13)

ω

and ω

From (6.12) and (6.13), u1 − u2 solves the following linear Goursat problem: gab(x, w1 (x))Dab(u1 − u2 ) = [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 + f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x))

in L T ,

(6.14)

u1 − u2 = 0 on G ωT . Due to Theorem 3.1, u1 − u2 satisfies the following energy inequality for t ∈ (0, T] u1 − u2 L t ,s−1 ≤

t Rs−1

LT ≡ c(h, γ s−1 , γ Γs−1 , T)

2 ω=1

Γ ,G ω ,1 P s−2

+

Lt t P s−2

,

(6.15)


g s,T = max Dw1A gab C b2s−5 (L T ×Vε ) + g C b2s−5 (L T ×Vε ) ,


187

where c is a non-decreasing function of T depending continuously on each of its arguments and LT γ s−1

=

LT T g(x, w1 (x)) s−1 ,

γ Γs−1

=

2

,G g(x, w1 (x)) Γs−2

ω

,1

,

ω=1

P = [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 + f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x)). (6.16) LT Proceeding as above (see (6.6b) and (6.7)), we derive the estimates of g(x, w1 (x)) s−1 and Γ ,G g(x, w1 (x)) s−2

ω

,1

Γ ,G . To estimate P s−2

ω

,1

we first proceed as in (6.6a) to have ω

,1

Gω

t ≤ c0 P s−2 .

Then we deduce that u1 − u2 satisfies the following inequality for t ≤ T: 2 G ωt Lt LT Γ u1 − u2 L t ,s−1 ≤ c(h, γ s−1 , γ s−1 , T) c0 P s−2 + t P s−2 .

(6.17)

ω=1

Due to (6.16), it holds that Lt Lt ≤ [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2 P s−2 Lt + f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x)) s−2 .

(6.18)

As u2 ∈ Es (L t ), we have Dabu2 ∈ Es−2 (L t ). Moser estimate (5.54) of Theorem 5.10 for m = s − 1 gives Lt [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2

≤ cT (s) max Dw1A gab C b2s−5 (L T

1 ×Vε1 )

A

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 u2 L t ,s .

Using (6.3) we gain Lt [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2

≤ cT (s) max Dw1A gab C b2s−5 (L T

1 ×Vε1 )

A

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 1/2 2s−3 × C1 2|u|G ωt ,2s−1 + t f C b2s−3 (L T ×Vε ×Wε∗ ) [1 + w2 L t ,s ] , ω=1

ω

1

(6.19a)

2

where cT (s) is a non-decreasing continuous function of T depending also on s. To estiLt , we use Moser inequality (5.52) of mate f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x)) s−2


Γ ,G P s−2

188


Theorem 5.10 with s replaced by s − 1 and where we take m = s − 1. Doing so we gain Lt f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x)) s−2

≤ cT1 (s) max DW1λ f C b2s−5 (L T

∗ 1 ×Vε1 ×Wε2 )

λ

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 ,

(6.20a)

where W1 = (W1λ ). Using similar tools as above we get the following estiGω

t and f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), mates for [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2

G ωt

Dw2 (x)) s−2

≤ cT (s)t1/2 max Dw1A gab C b2s−5 (L T ×Vε

1)

A

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 u2 L t ,s .

Using (6.3) we get Gω

t [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2

≤ cT (s)t1/2 max Dw1A gab C b2s−5 (L T ×Vε

1)

A

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 2 × C1 |u|G ωt ,2s−1 + t1/2 f C b2s−3 (L T ×Vε ×Wε∗ ) [1 + w2 L t ,s ]2s−3 . ω

ω=1

1

(6.19b)

2

Similarly we have Gω

t f(x, w1 (x), Dw1 (x)) − f(x, w2 (x), Dw2 (x)) s−2

≤ cT (s)t1/2 max DW1λ f C b2s−5 (L T ×Vε

∗ 1 ×Wε2 )

λ

Lt × [1 + w1 L t ,s + w2 L t ,s ]2s−5 w1 − w2 s−1 .

(6.20b)

Finally, due to (4.5) of Theorem 4.2 and (6.17), (6.18), (6.19a), (6.19b), (6.20a), and (6.20b), we gain the desired inequality (6.10).

Lemma 6.4. Let R > 0 be a real number such that R ≥ v1 L T ,s , where v1 is constructed in the first step. Let t ∈ (0, T]. Set B R,t = {w ∈ Es (L t ) : w = u on G ωt , w L t ,s ≤ R}. ω

Then B R,t endowed with the distance defined by the norm . L t ,s−1 is a non-empty and complete metric space.


Gω

t [gab(x, w2 (x)) − gab(x, w1 (x))]Dabu2 s−2


Proof.

189

B R,t is non-empty since v1 ∈ B R,t . It remains to show that B R,t is a closed subset

of the Banach space (Es (L t ), . L t ,s−1 ). Let then (wk)k∈N ⊂ B R,t be a sequence such that wk − w L t ,s−1 → 0. We need to show that w ∈ B R,t . By definition, it holds that wk L t ,s ≤ R and wk = u on G ωt . Since wk − w L t ,s−1 → 0, it follows from weak compactness arguments ω

as those used in [35] (see [35, Lemma 5.2, p. 193, proof of Theorem 6.2, p. 200, and proof of Theorem 8.1, p. 208]) that w ∈ Es (L t ), w = u on G ωt and w L t ,s ≤ R. Hence w ∈ B R,t . ω

Now we use Lemmas 6.1–6.4 to show that, for a sufficiently large R and T1 small enough, the mapping κ defined by the linear Goursat problem (6.1) is a contraction from % R = max v1 L T ,s , 2C 1

2 ω=1

& |u|G ωT ,2s−1 , ω

where C 1 is the constant of inequality (6.3) of Lemma 6.1, where w L T ,s is replaced by R. Let T1 ∈ (0, T] such that 1/2

C 1 T1 f C b2s−3 (L T ×Vε

∗ 1 ×Wε2 )

[1 + R]2s−3
0 is a non-decreasing function depending continuously on each of its arguments T, h, K0 , g C b2s−3 (L T ×Bε ) , 0

%" " " ∂2 f " " " | f|s,T,K0 = max " (∂u)2 "

C b2s−3 (L T ×YK0 )

" " 2 " ∂ f " " " ," ∂u∂ Du"

C b2s−3 (L T ×YK0 )

" " 2 " ∂ f " " " ," (∂ Du)2 "

& ,

C b2s−3 (L T ×YK0 )

with YK0 being a compact neighborhood of (0, 0) in V × R(n+1)N of the following form: YK0 = {z ∈ R N × R(n+1)N : |zΛ | ≤ cK0 , for Λ = 1, . . . , (n + 2)N}. c > 0 is a Sobolev constant of the following Sobolev inequality w C 0 (L T ) ≤ c w L T ,s , C 1 > 0 is a constant. Now setting ui = ϑ(Vi ) for i = 1, 2, we use similar tools as in the proof of Lemma 6.3 to show in view of (6.29) that u1 − u2 L T ,s−1 ≤ k2 (T, h, K0 , g C b2s−3 (L T ×Bε ) , | f|s,T,K0 )T 1/2 0

× ( V1 L T ,s + V2 L T ,s ) V1 − V2 L T ,s−1 ,

(6.33)

where the constant k2 > 0 is a non-decreasing function depending continuously on each of its arguments. Consider now v1 defined in the first step of the proof of item (i) above as the solution of the following linear Goursat problem mab Dabv1 = 0

in L T ,

v1 = u on G ωT , ω


ω=1


with

193

⎧ ⎪ −1 if (a, b) ∈ {(1, 2), (2, 1)}, ⎪ ⎪ ⎨

mab = 1 ⎪ ⎪ ⎪ ⎩ 0

if a = b ∈ {3, . . . , n + 1}, elsewhere.

It is known from Theorem 3.1 that v1 satisfies the following energy inequality: v1 L T ,s ≤ k3

2

|u|G ωT ,2s−1 ,

ω=1

(6.34)

ω

Let us now assume that f0 L T ,s−1 < d,

2 ' ' ' ' 'u' ω=1

ω G ω ,2s−1 T

< d,

(6.35)

where d is a positive real number to be chosen appropriately. We are going to choose K in such a way that, for d small enough, the mapping ϑ defined by (6.30) is a contraction from the non-empty complete metric space (B K,T , . L T ,s−1 ) into itself, where B K,T = {w ∈ Es (L T ) : w = u on G ωT and w L T ,s ≤ K}. ω

Choose d small enough such that 2ζ d < K0 , where ζ = max(2k1 , k3 ), with k1 and k3 being the constants of inequality (6.32) and inequality (6.34), respectively. We now set K = 2ζ d.

(6.36)

From (6.32) and (6.36), it is easy to see that v1 ∈ B K,T ,

w L T ,s ≤ K ⇒ u L T ,s ≤

K + k1 [C 1 T 1/2 | f|s,T,K0 [1 + 2ζ d]2s−3 (2ζ d)2 ], 2

V1 L T ,s ≤ K, V2 L T ,s ≤ K ⇒ u1 − u2 L T ,s−1 ≤ k2 T 1/2 (4ζ d) V1 − V2 L T ,s−1 . (6.37) We can then take d > 0 small enough such that k1 [C 1 T 1/2 | f|s,T,K0 [1 + 2ζ d]2s−3 (2ζ d)2 ] ≤ ζ d =

K , 2

1 k2 T 1/2 (4ζ d) < , 2

(6.38)

where k2 is the constant of inequality (6.33). From (6.37) and (6.38) we deduce that ϑ is a contraction from the non-empty metric space (B K,T , . L T ,s−1 ) into itself. ϑ admits therefore a fixed point u which is the desired global solution of the characteristic initial value problem (3.19) on L T such that u∈ Es (L T ).


where k3 > 0 is a non-decreasing function depending continuously on T.

194


Remark 6.5. Theorem 3.2 still holds if the function f, instead of being defined on U × V × R(n+1)N , is defined on U × V × W, where W is a strict non-empty open subset of R(n+1)N , under the following additional assumption U (Γ ) ⊂ W, with U = (D1 u, D2 u, D3 u, . . . , Dn+1 u), u is defined in the second step of the proof of item 2

1

(i) of Theorem 3.2.

In this section our concern is the resolution of the characteristic initial value problem for the EYMH system. (both the evolution problem and the constraints one are discussed). We begin by giving the complete form of the EYMH system (we refer, e.g., to [2, 8, 9, 11, 31] and references therein for some bibliographical links). After that, by the choice of harmonic gauge for the gravitational potentials and Lorentz gauge for Yang– Mills potentials, we reduce the complete EYMH system to a second-order hyperbolic quasilinear system. We then apply Theorem 3.2 to the reduced EYMH system and so the evolution problem is locally solved. The constraints problem is solved by generalizing the method made up by Rendall [37] to construct C ∞ initial data for the Goursat problem associated to the Vacuum Einstein and Einstein-perfect fluid models. Doing so and combining adequate energy inequalities, we gain the construction of initial data of finite differentiability order. Throughout the paper roman indices vary from 1 to 4 whereas Greek indices vary from 3 to 4.

7.1 The complete form of the EYMH system

The EYMH system reads as follows (see [2, 8, 31, 42]) Ri j − 12 Rgi j = ρi j (i F i j = J j ∇ (i ∇ Φ = H ∇ (i

(Einstein system), (Yang–Mills system),

(7.1)

(Higgs system),

where (gi j ) are the covariant components, in local coordinates (xi )i=1,...,4 on a 4 − d space– time M, of an unknown Lorentzian metric g of signature − + ++; (gi j ) will denote the inverse of (gi j ). (Ri j ) and R are, respectively, the Ricci tensor and the scalar curvature of


7 Application to the EYMH System


195

the metric g, i.e., k k l k l Ri j = Γikj,k − Γik, j + Γkl Γi j − Γ jl Γik,

R = gi j Ri j ,

(7.2)

where Γikj are the Christoffel symbols of the space–time metric g. Here and throughout the paper, commas “,” as indices denote usual partial derivatives. E.g., ∂Γilj ∂ xk

≡ Γilj,k.

F i j are the contravariant components of an unknown Yang–Mills field F , which is an antisymmetric 2-form of type Ad, defined on M with values in an N-dimensional tial A, which is a G-valued 1-form defined on M, as follows FiIj = ∇i AIj − ∇ j AiI + [Ai , Aj ] I

with [Ai , Aj ] I = C JI K AiJ AKj ,

(7.3)

where FiIj and AiI are the respective components of F and A in local coordinates (xi )i=1,...,4 on M and orthogonal basis (ε I ) I =1,...,N of G, ∇ denotes the covariant derivative w.r.t. the space–time metric g, [,] denote the Lie brackets of the Lie algebra G, and C JI K are the structure constants of the Lie group G. We assume that G admits an Ad-invariant nondegenerate scalar product, denoted by a dot “.”, which enjoys the following property (see [8, 9]): f.[k, l] = [ f, k].l

∀ f, k, l ∈ G.

(7.4)

Φ is the Higgs field, a G-valued function defined on M. (ρi j ) is the energy-momentum or the stress-energy tensor which is given by (see [2, 8, 31]) ρi j = Fik.F jk − 14 gi j Fkl .F kl + Φi j ,

(7.5)

(i Φ.∇ ( j Φ − 1 gi j (∇ (kΦ.∇ (kΦ + V(Φ 2 )), Φi j = ∇ 2

(7.6)

Φ 2 = Φ.Φ,

(7.7)

where

with

V is a C ∞ real-valued function defined on R (often called the self-interaction potential). (J k) is the Yang–Mills current defined by (kΦ]. J k(A, Φ, DΦ) = [Φ, ∇

(7.8)

( is the gauge covariant derivative. H (Φ) is the Higgs potential defined by (see [8]): ∇ H I (Φ) = V (Φ 2 )Φ I

where V is the derivative of V.

(7.9)


R-based Lie algebra G of a Lie group G. F is related to the unknown Yang–Mills poten-

196


Remark 7.1. (i) Due to the following consequences of Bianchi identities: ∇ i (Ri j − 12 Rgi j ) = 0,

(i ∇ ( j F i j = 0, ∇

it is easy to see that: if the EYMH system (7.1) is satisfied, then the stressenergy tensor ρi j and the current J a satisfy the following conservation laws: ∇i ρ i j = 0,

(i J i = 0. ∇

(7.10)

following algebraic structural condition [H (Φ), Φ] = 0.

(7.11)

The relation (7.11) is fulfilled by H (Φ) given by (7.9). (iii) If the YMH system is satisfied, then a direct calculation shows that the conservation laws (7.10) are fulfilled by the stress-energy tensor given by (7.5) and the current given by (7.8) (for H (Φ) given by (7.9)). It results that the EYMH system (7.1) is coherent.

7.2 The reduced EYMH system

Here, we use the harmonic gauge and the Lorentz gauge conditions to recast the EYMH system (7.1) as a second-order quasilinear hyperbolic system. Define k . Γ k = glm Γlm

(7.12)

A direct calculation shows that the Ricci tensor takes the following form (see [6, 24, 29]) Ri j = − 12 gkm gi j,mk + 12 (gki Γ,kj + gkj Γ,ik) + Qi j (g, Dg),

(7.13)

where Qi j is a rational function of its arguments depending quadratically on Dg and given by Qi j (g, Dg) = 12 (gki, j + gkj,i )Γ k + 12 gkm gnl (gnk, j gim,l + gnk,i g jm,l ) − 14 gkm gnl gkn,i glm, j − 12 gkm gnl gmn,k(gl j,i + gli, j − gi j,l ) + 14 gkm gnl gkm,l (gin, j + g jn,i − gi j,n) − 12 gkm gnl gki,n(gl j,m − gmj,l ).

(7.14)


(ii) Due to the expression (7.8) of (J a), the Higgs potential H must satisfy the


197

Taking the trace of Einstein equations and using (7.5) and (7.6), we derive the following form of the Einstein system Ri j = τi j ,

(7.15)

(i Φ.∇ ( j Φ + 1 gi j V(Φ 2 ). τi j = Fik.F jk − 14 gi j Fkl .F kl + ∇ 2

(7.16)

where

According to (7.13) and (7.15), the Einstein system has the following form (7.17)

We now move on to the reduction of the Yang–Mills-Higgs system. Using the definition of the gauge covariant derivative and familiar techniques of raising lowering indices, after some calculations, we get the following form of the YMH system: gik[[Ap,ki − Ak, pi ] + [Ak, Ap],i ] + g jp(gik g jl ),i [Al,k − Ak,l + [Ak, Al ]] j

i + g jpΓim F mj + g jpΓim F im + g jp[Ai , F i j ] = Jp(A, Φ, DΦ),

(7.18)

gi j Φ,i j − gi j Γilj Φ,l + [∇i Ai , Φ] + 2[Ai , ∇ i Φ] + [Ai , [Ai , Φ]] = H . Setting Δ = ∇k Ak = ∇ k Ak,

(7.19)

after a straightforward calculation, we deduce from (7.17), (7.18), and (7.19) that the EYMH system has the following form: − 12 gkm gi j,mk + 12 (gki Γ,kj + gkj Γ,ik) + Qi j = τi j , gik Ap,ik − (Δ, p + Γ,lp Al + Γ l Al, p − g,kip Ak,i ) + gik[Ak, Ap],i + g jp(gik g jl ),i [Al,k − Ak,l j

i + [Ak, Al ]] + g jpΓim F mj + g jpΓim F im + g jp[Ai , F i j ] = Jp(A, Φ, DΦ),

(7.20)

gi j Φ,i j − Γ l Φ,l + [Δ, Φ] + 2[Ai , ∇ i Φ] + [Ai , [Ai , Φ]] = H (Φ). If we now impose the harmonic gauge condition Γ k = 0 and the Lorentz gauge condition Δ = 0, then we obtain from (7.20) the reduced EYMH system that is )i j = τi j (g, A, Φ, Dg, D A, DΦ), R L Ap = Jp(A, Φ, DΦ), SΦ = H (Φ),

(7.21)


− 12 gkm gi j,mk + 12 (gki Γ,kj + gkj Γ,ik) + Qi j = τi j .

198


where )i j ≡ Ri j − 1 (gki Γ,kj + gkj Γ,ik) = − 1 gkm gi j,mk + Qi j (g, Dg), R 2 2 (i F i j + (Δ, p + Γ,lp Al + Γ l Al, p) L Ap ≡ g jp∇ = gik Ap,ik + g,kip Ak,i + gik[Ak, Ap],i + g jp(gik g jl ),i [Al,k − Ak,l + [Ak, Al ]]

(7.22)

j

i + g jpΓim F mj + g jpΓim F im + g jp[Ai , F i j ],

(i ∇ (i Φ + Γ l Φ,l − [Δ, Φ] = gi j Φ,i j + 2[Ai , ∇ i Φ] + [Ai , [Ai , Φ]]. SΦ ≡ ∇ It is clear that (7.21) is a second-order quasilinear hyperbolic system of the following

gkm (v)Dkm v = (X1 (v, Dv), Y1 (v, Dv), Z 1 (v, Dv)),

(7.23)

where the nonlinear terms are given by X1 (v, Dv) = 2(Qi j (g, Dg) − τi j (g, A, Φ, Dg, D A, DΦ)), Y1 (v, Dv) = Jp(A, Φ, DΦ) − (g,kip Ak,i + gik[Ak, Ap],i ) + g jp(gik g jl ),i [Al,k − Ak,l j

i + [Ak, Al ]] + g jpΓim F mj + g jpΓim F im + g jp[Ai , F i j ],

(7.24)

Z 1 (v, Dv) = H (Φ) − (2[Ai , ∇ i Φ] + [Ai , [Ai , Φ]]), where Qi j , τi j , Jp, and H are as above (see (??), (7.16), (7.8), and (7.9)), with g = (gkl ),

A = (Ap),

Φ = (Φ I ).

(7.25)

Remark 7.2. (i) Due to (7.22), any solution (gi j , Ap, Φ) of the reduced EYMH system (7.21) that satisfies the constraints Γ k ≡ gi j Γikj = 0 and Δ ≡ ∇i Ai = 0 is also a solution of the complete EYMH system (7.1). (ii) For the constraints Γ k = 0 and Δ = 0 to be satisfied everywhere, it is enough that they are satisfied on G 1 ∪ G 2 (see [25]): one uses the Bianchi identities to show that (Γ k, Δ) solves a second-order homogeneous linear system. (iii) The reduced EYMH system constitutes the evolution system associated to the EYMH system. (iv) The resolution of the constraints problem is the construction, from arbitrary choice of some components of the gravitational potentials and Yang–Mills potentials (called free data) on G 1 ∪ G 2 , of all initial data for the reduced


form with unknown v = (gi j , Ap, Φ):


199

EYMH such that the constraints Γ k = 0 and Δ = 0 are satisfied on G 1 ∪ G 2 for the solution of the corresponding evolution problem.

7.3 The existence and uniqueness theorem for the reduced EYMH system

The notation are the same as those introduced in Sections 3 and 6. Now set skl = gkl − γkl ,

Skl = gkl − γ kl ,

(7.26)

γkl (x3 , x4 ) = gkl (0, 0, x3 , x4 ),

γkl γ ki = δli .

(7.27)

It is easy to see that, for small s = (skl ), it holds Skl = −skl + O kl (s2 )

where skl = γ kpγ lq spq ,

(7.28)

and O kl (s2 ) are C ∞ functions that vanish to second order at s = 0, given by O kl (s2 ) = γ kpγ lq spi sqj gi j .

(7.29)

It follows that, for (skl ) small enough, (gkl ) is a hyperbolic metric and gkl are then rational functions (thus C ∞ functions) of their arguments si j . Then (7.23) takes the following form: gkm (x, u)Dkm u= (X(x, u, Du), Y(x, u, Du), Z (x, u, Du)),

(7.30)

which is a second-order quasilinear hyperbolic system with unknown u= (si j , Ap, Φ). In view of (7.24), direct calculations show that the nonlinear terms X(x, u, Du), Y(x, u, Du), and Z (x, u, Du) are such that Xi j (x, 0, 0) = 2Qi j (γ , Dγ ) − γi j V(0) − γ km Dkm γi j ,

Y(x, 0, 0) = 0,

Z (x, 0, 0) = 0.

(7.31)

We are now in the position to state the existence and uniqueness result for the evolution problem associated to the EYMH system.

Theorem 7.3. Let T ∈ (0, T0 ] be a real number and p an integer such that 3 < p. Let γi j (x3 , x4 ), i, j = 1, . . . , 4, be functions defined on Γ such that (γi j (x3 , x4 )) is hyperbolic at


where γkl is the restriction to Γ of the metric gkl and γ kl is the inverse of γi j , i.e.,

200


each point of Γ . Let

u= s , A , Φ ω

ωi j ω k ω

ω = 1, 2,

,

(7.32)

i, j,k∈{1,2,3,4}

be a vector function defined on G ωT such that ∀i, j, k ∈ {1, 2, 3, 4}, ∀ω ∈ {1, 2}, s = (s ) is continuous on G ωT ,

ω

ωi j

(γi j + s ) is hyperbolic at each point of G ωT , ωi j

G ωT is characteristic w.r.t. the metric s, ω

(7.33) (7.34)

s , A , Φ ∈ E 2 p−1 (G ωT ),

ωi j ω k ω ωi j

ωk

u = u on Γ .

ω

1

2

(7.35)

(i) Then there exists a real number T1 ∈ (0, T] such that the reduced EYMH system (7.30) has, in the domain L T1 , a unique solution u= (si j , Ak, Φ) ∈ E p(L T1 ), satisfying u= u on G ωT1 . Setting ω

gi j = γi j + si j ,

v = (gi j , Ak, Φ),

(7.36)

v solves the reduced EYMH system (7.23). (ii) Assume in addition to (7.33)–(7.35) that γ ∈ H2 p−1 (Γ ). There exists a positive real number d such that: if |γ |Γ2 p−1 < d,

2 |u|G ωT ,2 p−1 < d, ω=1

ω


(7.37)

Proof.

The proof of Theorem 7.3 is obtained by applying Theorem 3.2 to system

(7.30).

7.4 The constraints problem associated to the EYMH system

The task here is to solve the constraints problem, i.e., the construction of initial data for the reduced EYMH system such that the constraints Γ k = 0 and Δ = 0 are satisfied on G 1 ∪ G 2 for the solution of the corresponding evolution problem. The problem is solved in three main steps. Firstly, the method of Rendall [37] is used to construct C ∞ data satisfying the gauge conditions Γ k = 0 and Δ = 0 on G 1 ∪ G 2 . Secondly, energy inequalities are established in appropriate weighted Sobolev spaces for the corresponding linearized transport equations. Finally, the C ∞ result and energy inequalities are combined to derive, thanks to the fixed point theorem, the construction of all the remaining initial


[s ]Γ , [A ]Γ , [Φ ]Γ ∈ H2 p−1 (Γ ),


201

data of finite differentiability order. It is at this level that the precise coefficients of the respective transport equations are needed. Previous studies did not provide such details, even for the vacuum Einstein equations (see [37]). The construction of the data is done fully on G 1 and it will be clear that data on G 2 are constructed in quite a similar way. The novelty here is that the data are constructed for the EYMH model and are of finite differentiability order, whereas those of [15, 37] are constructed either for the vacuum Einstein, Einstein-perfect fluid models ([37]) or the EYM model ([15]) and are of infinite differentiability order. The construction will be made in a standard harmonic coordinates system and Lorentz gauge. The existence of such standard harmonic costandard harmonic coordinates system as given in [37].

Definition 7.4 (Standard harmonic coordinates). Let M be a four-dimensional manifold endowed with a Lorentzian metric μ, and N 1 and N 2 being two intersecting null hypersurfaces, S = N 1 ∩ N 2 . Consider a local coordinates system (xi ) in a neighborhood of N 1 ∪ N 2 . (xi ) is a standard harmonic system w.r.t. μ, N 1 , and N 2 if the following conditions are satisfied: k

k

(i) (xi ) is a harmonic system w.r.t. μ, i.e., μi j Γ i j = 0 for all k, where Γ i j are the Christoffel symbols of the metric μ. (ii)

N 1 and N 2 are locally defined by x1 = 0 and x2 = 0, respectively,

(iii) x1 is an affine parameter along the null geodesics that generate N 2 , (iv) x2 is an affine parameter along the null geodesics that generate N 1 , (v) x3 and x4 are constant along the null geodesics that generate N 1 or N 2 .

Consequence of the above definition (see [37, 38])

If (xi ) is a standard harmonic system w.r.t. μ, N 1 , and N 2 , then the following relations hold: On N 1 , μ2i = 0 for i = 1,

μ22,1 = 2μ12,2 .

(7.38)

μ1i = 0 for i = 2,

μ11,2 = 2μ12,1 .

(7.39)

On N 2 ,


ordinates system has been established by Rendall [37]. Let us recall the definition of a

202


Let us implement the method of Rendall [37] to construct C ∞ initial data for the EYMH system. We assume the following relevant conditions for the free data: g22 = g23 = g24 = 0 on G 1T ,

A2 = 0 on G 1T ,

Φ, A3 , and A4 are given C ∞ functions on G 1T .

(7.40)

We first give some remarks on the above assumptions (7.40).

Remark 7.5.

the general idea is to produce a space–time for which the given coordinates in R4 are standard coordinates. Rendall [37] used these conditions to sketch the handling of the vacuum Einstein and the Einstein-perfect fluid cases. (ii) In addition to the conditions in (i), the condition A2 = 0 on G 1T is assumed in order to facilitate the hierarchical construction process of initial data. (iii) For the construction of the initial data on G 2T , the following analog of conditions (7.40) will be assumed g11 = g13 = g14 = 0

on G 2T ,

A1 = 0 on G 2T ,

Φ, A3 , and A4 are given C ∞ functions on G 2T .

The first level of the construction process is now described.

7.4.1 Construction of (gαβ )α,β∈{3,4} and g12 on G 1T , arrangement of relations Γ 1 = 0 and g22,1 = 2g12,2 on G 1T Let

T ∈ (0, T0 ],

h

1 αβ

⎡

h

= ⎣ 1 33 h 1 34

h

⎤

1 34 ⎦

h

1 44

be a matrix function with determinant 1 at each point of G 1T . On G 1T , set gαβ = Ωh , 1 αβ

where Ω > 0 is an unknown scalar function defined on G 1T , called the conformity factor. For the sake of simplicity in the notation, we will use hαβ instead of h . One deduces 1 αβ

from the free data given above in (7.40) that the following algebraic relations hold on G 1T : g12 g12 = 1,

g11 = g1α = 0,

g2β gαβ = −g12 g1α ,

gλβ gαβ = δλα .

(7.41)


(i) The conditions g22 = g23 = g24 = 0 on G 1T are in accordance with (7.38) since


203

At this level, the expressions of R22 and τ22 are needed. Following the work of Rendall [37], a straightforward calculation shows that on G 1T the following equalities hold: βλ

R22 = 14 g12 gαβ gαβ,2 (2g12,2 − g22,1 ) + 14 g,2 gλβ,2 − 12 (gαβ gαβ,2 ),2 , τ22 = Ω −1 hαβ Aα,2 .Aβ,2 + (Φ,2 )2

(7.42)

where (Φ,2 )2 = Φ,2 .Φ,2 .

If, in addition, we assume g22,1 = 2g12,2 on G 1T , then Γ 1 = 0 is equivalent to g12,2 =

Ω,2 1 g12 . 2 Ω

(7.43)

1 αβ g g 4 ,2 αβ,2

− 12 (gαβ gαβ,2 ),2 = τ22 ,

(7.44)

provides the following nonlinear second-order ordinary differential equation (ODE) with the conformity factor Ω as unknown Ω,2 2 1 Ω,2 αβ − + hαβ,2 h,2 − 2 = Ω −1 hαβ Aα,2 .Aβ,2 + (Φ,2 )2 . Ω 2 Ω ,2

(7.45)

If we set Ω = ev , then (7.45) reads 2v,22 = f(x, v, v,2 ),

(7.46)

where αβ

f(x, v, v,2 ) = −(v,2 )2 − 2e−v hαβ Aα,2 .Aβ,2 + 12 hαβ,2 h,2 − 2(Φ,2 )2 .

(7.46a)

Assume the following for the free data hαβ , A3 , A4 , Φ ∈ C ∞ (G 1T ),

(7.47)

and let v0 , v1 ∈ C ∞ (Γ )

where Γ ≡ G 1T ∩ G 2T .

Then there exists T1 ∈ (0, T] such that (7.46) has, on G 1T1 , a unique solution v ∈ C ∞ (G 1T1 ) satisfying v = v0 and v,2 = v1 on Γ . (One uses known local existence and uniqueness results concerning nonlinear ODEs depending on parameters with C ∞ coefficients and initial data.) Consider now Equation (7.43) of unknown g12 , which can be written as follows: g12,2 = 12 g12 v,2 .

(7.48)

Let w0 ∈ C ∞ (Γ ). Then (7.48) has, on G 1T1 , a unique solution g12 ∈ C ∞ (G 1T1 ) satisfying g12 = w0 on Γ . (This is because (7.48) is a linear ODE with C ∞ coefficients and initial data.)


The equation

204


The condition g22,1 − 2g12,2 = 0 on G 1T1 is now arranged. On G 1T1 , the reduced )22 = τ22 is equivalent to the following homogenous ODE with unknown equation R g22,1 − 2g12,2 : (g12 )2 g12,2 (g22,1 − 2g12,2 ) − g12 (g22,1 − 2g12,2 ),2 = 0.

(7.49)

Set g22,1 = 2g12,2 on Γ . Then g22,1 − 2g12,2 = 0 on G 1T1 and so Γ 1 = 0 on G 1T1 . Remark 7.6. In the same way, given a matrix function ⎡

(kαβ ) ≡ (h ) = ⎣ 2 33 2 αβ h 2 34

h

⎤

2 34 ⎦

h

2 44

with determinant 1 at each point of G 2T , set gαβ = Ωkαβ on G 2T , where Ω > 0 is an unknown function defined on G 2T , called the conformity factor. Analogously to (7.40), assume g11 = g13 = g14 = 0 on G 2T ,

A1 = 0 on G 2T ,

Φ, A3 and A4 are given on G 2T . Then g12 and gαβ are constructed on G 2T1 such that gαβ , g12 ∈ C ∞ (G 2T1 ) with the relations g11,2 − 2g12,1 = 0, Γ 2 = 0 on G 2T1 if g11,2 = 2g12,1 on Γ .

We now proceed to the construction of the data g13 , g14 , and A1 on G 1T1 such that g13 , g14 , A1 ∈ C ∞ (G 1T1 ). We also arrange the relations Γ α = 0 and Δ = 0 on G 1T1 , α = 3, 4. 7.4.2 Construction of g1α and A1 , arrangement of relations Γ α = 0 and Δ = 0 on G 1T1 , α = 3, 4 We seek for a combination of R2α , Γ α , Γ,2α , L A2 , Δ, and Δ,2 that will provide a system of ODEs on G 1 with unknowns g1α and A1 . After performing tedious and lengthy calculations (following the work of Rendall [37]), we have the following result: β

R2α + 12 gαβ Γ,2 + (g12 g12,2 gαβ + 12 gαβ,2 )Γ β = g12 g1α,22 + (g12 )2 g12,2 g1α,2 − gαβ,2 gβλ g12 g1λ,2 βλ

βλ

+ {(g12 )2 g12,2 gαβ g,2 + 12 [gαβ g12 g,22 − g12 gβλ gαβ,22 ]}g1λ + cα ,

(7.50)

L A2 − 2Δ,2 − 2g12 g12,2 Δ + 2(g12 g12,2 Aν + Aν,2 )Γ ν + 2Aν Γ,2ν = −2g12 A1,22 − 2(g12 )2 g12,2 A1,2 + 2g12 gαλ Aα,2 g1λ,2 + K λ g1λ + Ag ,

(7.51)


h


205

where cα = 12 (g12 )g12,2 [−2g12 g12,α + gμθ (2gαμ,θ − gμθ,α )] + 14 gαβ,2 [−2gβλ g12 g12,λ + gβλ gμθ (2gλμ,θ − gμθ,λ )] + 12 (gλβ gαβ,2 ),λ − 3(g12 g12,2 ),α − 12 (g12 )2 g12,2 g12,α βλ

+ 12 g12 (g22,1α + g12,2α ) + 12 g,2 (gλβ,α + gλα,β ) + 12 gαβ [−2gβλ g12 g12,λ + gβλ gμθ (2gλμ,θ − gμθ,λ )],2 ,

(7.50a)

K λ = 4(g12 )2 g12,2 gαλ Aα,2 + [2g12 gαλ gβμ gμβ,2 Aα + 2g12 gαλ Aα,2 ],2 Ag = −2(g12 gαλ Aα ),2 g12,λ − 2g12 gαλ Aα g12,2λ − gβα ([Aβ , Aα,2 ] − g12 g12,β Aα,2 ) + (g12 gαλ Aα ),2 g12 [gβμ (gμλ,β + gλβ,μ − gμβ,λ )] + g12 gαλ Aα (g12 [gβμ (gμλ,β + gλβ,μ − gμβ,λ )]),2 αβ

− {gαβ [g12 g12,β + 12 gλμ (gμβ,λ + gλμ,β − gβλ,μ )] + g,β }Aα,2 + (2g12 gαλ g12,λ Aα − [gαδ gβμ (gμδ,β + gδβ,μ − gμβ,δ )]Aα ),2 .

(7.51a)

All the coefficients are known on G 1T1 . On G 1T1 the system β

R23 + 12 g3β Γ,2 + (g12 g12,2 g3β + 12 g3β,2 − Aβ A3,2 )Γ β + A3,2 .Δ = τ23 , β

R24 + 12 g4β Γ,2 + (g12 g12,2 g4β + 12 g4β,2 − Aβ A4,2 )Γ β + A4,2 .Δ = τ24 ,

(7.52)

L A2 − 2Δ,2 − 2g12 g12,2 Δ + 2(g12 g12,2 Aν + Aν,2 )Γ ν + 2Aν Γ,2ν = J2 , is equivalent to the following second-order linear system of ODEs with unknown (A1 , g13 , g14 ) g12 g13,22 + κ3λ g1λ,2 + 3 .A1,2 + χ3λ g1λ + 3 = 0, g12 g14,22 + κ4λ g1λ,2 + 4 .A1,2 + χ4λ g1λ + 4 = 0, −2g12 A1,22 − 2(g12 )2 g12,2 A1,2 + aλ g1λ + b = 0, where all the coefficients are known on G 1T1 and given as follows κ33 = (g12 )2 g12,2 − g3β,2 gβ3 ,

κ34 = −g3β,2 gβ4 ,

κ44 = (g12 )2 g12,2 − g4β,2 gβ4 ,

κ44 = (g12 )2 g12,2 − g4β,2 gβ4 ,

3 = 2g12 A3,2 , 4 = 2g12 A4,2 ,

κ43 = −g4β,2 gβ3 ,

(7.53)


− 2(g12 gαλ Aα ),2 gβμ gμβ,2 − 2g12 gαλ Aα [gβμ gμβ,2 ],2 ,

206

M. Dossa and C. Tadmon βλ

βλ

χαλ = (g12 )2 g12,2 gαβ g,2 + 12 [gαβ g12 g,22 − g12 gβλ gαβ,22 ] − 2gνλ g12 Aα,2 .Aν,2 , λ

αλ

(7.54) 12 αλ βμ

12 αλ

a = 4(g ) g12,2 g Aα,2 + [2g g g gμβ,2 Aα + 2g g Aα,2 ],2 12 2

− 2(g12 gαλ Aα ),2 gβμ gμβ,2 − 2g12 gαλ Aα [gβμ gμβ,2 ],2 , α = cα + gβλ (Aα,λ − Aλ,α + [Aλ , Aα ]).Aβ,2 − (Φ,2 ).(Φ,α + [Aα , Φ]), b = Ag − J2 . The following statement provides the construction of (g13 , g14 , A1 ) in G 1T1 . Let

Then system (7.53) has, on G 1T1 , a unique solution (g13 , g14 , A1 ) ∈ C ∞ (G 1T1 ) satisfying (g13 , g14 , A1 ) = (a0 , b0 , c0 ) on Γ

and

(g13,2 , g14,2 , A1,2 ) = (a1 , b1 , c1 ) on Γ .

Now, on G 1T1 , the reduced system )2α = τ2α , R L A2 = J2 , is equivalent to β

g3β Γ,2 + (g12 g12,2 g3β + 12 g3β,2 − Aβ .A3,2 )Γ β + A3,2 .Δ = 0, β

g4β Γ,2 + (g12 g12,2 g4β + 12 g4β,2 − Aβ .A4,2 )Γ β + A4,2 .Δ = 0, β 2Aβ Γ,2

(7.55)

− 2Δ,2 − 2g12 g12,2 Δ + 2(g12 g12,2 Aβ + Aβ,2 )Γ β = 0.

Thus, if Γ β = 0 and Δ = 0 on Γ then Γ β = 0 and Δ = 0 on G 1T1 . The condition Γ β = 0 and Δ = 0 on Γ is arranged thanks to a convenient choice of the data a0 , b0 , c0 , a1 , b1 , and c1 on Γ .

Remark 7.7. (i) In the same way g2α and A2 are constructed on G 2T1 such that g2α , A2 ∈ C ∞ (G 2T1 ), the relations Γ β = 0, Δ = 0 are established on G 2T1 if Γ β = 0 and Δ = 0 on Γ , for an appropriate choice of data on Γ . (ii) The determination of g1α and A1 on G 1T1 , with Γ β = 0 and Δ = 0 on G 1T1 , provides the determination of g2β , g2λ,1 , and A2,1 on G 1T1 thanks to the following


a0 , a1 , b0 , b1 , c0 , c1 ∈ C ∞ (Γ ).


207

relations that hold on G 1T1 via a direct computation g2β = −g12 gβα g1α ,

Δ = g12 (A2,1 + A1,2 ) + g2α Aα,2 + gαβ Aα,β + Γ α Aα , βμ

g12 g2λ,1 = gλβ Γ β − g12 g1λ,2 − gλβ g,2 g12 g1μ

(7.56)

− 12 [−2g12 g12,λ + gμθ (2gλμ,θ − gμθ,λ )]. (iii) Similarly, the determination of g2α and A2 on G 2T1 with Γ β = 0 and Δ = 0 on G 2T1 provides the determination of g1β , g1λ,2 , and A1,2 on G 2T1 thanks to the following relations that hold on G 2T1 via a direct computation: Δ = g12 (A2,1 + A1,2 ) + g1α Aα,1 + gαβ Aα,β + Γ α Aα , βμ

g12 g1λ,2 = gλβ Γ β − g12 g2λ,1 − gλβ g,1 g12 g2μ

(7.57)

− 12 [−2g12 g12,λ + gμθ (2gλμ,θ − gμθ,λ )].

The last level of the construction process is now described. 7.4.3 Construction of g11 on G 1T1 such that g11 ∈ C ∞ (G 1T1 ) and arrangement of relation Γ 2 = 0 on G 1T1 )αβ = ταβ which are equivalent to Rαβ = ταβ since We now consider the reduced equations R )αβ = Rαβ − 1 (gkα Γ,βk + gkβ Γ,αk ) R 2

and

Γ 1 = Γ 3 = Γ 4 = 0 on G 1T1 .

We seek for a combination of gαβ Rαβ and Γ 2 that will provide an ODE with unknown g11 . A direct and lengthy calculation shows that on G 1T1 the following combinations hold gαβ Rαβ − 2Γ,22 − 2g12 g12,2 Γ 2 = −2(g12 )2 g11,22 + 4(g12 )3 g12,2 g11,2 + {4(g12 )4 (g12,2 )2 + 12 (g12 )2 (gαβ gαβ,2 ),2 }g11 + 14 gαβ (Nαβ + Mαβ ) − 2P − 2g12 g12,2 S,

(7.58)

gαβ ταβ = K, where all the coefficients are known on G 1T1 and given by Nαβ = −gαβ,2 [(g12 )2 g22,1 g2μ g1μ − g12 g2μ g2μ,1 ] − 2(g12 )2 g12,2 (g1β,α + g1α,β ) − g12 (2g2μ g12,2 + gμλ g2λ,1 )(gβμ,α + gμα,β − gαβ,μ ) + g12 (g2β,1α + g2α,1β ) + gαβ,2 (g12 g2μ g1μ ),2 + g12 g2μ g1μ gαβ,22 2μ

12 + g,2 (g1β,α + g1α,β ) + g,2 (gμβ,α + gμα,β − gαβ,μ )

(7.59)


g1β = −g12 gβα g2α ,

208


+ g12 (g1β,2α + g1α,2β ) + g2μ (gβμ,2α + gμα,2β − gαβ,2μ ) λμ

2λ − g,λ gαβ,2 + g,λ (gμβ,α + gμα,β − gαβ,μ ) − g2λ gαβ,2λ

+ gλμ (gμβ,λα + gμα,λβ − gαβ,λμ ) − [2g12 g12,α + gλμ (gμλ,α + gμα,λ − gαλ,μ )],β , Mαβ = −g12 gαβ,2 [−g12 g22,1 g2μ g1μ + 2g2λ g2λ,1 + gλμ (g1μ,λ − g1λ,μ )] + [g12 g2μ g1μ gαβ,2 + g12 (g1β,α + g1α,β ) + g2μ (gμβ,α + gμα,β − gαβ,μ )] × (3g12 g22,1 + gλμ gλμ,2 ) × [−g2λ gαβ,2 + gμλ (gμβ,α + gμα,β − gαβ,μ )] − (g12 )2 (g12,β + g2β,1 − g1β,2 )(g12,α + g2α,1 − g1α,2 )

(7.60)

+ g12 gμα,2 [g2μ (g2β,1 + g12,β − g1β,2 ) + gλμ (g1λ,β − g1β,λ )] − 2g12 g2λ g1λ gθμ gθβ,2 gαμ,2 − g12 gλμ gλβ,2 (g1μ,α + g1α,μ ) − gθμ gθβ,2 g2λ (gλμ,α + gλα,μ − gαμ,λ ) − [g12 (g12,β + g1β,2 − g2β,1 ) + g2μ gμβ,2 ] × [g12 (g12,α + g1α,2 − g2α,1 ) + g2λ gλα,2 ] − g12 gλθ gθα,2 (g1β,λ + g1λ,β ) − g2μ gλθ gθα,2 (gμβ,λ + gμλ,β − gλβ,μ ) + mαβ , with mαβ = g12 gλβ,2 [g2λ (g2α,1 + g12,α − g1α,2 ) + gλμ (g1μ,α − g1α,μ )] − [−g2μ gλβ,2 + gθμ (gθλ,β + gθβ,λ − gλβ,θ )] × [−g2λ gαμ,2 + gδλ (gδμ,α + gδα,μ − gαμ,δ )], S = 2g g g1λ,2 + 12 2λ

(7.60a)

1 12 λμ g g (2g1λ,μ ) 2

+ 12 g2μ [gλ2 (2gμλ,2 − gλ2,μ ) + gλθ (2gμλ,θ − gλθ,μ )], P = [2g12 g2λ g1λ,2 + g12 gλμ (g1λ,μ )],2 + 12 {g2μ [gλ2 (2gμλ,2 − gλ2,μ ) + gλθ (2gμλ,θ − gλθ,μ )]},2 , αβ 2λ

αβ μλ

K = 2g g F2α .Fλβ + g g Fμα .Fλβ − F12 .F

12

− F34 .F

(7.61)

34

− F2λ .[g21 gλ2 F12 + g23 gλ2 F32 + g23 gλ4 F34 + g24 gλ2 F42 + g24 gλ3 F43 ] + gαβ (Φ,α + [Aα , Φ]).(Φ,β + [Aβ , Φ]) + V(Φ 2 ). The equation gαβ Rαβ − 2Γ,22 − 2g12 g12,2 Γ 2 = gαβ ταβ ,

(7.62)


+ [2g12 g12,λ + gμθ (gμθ,λ + gθλ,μ − gμλ,θ )]


209

is equivalent to the following second-order ODE on G 1T1 with unknown g11 , − 2(g12 )2 g11,22 + 4(g12 )3 g12,2 g11,2 + χg11 + ψ = 0,

(7.63)

where χ = 4(g12 )4 (g12,2 )2 + 12 (g12 )2 (gαβ gαβ,2 ),2 ,

ψ = 14 gαβ (Nαβ + Mαβ ) − 2V − 2g12 g12,2 S − K. (7.64)

Let d0 , d1 ∈ C ∞ (Γ ). Then (7.63) has in G 1T1 a unique solution g11 ∈ C ∞ (G 1T1 ) satisfying g11 = d0 and g11,2 = d1 on Γ . Now, on G 1T1 the reduced system )αβ = ταβ , R

(7.65)

implies the following homogenous ODE on G 1T1 with unknown Γ 2 : Γ,22 + g12 g12,2 Γ 2 = 0.

(7.66)

Assume Γ 2 = 0 on Γ . Then Γ 2 = 0 on G 1T1 . Remark 7.8. In the same way g22 is constructed on G 2T1 such that g22 ∈ C ∞ (G 2T1 ). The relation Γ 1 = 0 is also established on G 2T1 , if Γ 1 = 0 on Γ .

7.4.4 Compatibility conditions for C ∞ data on Γ ≡ G 1T1 ∩ G 2T1 We have presented Rendall’s method through which, given a positive real number 0 < T ≤ T0 , appropriate scalar functions h

ω αβ

on G ωT , ω = 1, 2, and some adequate condi-

tions on Γ , C ∞ initial data for the reduced EYMH system are constructed on G ωT . We have established that the solution of the evolution problem with those initial data satisfies the relations Γ i = 0 and Δ = 0 on G 1T1 ∪ G 2T1 for some T1 ∈ (0, T], provided that certain conditions are fulfilled on Γ . In fact, setting gαβ = Ωhαβ , where hαβ = h

1 αβ

on G 1T and hαβ = h

2 αβ

on G 2T , and (h ) is a symmetric positive definite matrix function with determinant 1 ω αβ

at each point of G ωT , ω = 1, 2, and Ω is an unknown positive scalar function, we have constructed C ∞ initial data as follows:


The following statement provides the construction of g11 on G 1T1 such that g11 ∈ C ∞ (G 1T1 ).

210


• gαβ , g12 on G 1T1 such that Γ 1 = 0 and g22,1 = 2g12,2 on G 1T1 under the following conditions ⎧ ⎪ Φ, A3 , and A4 are given C ∞ functions on G 1T , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ G 1T , A2 = 0 on G 1T , (i) ⎪ ⎪ g12 , Ω, and Ω,2 are given C ∞ functions on Γ , ⎪ ⎪ ⎪ ⎪ ⎩ g22,1 = 2g12,2 on Γ .

g22 = g23 = g24 = 0 on

Γ 1 = 0 and

• gαβ , g12 on G 2T1 such that Γ 2 = 0 and g11,2 = 2g12,1 on G 2T1 under the following

g11 = g13 = g14 = 0 on

Γ 2 = 0 and

• g1α , A1 on G 1T1 such that Γ α = 0 and Δ = 0 on G 1T1 under the following supplementary conditions (in addition to condition (i)): (iii) g1α,2 and A1,2 are given C ∞ functions on Γ ,

Γ α = 0 and Δ = 0 on Γ .

• g2α , A2 on G 2T1 such that Γ α = 0 and Δ = 0 on G 2T1 under the following supplementary conditions (in addition to condition (ii)): (iv) g2α,1 and A2,1 are given C ∞ functions on Γ ,

Γ α = 0 and Δ = 0 on Γ .

• g11 on G 1T1 such that Γ 2 = 0 on G 1T1 under the following supplementary conditions (in addition to conditions (i) and (iii)): (5i) g11,2 is a given C ∞ function on Γ ,

Γ 2 = 0 on Γ .

• g22 on G 2T1 such that Γ 1 = 0 on G 2T1 under the following supplementary conditions (in addition to conditions (ii) and (iv)): (6i) g22,1 is a given C ∞ function on Γ ,

Γ 1 = 0 on Γ .

We now show how the above adequate conditions (i), (ii), (iii), (iv), (5i), and (6i) are arranged. Begin by taking g12 = −1 on Γ (this is a property of any metric in standard coordinates, see [37, p. 232]). Then choose (h ), a C ∞ symmetric positive definite matrix ω αβ


conditions ⎧ ⎪ Φ, A3 , and A4 are given C ∞ functions on G 2T , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ G 2T , A1 = 0 on G 2T , (ii) ⎪ ⎪ g12 , Ω, and Ω,1 are given C ∞ functions on Γ , ⎪ ⎪ ⎪ ⎪ ⎩ g11,2 = 2g12,1 on Γ .


211

function on G ωT with determinant 1 at each point of G ωT , such that h

1 αβ

=h

2 αβ

on Γ .

On G 1T ∪ G 2T , set gαβ = Ωhαβ , where hαβ = h

1 αβ


2 αβ

) is a given C ∞ function on Γ . Take also on Γ , where Ω g22 = g23 = g24 = 0 on G 1T ,

A2 = 0 on G 1T ,

g11 = g13 = g14 = 0 on G 2T ,

A1 = 0 on G 2T .

) on G 2T . Let Ω = Ω

(7.40a)

g1α = 0, g22 = g2α = 0 on Γ , Ω, and hαβ are known on Γ . Next choose Φ , A and A which are C ∞ given functions on G ωT such that ω

ω3

ω4

Φ = Φ on Γ , 1

2

A = A on Γ , 13

23

A = A on Γ . 13

23

)1 and Ω,2 = Ω )2 on Γ , where Ω )1 and Ω )2 are two given C ∞ functions on Γ . Then Let Ω,1 = Ω Equations (7.43) and (7.44) as well as their corresponding analogs on G 2T , i.e., g12,1 =

Ω,1 1 g12 on G 2T , 2 Ω

(7.43a)

and 1 αβ g g 4 ,1 αβ,1

− 12 (gαβ gαβ,1 ),1 = τ11

on G 2T ,

(7.44a)

are integrated and it holds that (see [37, p. 233]) g11,2 = g22,1 = 14 gαβ gαβ,1 = 14 gαβ gαβ,2

on Γ .

This ensures that g11,2 and g22,1 are known on Γ and Γ 1 = Γ 2 = 0 on Γ ≡ G 1T ∩ G 2T . Finally, let g13,2 = ) b , g14,2 = ) b , A1,2 = ) A on Γ , where ) b ,) b ,) A ∈ C ∞ (Γ ). There is only one 23

24

21

23 24 2 1

way to choose g2β,1 on Γ such that Γ 3 = Γ 4 = 0 on Γ . In fact, through a direct computation, on G 1T it holds that βλ

g12 g2α,1 = gαβ Γ β − g12 g1α,2 − gαβ g,2 g12 g1λ − 12 [−2g12 g12,α + gμθ (2gαμ,θ − gμθ,α )],

(7.67)

and on G 2T it holds that βλ

g12 g1α,2 = gαβ Γ β − g12 g2α,1 − gαβ g,1 g12 g2λ − 12 [−2g12 g12,α + gμθ (2gαμ,θ − gμθ,α )].

(7.68)


Then all the components gab of the metric are determined on Γ since gαβ = Ωhαβ , g11 =

212


Hence, in view of (7.40a), on Γ it holds that g12 g2α,1 = gαβ Γ β − g12 g1α,2 − 12 [−2g12 g12,α + gμθ (2gαμ,θ − gμθ,α )].

(7.69)

We now proceed to arrange the condition Δ = 0 on Γ . As A2 = 0 on G 1T , A1 = 0 on G 2T , Aα are given on G 1T ∪ G 2T , there is only one way to choose A2,1 on Γ such that Δ = 0 on Γ . In fact, a straightforward calculation shows that on G 1T it holds 1 1 g12 A2,1 = Δ − g12 A1,2 + (2g12 Γ12 + gαβ Γαβ )A1 + g12 gαλ Aα (g1λ,2 + g2λ,1 ) β

λ − g12 gαλ Aα g12,λ − gαβ (Aβ,α − Γαβ Aλ ).

(7.70)

Hence, in view of (7.40a), on Γ it holds that g12 A2,1 = Δ − g12 A1,2 + g12 gαλ Aα (g1λ,2 + g2λ,1 ) λ − g12 gαλ Aα g12,λ − gαβ (Aβ,α − Γαβ Aλ ).

(7.71)

It follows from the above discussion that all necessary data are given on Γ and all necessary assumptions ((i)–(6i)) are fulfilled. We can now summarize the C ∞ resolution of the Goursat problem for the EYMH system in the following theorem, taking into consideration the work of Rendall [37]. Theorem 7.9 (C ∞ result). Let T ∈ (0, T0 ] be a real number and ω ∈ {1, 2}. Let h , h , h

ω 33 ω 34 ω 44

be C ∞ scalar functions on G ωT such that ⎛

h

h

h

h

⎝ ω 33 ω 43

⎞

ω 34 ⎠ ω 44

is a symmetric positive definite matrix with determinant 1 at each point of G ωT and ) , and A ) be C ∞ functions on G ωT such that ), A (h , h , h ) = (h , h , h ) on Γ . Let Φ 1 33 1 34 1 44

2 33 2 34 2 44

ω

ω3

ω4

),A ) ) = (Φ ),A ) ) on Γ . Let C ∞ functions Ω, ), A ) Ω )1 , Ω )2 , ) ), A b ,) b , and ) A be given on Γ . (Φ 1

13 14

2

23 24

23 24

21

Then there exists T1 ∈ (0, T], a unique C ∞ scalar function Ω on G 1T1 ∪ G 2T1 , a unique

C ∞ Lorentz metric gi j on L T1 , a unique C ∞ Yang–Mills potential Ak on L T1 and a unique C ∞ Higgs function Φ on L T1 such that: (1) gαβ = Ωhαβ on G 1T1 ∪ G 2T1 , where hαβ = h

1 αβ


(2) u= (gi j , Ak, Φ) satisfies the EYMH equations on L T1 ,

2 αβ

on G 2T ,


− [2g12 Γ2α Aβ gαλ + 2(g12 )2 g12,2 gαλ Aα − g12 gαλ Aα,2 ]g1λ


213

(3) the given coordinates on R4 are standard coordinates for gi j and the Lorentz gauge condition ∇k Ak = 0 is satisfied on L T1 , with A2 = 0 on G 1T1 and A1 = 0 on G 2T1 , (4) u= (gi j , Ak, Φ) induce the given data on G 1T1 ∪ G 2T1 , )1 ; Ω,2 = Ω )2 ; g13,2 = ) ) Ω,1 = Ω b ; g14,2 = ) b and A1,2 = (5) on Γ it holds that : Ω = Ω; 23

) A.

24

21

7.4.5 Construction of gi j and Ak in E 2 p−1 (G ωT ), arrangement of relations Γ i = 0 and Δ = 0 on G ωT for 4 ≤ p ∈ N, i, j, k = 1, . . . , 4

ceed to the resolution of the constraints problem in appropriate weighted Sobolev spaces. We just show how, through the conformity factor, gαβ are constructed in E 2 p−1 (G 1T1 ), for 4 ≤ p ∈ N. It will be clear that the remaining data are constructed in quite a similar manner. For m ∈ N, define Km (G 1T ) = {v ∈ E m (G 1T ) : v,2 ∈ E m (G 1T )}. Km (G 1T ) endowed with the natural norm v Km (G 1T ) = |v|G 1T ,m + |v,2 |G 1T ,m

(7.72)

is a Banach space such that Km (G 1T ) → E m (G 1T ). The following lemmas will be needed. The first lemma below provides appropriate energy inequalities for the following second-order linear Cauchy problem on G 1T 2v,22 = f in G 1T , v = v0 , v,2 = v1

on Γ ≡ G 1T ∩ G 2T .

(7.73)

Lemma 7.10. Let v be a solution of (7.73). Assume v, f ∈ C ∞ (G 1T ), v0 , v1 ∈ C ∞ (Γ ). Then v satisfies the following energy inequality for m ≥ 0 G1

v Km (G 1T ) ≤ K(T)[|v0 |Γm + |v1 |Γm + T| f|mT ], where K is a continuous non-decreasing function of T. Proof.

(7.74)

One uses similar arguments as those developed in [35]. The main tools consist

of the Sobolev inequalities of Section 4 and Gronwall lemma. Let v be a solution of (7.73) in C ∞ (G 1T ). Our purpose is to estimate v Km (G 1T ) in terms of the norms of the initial


Combining the C ∞ result obtained above with appropriate energy inequalities, we pro-

214


data v0 and v1 and the norm of the coefficient f. Assume v0 , v1 ∈ C ∞ (Γ ) and f ∈ C ∞ (G 1T ). Let α = (α1 , α2 , α3 ) ∈ N3 be a multi-index such that |α| ≤ m, where |α| = α1 + α2 + α3 . Let t ∈ (0, T]. Integrating the relation ((∂ α v)2 ),2 = 2(∂ α v)(∂ α v,2 ) on G 1t and estimating, we have Γt1

(∂ α v)2 dΓ ≤

Γ

(∂ α v0 )2 dΓ + 2

t Γσ1

0

(∂ α v)2 dΓ

1/2

1/2

Γσ1

(∂ α v,2 )2 dΓ

dσ ,

(7.75)

where dΓ = dx3 dx4 .

α

1 |α|≤m Γt

(∂ v) dΓ ≤

2

2

(∂ v,2 ) dΓ 2

1/2

0 |α|≤m

1/2 α

Γσ1

t

(∂ v0 ) dΓ + 2

|α|≤m Γ

×

α

α

Γσ1

(∂ v) dΓ 2

dσ .

(7.76)

Using Cauchy–Schwarz inequality, (7.76) yields 1 |α|≤m Γt

(∂ α v)2 dΓ ≤

|α|≤m Γ

⎛

×⎝

(∂ α v0 )2 dΓ + 2

⎛

t

⎝

⎞1/2

1 |α|≤m Γσ

0

(∂ α v)2 dΓ ⎠

⎞1/2

1 |α|≤m Γσ

(∂ α v,2 )2 dΓ ⎠

dσ .

(7.77)

By the definition of the norms (7.77) reads Γ 1 ,G 1 2

) ≤ (|v0 |Γm )2 + 2

(|v|mt

t

Γ 1 ,G 1

|v|mσ

Γ 1 ,G 1

|v,2 |mσ

dσ .

(7.78)

0

Similarly, by integrating the relation ((∂ α v,2 )2 ),2 = (∂ α v,2 )(∂ α f) on G 1t and estimating, we obtain Γ 1 ,G 1 2

(|v,2 |mt

) ≤ (|v1 |Γm )2 +

t

Γ 1 ,G 1

| f|mσ

Γ 1 ,G 1

|v,2 |mσ

dσ .

(7.79)

y(σ ) dσ ,

(7.80)

0

Adding (7.79) to (7.78) we have

t

y2 (t) ≤ c2 +

t

y2 (σ ) dσ +

0

Γ 1 ,G 1

| f|mσ

0

where Γ 1 ,G 1 2

y2 (t) = (|v|mt

Γ 1 ,G 1 2

) + (|v,2 |mt

) ,

c2 = (|v0 |Γm )2 + (|v1 |Γm )2 .

(7.81)


Equation (7.75) implies


Now, applying Gronwall Lemma to inequality (7.80), we gain t 1 1 t Γσ1 ,G 1 dt c + | f|m dσ . y(t) ≤ exp 2 0 2 0

215

(7.82)

In view of (7.81) and (7.82) we get Γ 1 ,G 1 |v|mt

We estimate

T 0

+

Γ 1 ,G 1 |v,2 |mt

Γ 1 ,G 1

| f|mσ

√ 1 1 T Γ 1 ,G 1 t |v0 |Γm + |v1 |Γm + ≤ 2 exp | f|mσ dσ . 2 2 0

(7.83)

dσ by using the following Sobolev inequality:

Γ 1 ,G 1

| f|mσ

G1

dσ ≤ c0 T| f|mT ,

(7.84)

0

where c0 denote any constant which does not depend on T and f (see (4.17)). We gain, in view of (7.83) and (7.84), Γ 1 ,G 1

|v|mt

Γ 1 ,G 1

+ |v,2 |mt

G1

≤ c0 exp( 12 t)[|v0 |Γm + |v1 |Γm + T| f|mT ].

(7.85)

Using (7.72) and (7.85) we get G1

v Km (G 1T ) ≤ K(T)[|v0 |Γm + |v1 |Γm + T| f|mT ],

(7.86)

where K(T) is a non-decreasing continuous function of T, of the following form: K(T) = c0 exp( 12 T).

(7.87)

Using Lemma 7.10 and known C ∞ results on the second-order linear ODEs, we derive the following lemma which provides an existence and uniqueness result for (7.73) in Km (G 1T ). Lemma 7.11. Let T ∈ (0, T0 ]. Let m be an integer. Assume f ∈ E m (G 1T ), v0 , v1 ∈ Hm (Γ ). Then (7.73) has, in G 1T , a unique solution v ∈ Km (G 1T ) satisfying the following energy inequality for t ∈ (0, T] G1

v Km (G 1t ) ≤ K(T)[|v0 |Γm + |v1 |Γm + t| f|mt ].

(7.88)

Proof.

The proof of this existence and uniqueness result is classical. It uses the energy

inequality (7.74) in C ∞ approximation of data.


T

216


We are now in a position to construct the conformity factor in E 2 p−1 (G 1T1 ), for 4 ≤ p ∈ N. Proposition 7.12. Let T ∈ (0, T0 ]. Let m ≥ 2 be an integer. Assume hαβ , A3 , A4 , Φ ∈ Km (G 1T ). Take v0 , v1 ∈ Hm (Γ ), where Γ ≡ G 1T ∩ G 2T . (i) Then there exists T1 ∈ (0, T] such that (7.46) has, in G 1T1 , a unique solution v ∈ E m (G 1T1 ) satisfying v = v0 and v,2 = v1 on Γ . (ii) There exists a positive real number r such that if |vΛ |Γm

Λ=0

+

4

|hαβ,2 |

G 1T ,m

α,β=3

+

4

|Aα,2 |G 1T ,m + |Φ,2 |G 1T ,m < r,

α=3

then the solution in (i) is global, i.e., T1 = T. Proof.

[Proof of item (i)] Let m ≥ 2 be an integer. Let w ∈ Km (G 1T ) be such that w = v0

and w,2 = v1 on Γ . It holds that w ∈ E m (G 1T ) and w,2 ∈ E m (G 1T ). Set Km (G 1T ) = {w ∈ Km (G 1T ) : w = v0 and w,2 = v1 on Γ }, and consider the mapping F : Km (G 1T ) −→ Km (G 1T ),

w −→ F (w) = v,

(7.89)

where v solves the following second-order linear Cauchy problem on G 1T 2v,22 = f(x, w, w,2 )

in G 1T ,

v = v0 and v,2 = v1

on Γ ,

(7.90)

where (see (7.46a)) αβ

f(x, w, w,2 ) = −(w,2 )2 − 2e−w hαβ Aα,2 .Aβ,2 + 12 hαβ,2 h,2 − 2(Φ,2 )2 .

(7.90a)

Using similar arguments as in the proof of Theorem 3.2 in Section 6, one shows that there exist two real numbers T1 (small enough) and R (large enough) such that F is a contraction from a ball B R,T1 of Km (G 1T1 ) into itself. By the fixed point theorem, it follows that there exists a unique v ∈ B R,T1 such that v = F (v). The uniqueness of v in Km (G 1T1 ) follows from the energy inequality (7.74) by classical arguments.


1


Proof.

217

[Proof of item (ii)] It is similar and even simpler than the proof of item (ii) of

Theorem 3.2, since the expression of the nonlinear term f(x, v, v,2 ) in (7.46) is known (see (7.46a)). Set A=

4

|Aα,2 |G 1T ,m ,

B=

α=3

V=

1

4

|hαβ |G 1T ,m ,

α,β=3

|vΛ |Γm ,

H=

4

|hαβ,2 |G 1T ,m ,

α,β=3

ϕ = |Φ,2 |G 1T ,m .

(7.91)

Λ=0

ity (4.4) of Theorem 4.2, Lemma 7.11 implies that the solution of (7.90) satisfies the following energy inequality v Km (G 1T ) ≤ k1 (T)[|v0 |Γm + |v1 |Γm + T| f(x, w, w,2 )|G 1T ,m ],

(7.92)

where k1 (T) is a non-decreasing continuous function of T. In view of (7.90a) and (7.91), and using the fact that E m (G 1T ) is a Banach algebra for m ≥ 2 (see (4.10)), we estimate | f(x, w, w,2 )|G 1T ,m to have | f(x, w, w,2 )|G 1T ,m ≤ w 2Km (G 1 ) + 2B A2 |e−w |G 1T ,m + 12 H 2 + 4ϕ 2 . T

(7.93)

The estimate of |e−w |G 1T ,m is done via Moser inequality (5.7) for M = G 1T , m = s, p = 2, to gain |e−w |G 1T ,m ≤ cT (m)[1 + |w|G 1T ,m ]m ,

(7.94)

where cT (m) is a non-decreasing continuous function of T, depending also on m. In view of the above inequalities (7.92)–(7.94), it follows that v Km (G 1T ) ≤ k1 (T)(|v0 |Γm + |v1 |Γm ) + Tk1 (T)[ w 2Km (G 1 ) + 2cT (m)B A2 (1 + w Km (G 1T ) )m + 12 H 2 + 4ϕ 2 ]. T

(7.95)

Now setting v = F (w), u= F (z), with w Km (G 1T ) ≤ K0 , z Km (G 1T ) ≤ K0 , we gain, in view of Lemma 7.11 and inequality (4.4) of Theorem 4.2, v − u Km (G 1T ) ≤ k2 (T)[T| f(x, w, w,2 ) − f(x, z, z,2 )|G 1T ,m ],

(7.96)


Let K0 > 0 be a real number and w ∈ Km (G 1T ) such that w Km (G 1T ) ≤ K0 . In view of inequal-

218


where k2 (T) is a non-decreasing continuous function of T. The estimate of | f(x, w, w,2 ) − f(x, z, z,2 )|G 1T ,m is done via Moser inequality (5.4) for M = G 1T to gain | f(x, w, w,2 ) − f(x, z, z,2 )|G 1T ,m ≤ ( w Km (G 1T ) + z Km (G 1T ) ) w − z Km (G 1T ) + dT (m)B A2 (1 + w Km (G 1T ) + z Km (G 1T ) )m w − z Km (G 1T ) ,

(7.97)

where dT (m) a non-decreasing continuous function of T, depending also on m. Inserting (7.97) into (7.96) yields

≤ k2 (T)T( w Km (G 1T ) + z Km (G 1T ) ) w − z Km (G 1T ) + dT (m)k2 (T)T B A2 [1 + w Km (G 1T ) + z Km (G 1T ) ]m w − z Km (G 1T ) .

(7.98)

Let us now assume that V + H + A + ϕ < r,

(7.99)

i.e., 1 Λ=0

|vΛ |Γm +

4

|hαβ,2 |G 1T ,m +

α,β=3

4

|Aα,2 |G 1T ,m + |Φ,2 |G 1T ,m < r,

α=3

where r > 0 is a real to be chosen conveniently. We are going to choose R in such a way that for r small enough, the mapping F is a contraction from a non-empty complete metric space into itself. Set for R > 0, B R,T = {w ∈ Km (G 1T ) : w Km (G 1T ) ≤ R}.

(7.100)

Considering the following linear Cauchy problem 2y,22 = 0 in G 1T , y = v0 and y,2 = v1

on Γ ,

(7.101)

it follows from Lemma 7.11 that (7.101) has, in G 1T , a unique solution y ∈ Km (G 1T ) satisfying the following energy inequality: y Km (G 1T ) ≤ k3 (T)(|v0 |Γm + |v1 |Γm ), where k3 (T) is a non-decreasing continuous function of T.

(7.102)


v − u Km (G 1T )


219

Choose R = 2rκ

with κ = max(k1 , k3 ).

(7.103)

It follows from (7.99) and (7.102) and (7.103) that B R,T defined by (7.100) is not empty, since y ∈ B R,T . Hence B R,T , endowed with the distance defined by the norm . Km (G 1T ) , is a non-empty complete metric space since it is a close subset of the Banach space Km (G 1T ). Equations (7.95), (7.98), (7.99), and (7.103) imply that w Km (G 1T ) ≤ R w Km (G 1T ) ≤ R, z Km (G 1T ) ≤ R

(7.104)

⇒ v − u Km (G 1T ) ≤ k2 (T)T[4rκ + dT (m)Br 2 [1 + 4rκ]m ] w − z Km (G 1T ) . We can then choose r small enough such that 9 R Tk1 (T) (2rκ)2 + 2cT (m)Br 2 (1 + 2rκ)m + r 2 ≤ rκ = , 2 2 k2 (T)T[4rκ + dT (m)Br 2 [1 + 4rκ]m ] < 12 .

(7.105)

It follows from (7.104) and (7.105) that F is a contraction from the non-empty metric space (B R,T , . Km (G 1T ) ) into itself. F admits therefore a fixed point which is the desired global solution of (7.46) such that v ∈ E m (G 1T1 ), v = v0 and v,2 = v1 on Γ .

7.4.6 Compatibility conditions on Γ for data of finite differentiability order Taking into consideration what has been done for the C ∞ case, given a positive real number 0 < T ≤ T0 , an integer p ≥ 4, appropriate scalar functions h

ω αβ

on G ωT , ω = 1, 2,

and some adequate conditions on Γ , initial data for the reduced EYMH system can be constructed, mutatis mutandis, on G ωT such that they belong to E 2 p−1 (G ωT1 ), for some T1 ∈ (0, T]. Begin by taking g12 = −1 on Γ . (This is a property of any metric in standard coordinates, see [37, p. 232].) Then choose (h ), a symmetric positive definite matrix ω αβ

function on G ωT with determinant 1 at each point of G ωT , such that h

ω αβ

Set gαβ = Ωhαβ , where hαβ = h

1 αβ

∈ K2 p−1 (G ωT ),

h

1 αβ


=h

2 αβ

2 αβ

on Γ .

) on Γ , where Ω ) is a on G 2T . Let Ω = Ω

) ∈ H2 p−1 (Γ ). Take also (7.40a). Then all the components given function on Γ such that Ω


⇒ v Km (G 1T ) ≤ rκ + Tk1 (T)[(2rκ)2 + 2cT (m)Br 2 (1 + 2rκ)m + 92 r 2 ],

220


gαβ of the metric are determined on Γ since gαβ = Ωhαβ , g11 = g1α = 0, g22 = g2α = 0 on Γ , Ω, and hαβ are known on Γ . Next choose Φ , A , and A which are given functions on G ωT such that ω

ω3

ω4

Φ , A , A ∈ K2 p−1 (G ωT ), ω

Φ = Φ on Γ ,

ω3 ω4

1

A = A on Γ , 13

2

A = A on Γ . 13

23

23

)1 and Ω,2 = Ω )2 on Γ , where Ω )1 and Ω )2 are two given functions on Γ such that Let Ω,1 = Ω )1 , Ω )2 ∈ H2 p−1 (Γ ). Ω

and it holds that (see [37, p. 233]) g11,2 = g22,1 = 14 gαβ gαβ,1 = 14 gαβ gαβ,2 on Γ . This ensures that g11,2 and g22,1 are known on Γ and Γ 1 = Γ 2 = 0 on Γ ≡ G 1T ∩ G 2T . Finally, let g13,2 = ) b , g14,2 = ) b , and A1,2 = ) A on Γ , where ) b ,) b ,) A ∈ H2 p−1 (Γ ). Then, there 23

21

24

23 24 2 1

is only one way to choose g2β,1 on Γ such that Γ 3 = Γ 4 = 0 on Γ , thanks to the equalities (7.67) and (7.68) that hold, respectively, on G 1T and G 2T through direct calculations. Hence, equality (7.69) holds on Γ . The condition Δ = 0 on Γ is arranged as in the C ∞ case. As A2 = 0 on G 1T , A1 = 0 on G 2T , Aα are given on G 1T ∪ G 2T , there is only one way to choose A2,1 on Γ such that Δ = 0 on Γ . This is done by using (7.70) and (7.71). It follows from the above discussion that all necessary data are given on Γ and all necessary assumptions fulfilled. What has been proved for the resolution of the Goursat problem associated to the EYMH system in weighted Sobolev spaces can be summed up in the following theorem, where the constraints problem is solved by the method just described in Sections 7.4.5 and 7.4.6, and the evolution problem is solved by applying Theorem 7.3 with ⎛

0 −1

⎜ ⎜−1 (γi j (x , x )) = ⎜ ⎜ ⎝ 0

0

0

0

0

Ωh33 (0, x3 , x4 )

0

0

Ωh34 (0, x3 , x4 )

3

4

0

⎞

⎟ ⎟ ⎟. 3 4 ⎟ Ωh34 (0, x , x )⎠ Ωh44 (0, x3 , x4 ) 0

Theorem 7.13 (Finite differentiability result). Let T ∈ (0, T0 ] be a real number, p ≥ 4 an integer, and ω ∈ {1, 2}. Let h , h , and h ω 33

ω 34

ω 44

be scalar functions defined on G ωT such


Then Equations (7.43) and (7.44) as well as Equations (7.43a) and (7.44a) are integrated


that h , h , h ∈ ω 33

ω 34

ω 44

K2 p−1 (G ωT ),

h

h

h

h

221

ω 33 ω 34

is a symmetric positive definite matrix function

ω 43 ω 44

with determinant 1 at each point of G ωT and (h , h , h ) = (h , h , h ) on Γ , with 1 33 1 34 1 44

2 33 2 34 2 44

) , and A ) be defined on G ωT such that Φ ), ), A ), A [h ]Γ ∈ C 2 p−1 (Γ ). Let functions Φ ω

ω αβ

ω3

ω4

ω

ω3

) ∈ K2 p−1 (G ωT ) and (Φ ),A ) ) = (Φ ),A ) ) on Γ . Let Ω, ), A ), A ) Ω )1 , Ω )2 , ) A b ,) b ,) A ∈ H2 p−1 (Γ ), with ω4

1

) ∈ C 2 p−1 (Γ ). Ω

13 14

2

23 24

23 24 2 1

(i) Then there exists T1 ∈ (0, T], a unique positive scalar function Ω defined on G 1T1 ∪ G 2T1 such that its restriction to G ωT1 belongs to K2 p−1 (G ωT1 ) and there E p(L T1 ), a unique Yang–Mills potential Ak ∈ E p(L T1 ), and a unique Higgs function Φ ∈ E p(L T1 ) such that: (1) gαβ = Ωhαβ on G 1T1 ∪ G 2T1 , where hαβ = h

1 αβ


2 αβ

(2) u= (gi j , Ak, Φ) satisfies the EYMH equations on L T1 ,

on G 2T ,

(3) the given coordinates on R4 are standard coordinates for gi j and the Lorentz gauge condition ∇k Ak = 0 is satisfied on L T1 , with A2 = 0 on G 1T1 and A1 = 0 on G 2T1 , (4) u= (gi j , Ak, Φ) induce the given data on G 1T1 ∪ G 2T1 , )1 ; Ω,2 = Ω )2 ; g13,2 = ) ) Ω,1 = Ω b ; g14,2 = ) b , (5) on Γ it holds that : Ω = Ω; 23

A. and A1,2 = )

24

21

(ii) There exists a positive real number d such that: if ) Γ2 p−1 + |Ω )1 |Γ2 p−1 + |Ω )2 |Γ2 p−1 + |) |Ω| b |Γ2 p−1 + |) b |Γ2 p−1 + |) A |Γ2 p−1 + 23

+

4 α,β=3

|h

2 αβ,1

|G 2T ,2 p−1 +

4 α=3

|A |G 1T ,2 p−1 + 1 α,2

21

24

4 α=3

4 α,β=3

|h

1 αβ,2

|G 1T ,2 p−1

|A |G 2T ,2 p−1 2 α,1

+ |Φ |G 1T ,2 p−1 + |Φ |G 2T ,2 p−1 < d, 1 ,2

2 ,1


7.5 Findings and other possible applications of the main Theorem 3.2

In view of Theorems 7.3 and 7.13 it follows that the Goursat problem for the EYMH system has been successfully solved in some weighted Sobolev spaces defined in Section 3.1, thanks to the main Theorem 3.2 establishing existence and uniqueness results for the quasilinear Goursat problem. It is worth recalling that the constraints problem for


exists the following functions defined on L T1 : a unique Lorentz metric gi j ∈

222


the EYMH system has been solved through a judicious adaptation of the method set up by Rendall [37] for the vacuum Einstein equations. As for the evolution problem associated to the EYMH system, it is solved via a straightforward application of Theorem 3.2. In the present Section 7.5 we give, apart from the EYMH system, other situations in which Theorem 3.2 could also be used. The main existence and uniqueness results provided by Theorem 3.2 for the quasilinear Goursat problem can naturally be applied to: • various physical or geometric wave maps, are the second-order hyperbolic system such as the Yang–Mills–Higgs–Dirac system (see [9]). Suitably extended to general hyperbolic Leray systems or other integrodifferential equations, Theorem 3.2 could also be applied to: • Einstein equations with matter source such as relativistic fluids models or kinetic models like Einstein–Vlasov equations or Einstein–Boltzmann equations, • the regular conformal field equations of Choquet-Bruhat and Novello [10], with initial data prescribed on two intersecting characteristic hypersurfaces one of which is a portion of the past null infinity.

Appendix. Proof of Theorem 5.9 By definition, we have 2 ( f(x, u(x)) G ωt ,m )2 ]1/2 . f(x, u(x)) L t ,m = [(| f(x, u(x))| L t ,m ) + 2

ω=1

Due to (5.33) of Theorem 5.6, it holds that | f(x, u(x))| L t ,m ≤ ct (m, s) f C bm (L t ×W) [1 + |u| L t ,s ]m .

(A1)

From f C bm (L t ×W) ≤ f C b2m−1 (L t ×W) since m ≥ 1,

|u| L t ,s ≤ u L t ,s ,

(A2)

we get | f(x, u(x))| L t ,m ≤ ct (m, s) f C b2m−1 (L t ×W) [1 + u L t ,s ]2m−1 .

(A3)


• any gauge field equations for which the fundamental evolution equations


223

To prove (5.50) of Theorem 5.9, it is sufficient to show that for all ω = 1, 2, it holds that f(x, u(x)) G ωt ,m ≤ ct (m, s) f C b2m−1 (L t ×W) [1 + u G ωt ,s ]2m−1 .

(A4)

We know by definition that

f(x, u(x)) G ωt ,m =

m−1 ⎡

2(m−k)−1

f(x, u(x))} H 2(m−k)−1 (Γσω ,G ωt ) )

2

,

⎤1/2

DG ωt [Dωk { f(x, u(x))}] 2L 2 (Γσω ) ⎦ p

.

(A5)

p=0

For all k ∈ {0, . . . , m − 1}, Dωk { f(x, u(x))} is a linear combination of terms gkω ,γ ,λ of the form gkω ,γ ,λ = Dωkω Duγ f(x, u)|u=u(x) .

!

Dωλ(q,ρ) uq ,

(A6)

q,ρ

where γ = (γ1 , . . . , γl ) with |γ | + kω ≤ k; q takes possible non-negative integer values for which γq > 0; ρ takes possible values 1, . . . , γq ; λ = (λ1 , . . . , λn+1 ) where λa(q, ρ) ∈ N are such that kω +

|λ(q, ρ)| = k.

(A7)

q,ρ

To prove (5.50), it is sufficient to show that for all k ∈ {0, . . . , m − 1}, for all kω , γ , and λ satisfying (A6) and (A7), it holds that ess sup gkω ,γ ,λ H 2(m−k)−1 (Γσω ,G ωt ) ≤ ct (m, s) f C b2m−1 (L t ×W) [1 + u G ωt ,s ]2m−1 . 0≤σ ≤t

(A8)

We set j = |γ | = number of possible values taken by (q, ρ). First case: j = 0. In this case we have gkω ,γ ,λ = gk,0,0 = Dωk f(x, u) and then ess sup gk,0,0 H 2(m−k)−1 (Γσω ,G ωt ) = ess sup Dωk f(x, u) H 2(m−k)−1 (Γσω ,G ωt ) . 0≤σ ≤t

0≤σ ≤t

(A9)

Second case: j ≥ 1. Denote Λ the set of all (q, ρ) and λ0 = max λ(q, ρ). (q,ρ)∈Λ

(A10)


Dωk { f(x, u(x))} H 2(m−k)−1 (Γσω ,G ωt ) = ⎣

12

(ess sup Dωk { 0≤σ ≤t k=0

224


First sub-case: λ0 < s/2 For all (q, ρ) ∈ Λ, we have λ(q, ρ) ≤ λ0 < s/2. As n< 2s, it is clear that n− 1 < 2(s − λ(q, ρ)) − 1. 2

(A11)

Thus, for all (q, ρ) ∈ Λ, for almost all σ ∈ [0, t], it holds that D λ(q,ρ) uq ∈ H 2(s−λ(q,ρ))−1 (Γσω , G ωt ).

(A12)

Applying inequality (4.3) of Lemma 4.1 for Kt = G ωt and (N, p) = (n, 2), for almost all σ ∈ [0, t], we get

D λ(q,ρ) uq ∈ H s (Γσω , G ωt )

(q,ρ)∈Λ

" " " " " ! " λ(q,ρ) " D uq " " " "(q,ρ)∈Λ "

and

≤c

!

D λ(q,ρ) uq H 2(s−λ(q,ρ))−1 (Γσω ,G ωt ) .

(A13)

(q,ρ)∈Λ

H s (Γσω ,G ωt )

Since 2(m − k) − 1 < 2(m − k) − 1 + s + (n − 1)/2, by virtue of (A6) and inequality (4.3) of Lemma 4.1 for Kt = G ωt and (N, p) = (n, 2), we gain gkω ,γ ,λ H 2(m−k)−1 (Γσω ,G ωt ) ≤ c Dωkω Duγ f(x, u) H 2(m−k)−1 (Γσω ,G ωt )

!


(A14)

(q,ρ)∈Λ

Thus ess sup gkω ,γ ,λ H 2(m−k)−1 (Γσω ,G ωt ) ≤ c u G ωt ,s ess sup Dωkω Duγ f(x, u) H 2(m−k)−1 (Γσω ,G ωt ) . j

0≤σ ≤t

0≤σ ≤t

(A15)

Second sub-case: λ0 ≥ s/2 We set Λ1 = {(q, ρ) ∈ Λ : λ(q, ρ) ≥ s/2}. From (A7) and k ≤ m − 1 < s, it is clear that Λ has one unique element (q1 , ρ1 ) and all (q, ρ) ∈ Λ\Λ1 satisfies (A12). Therefore, due to 1

inequality (4.3) of Lemma 4.1 for Kt = G ωt and (N, p) = (n, 2), for almost all σ ∈ [0, t], it holds that

!

D λ(q,ρ) uq ∈ H s (Γσω , G ωt ),

(A16)

(q,ρ)∈Λ\Λ1

and

" " " ! " " " λ(q,ρ) " " D u q" " "(q,ρ)∈Λ\Λ1 "

≤c H s (Γσω ,G ωt )

!


(A17)

(q,ρ)∈Λ\Λ1

Let us for a moment assume that for almost all σ ∈ [0, t], it holds that Dωkω Duγ f(x, u) ∈ H s (Γσω , G ωt ).

(A18)


!


225

Applying once more inequality (4.3) of Lemma 4.1 for Kt = G ωt and (N, p) = (n, 2), we gain Dωkω Duγ f(x, u) ×

!

D λ(q,ρ) uq ∈ H s (Γσω , G ωt ),

(A19)

(q,ρ)∈Λ\Λ1

and

" " " " ! " " k γ λ(q,ρ) " D ω D f(x, u) × D uq " " " ω u " s ω ω " (q,ρ)∈Λ\Λ1 H (Γσ ,G t ) ! kω γ ≤ c Dω Du f(x, u) H s (Γσω ,G ωt ) × D λ(q,ρ) uq H 2(s−λ(q,ρ))−1 (Γσω ,G ωt ) .

(A20)

It follows, from (A6) and (A20) by applying inequality (4.3) of Lemma 4.1 for Kt = G ωt and (N, p) = (n, 2), that gkω ,γ ,λ H 2(m−k)−1 (Γσω ,G ωt ) ≤ c Dωλ(q1 ,ρ1 ) uq1 H 2(m−k)−1 (Γσω ,G ωt ) × Dωkω Duγ f(x, u) H s (Γσω ,G ωt ) ×

!


(A21)

(q,ρ)∈Λ\Λ1

As m ≤ s and λ(q1 , ρ1 ) ≤ k ≤ m − 1 (due to (A7)), we finally gain ess sup gkω ,γ ,λ H 2(m−k)−1 (Γσω ,G ωt ) ≤ c u G ωt ,s ess sup Dωkω Duγ f(x, u) H s (Γσω ,G ωt ) . j

0≤σ ≤t

0≤σ ≤t

(A22)

We must note that in this second sub-case, it holds that kω + |γ | + s ≤ 2m − 1.

(A23)

In fact, due to (A7) we have s ≤ λ(q1 , ρ1 ) ≤ k − kω ≤ m − 1, 2

kω + |γ | − 1 + λ(q1 , ρ1 ) ≤ k ≤ m − 1.

Therefore kω + |γ | + s ≤ k − λ(q1 , ρ1 ) + 1 + s ≤ m −

s s + s ≤ m + ≤ 2m − 1, 2 2

and kω + |γ | + 2(m − k) − 1 ≤ k − λ(q1 , ρ1 ) + 1 + 2m − 2k − 1 ≤ 2m − 1 − k + 1 ≤ 2m − 1. So that we gain kω + |γ | + q ≤ 2m − 1 with q = max{s, 2(m − k) − 1}.


(q,ρ)∈Λ\Λ1

226


From (A15), (A22), and (A23), we have to estimate in terms of f C b2m−1 (L t ×W) and u G ωt ,s , the following quantities ess sup Dωkω Duγ f(x, u) H 2(m−k)−1 (Γσω ,G ωt ) 0≤σ ≤t

for kω + |γ | ≤ k ≤ m − 1,

and ess sup Dωkω Duγ f(x, u) H s (Γσω ,G ωt )

for kω + |γ | + s ≤ 2m − 1.

0≤σ ≤t

γ

Let us now estimate ess sup0≤σ ≤t Dωkω Du f(x, u) H 2(m−k)−1 (Γσω ,G ωt ) for kω + |γ | ≤ k ≤ m − 1. Recall that 2(m−k)−1

f(x, u) H 2(m−k)−1 (Γσ1 ,G 1t ) = ⎣

⎤1/2 p DG 1 (D1k1 Duγ t

f(x, u)) 2L 2 (Γσ1 ) ⎦

,

p=0

with DG 1t = (D2 , . . . , Dn+1 ). Let β = (β2 , . . . , βn+1 ) ∈ Nn such that |β| ≤ 2(m − k) − 1. Setting γ

∂ β = (∂ |β| )/((∂ x2 )β2 . . . (∂ xn+1 )βn+1 ), it follows that ∂ β (D1k1 Du f(x, u)) is a linear combination with constant coefficients of terms of the form α γ +θ f(x, u)|u=u(x) hk1 ,α,θ,ν = D1k1 D(x i ) Du

!

∂ ν(w,z) uw ,

(A24)

where θ = (θ2 , . . . , θn+1 ), α = (α2 , . . . , αn+1 ), (xi ) = (x2 , . . . , xn+1 ), w takes possible nonnegative integer values for which θw is positive. z takes possible values 1, . . . , θw , ν(w, z) = (ν2 (w, z), . . . , νn+1 (w, z)) with νη (w, z) ∈ N such that α+

ν(w, z) = β.

(A25)

w,z

We set d = |θ | is the number of possible values taken by (w, z),

a = |α|.

(A26)

It is clear that d ≤ |β| ≤ 2(m − k) − 1. γ

To evaluate ess sup0≤σ ≤t D1k1 Du f(x, u) H 2(m−k)−1 (Γσ1 ,G 1t ) for k1 + |γ | ≤ k ≤ m − 1, it is sufficient to evaluate ess sup0≤σ ≤t hk1 ,α,θ,ν L 2 (Γσ1 ) for k1 + |γ | ≤ k ≤ m − 1 and for all β = (β2 , . . . , βn+1 ) ∈ Nn with |β| ≤ 2(m − k) − 1. First case: d = 0 In this case we have θ = 0, ν = 0. It holds that α γ hk1 ,α,0,0 = D1k1 D(x i ) Du f(x, u)|u=u(x) ,

and k1 + |α| + |γ | ≤ k1 + |β| + |γ | ≤ k + 2(m − k) − 1 = 2m − k − 1.


⎡ D1k1 Duγ


227

As 0 ≤ k ≤ m − 1, we get m ≤ 2m − k − 1 ≤ 2m − 1. Hence k1 + |α| + |γ | ≤ 2m − 1. First sub-case: k1 + |α| + |γ | = 2m − 1. Here k = k1 = 0. Therefore ess sup hk1 ,α,θ,ν L 2 (Γσ1 ) = ess sup h0,α,0,0 L 2 (Γσ1 ) ≤ c f C b2m−1 (L t ×W) . 0≤σ ≤t

0≤σ ≤t

(A27)

Second sub-case: k1 + |α| + |γ | < 2m − 1. γ

α In this case, as D1k1 D(x i ) Du f(x, 0) = 0, we have by use of the mean value formula, l

uw

w=1

1

0

α γ D1k1 D(x i ) Du f(x, θ u(x)) dθ .

(A28)

From this we gain ess sup hk1 ,α,0,0 L 2 (Γσ1 ) ≤ c f C b2m−1 (L t ×W) u G 1t ,s . 0≤σ ≤t

(A29)

Second case: d ≥ 1 In this case, the set Q of all (w, z) is not empty. We set Q1 = {(w, z) ∈ Q : |ν(w, z)| ≥ s}. It holds that either Q1 = ∅ or Q1 = {(w1 , z1 )}. In fact let us assume that Q1 has at least two elements. From (A25), we get

2(m − k) − 1 ≥ |β| ≥

|ν(w, z)| ≥

(w,z)∈Q

|ν(w, z)| ≥ 2s.

(w,z)∈Q1

But 2(m − k) − 1 ≤ 2m − 1 ≤ 2s − 1. So we deduce that Q1 at most one element. By definition of Q1 , it holds that ∀(w, z) ∈ Q\Q1 , |ν(w, z)| < s. We thus gain ∀(w, z) ∈ Q\Q1 ,

n− 1 n < < s < 2s − |ν(w, z)| − 1. 2 2

(A30)

So for almost all σ ∈ [0, t], by applying inequality (4.2) of Lemma 4.1 for Kt = Γσ1 and ( j, N, p) = (0, n, 2), we get ∀(w, z) ∈ Q\Q1 ,

∂ ν(w,z) uw ∈ H s (Γσ1 , G 1t ) → C 0 (Γσ1 ).

(A31)

Let (w1 , z1 ) be the unique element of Q1 if Q1 = ∅ or (w1 , z1 ) ∈ Q if Q1 is empty. We set Q2 = Q\{(w1 , z1 )}; it is clear that Q2 ⊂ Q\Q1 . From (A24) and the Applying inequality (4.1) of Lemma 4.1 for Kt = Γσ1 and ( j, N, p) = (0, n, 2), we have hk1 ,α,θ,ν L 2 (Γσ1 ) ≤ c f C b2m−1 (L t ×W)

!

uw H 2s−1 (Γσ1 ,G 1t ) uw1 H |ν(w1 ,z1 )| (Γσ1 ,G 1t ) .

(w,z)∈Q2

Due to |ν(w1 , z1 )| ≤ 2(m − k) − 1 ≤ 2s − 1,

(A32)


hk1 ,α,0,0 =

228


we get ess sup hk1 ,α,θ,ν L 2 (Γσ1 ) ≤ c f C b2m−1 (L t ×W) u dG 1 ,s . t

0≤σ ≤t

(A33)

Equations (A27), (A29), and (A33) imply that ess sup D1k1 Duγ f(x, u) H 2(m−k)−1 (Γσ1 ,G 1t ) 0≤σ ≤t

≤ c f C b2m−1 (L t ×W) [1 + u G 1t ,s ]d for k1 + |γ | ≤ k ≤ m − 1.

(A34)

By the same tools, we gain the following inequalities:

0≤σ ≤t

for k2 + |γ | ≤ k ≤ m − 1; ess sup D1k1 Duγ 0≤σ ≤t

(A35)

f(x, u) H s (Γσ1 ,G 1t ) ≤ c f C b2m−1 (L t ×W) [1 + u G 1t ,s ]2m−|γ |−1 u G 1t ,s for k1 + |γ | + s ≤ 2m − 1;

(A36)

ess sup D2k2 Duγ f(x, u) H s (Γσ2 ,G 2t ) ≤ c f C b2m−1 (L t ×W) [1 + u G 2t ,s ]2m−|γ |−1 u G 2t ,s 0≤σ ≤t

for k2 + |γ | + s ≤ 2m − 1.

(A37)

Equation (A8) follows immediately from (A9), (A15), (A22), (A34), (A35), (A36), and (A37). Equations (A4) is a direct consequence of (A8). The proof of (5.51a) is similar, mutatis mutandis, to that of (A4).

Acknowledgement The second author (C. Tadmon) was partially supported by the SARIMA project of the French Ministry of Foreign Affairs. He wishes to acknowledge the hospitality of Professor Anne Nouri ´ where a portion of this work was and the LATP (Laboratoire d’Analyse, Topologie, Probabilites) done at “Universite´ de Provence, Marseille, France”.

References [1] Adams, R. A. Sobolev Spaces. New York: Academic Press, 1975. [2] Balakin, A., H. Dehnen, and A. E. Zayats. “Effective metrics in the non-minimal Einstein– Yang–Mills–Higgs theory.” Annals of Physics 323, no. 9 (2008): 2183–207. [3] Cabet, A. “Local existence of a solution of a semilinear wave equation with Gradient in a neighborhood of Initial Characteristic Hypersurfaces of a Lorentzian Manifold.” Communications in Partial Differential Equations 33, no. 12 (2008): 2105–56.


ess sup D1k2 Duγ f(x, u) H 2(m−k)−1 (Γσ2 ,G 2t ) ≤ c f C b2m−1 (L t ×W) [1 + u G 2t ,s ]d


229

[4] Caciotta, G. and F. Nicolo. “Global characteristic problem for Einstein vacuum equations with small initial data: (I) The initial constraints.” Journal of Hyperbolic Differential Equations 2, no. 1 (2005): 201–77. (II The existence proof, Arxiv: gr-qc/0608038v1, February 7, 2008.) ` ´ ´ [5] Cagnac, F. “Probleme de Cauchy sur un cono¨ıde caracteristique pour des equations quasi´ lineaires.” Annali di Matematica Pura ed Applicata, IV, CXXIX (1980): 13–41. [6] Choquet-Bruhat, Y., D. Christodoulou, and M. Francaviglia. “Cauchy data on a manifold.” ´ vol. XXIX, no. 3 (1978): 241–55. Annales de l’ Institut Henri Poincare, [7] Choquet-Bruhat, Y. “Yang–Mills fields on Lorentzian manifolds.” Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Series, Amsterdam: North-

´ e´ Mathematique ´ de la Societ de France, N. S 46 (1991): 73–97; Analyse globale et physique ´ mathematique, Lyon, 1989. [9] Choquet-Bruhat, Y. and D. Christodoulou. “Existence of global solutions of the Yang–Mills, Higgs and spinor field equations in 3 + 1 dimensions.” Annales Scientifiques de l’Ecole ´ Normale Superieure 14, no. 4 (1981): 481–506. ` ´ ´ [10] Choquet-Bruhat, Y. and M. Novello. “Systeme conforme regulier pour les equations ´ ´ ´ d’Einstein.” Comptes Rendus Mathematique Academie des Sciences Paris, t. 305, Serie II, no. 3 (1987): 155–60. [11] Christodoulou, D. “The formation of black holes in general relativity.” (2008): preprint arXiv: gr-qc/0805.3880v1, 594pp. [12] Christodoulou, D. “Characteristic initial value problem for quasilinear hyperbolic systems ´ ´ ´ ees ´ partielles hyperboliques of second order.” Seminaire Vaillant sur les equations aux deriv et holomorphes, Paris VI, 1980–1981. ¨ ` ´ [13] Christodoulou, D. and H. Muller zum Hagen. “Probleme de valeur initiale caracteristique ´ ` ´ pour des systemes quasi lineaires du second ordre.” Comptes Rendus Mathematique ´ ´ Academie des Sciences Paris, t. 293, Serie I, no. 1 (1981): 39–42. [14] Chrusciel, P. T. and J. Shatah. “Global existence of solutions of the Yang–Mills equations on globally hyperbolic four dimensional Lorentzian manifolds.” Asian Journal of Mathematics 1, no. 3 (1997): 530–48. [15] Damour, T. and B. G. Schmidt. “Reliability of perturbation theory in General Relativity.” Journal of Mathematical Physics 31, no. 10 (1990): 2441–53. [16] Dautcourt, G. “Zum charakteristischen Anfangswertproblem der Einsteinschen Feldgleichungen.” Annalen der Physik 467, no. 5 (1963): 302–24. ` ´ [17] Dionne, P. A. “Problemes de Cauchy hyperboliques bien poses.” Journal d’Analyse ´ ´ Mathematique de Jerusalem 10 (1962): 1–90. ´ es ´ de R N .” [18] Dossa, M. “Remarques sur les constantes de Sobolev de quelques sous-variet ´ Nouvelles series ´ ´ Annales de la Faculte´ des Sciences Yaounde, Mathematiques-Physique, ´ Serie I, t. 1 (1988): 51–69.


Holland, 289–313, 1991. ´ [8] Choquet-Bruhat, Y. “Yang–Mills-Higgs fields in three space–time dimensions.” Memoires

230


` ´ ´ [19] Dossa, M. “Probleme de Cauchy sur un cono¨ıde caracteristique pour des equations ` ´ ` quasi-lineaires du second ordre.” These de Doctorat d’Etat, 2eme partie, Universite´ de

Yaounde´ (Cameroun), 1992. ` ´ [20] Dossa, M. “Espaces de Sobolev non isotropes a` poids et problemes de Cauchy quasi-lineaires ´ Physique Theorie ´ ´ sur un cono¨ıde caracteristique.” Annales de l’Institut Henri Poincare, 66, no. 1 (1997): 37–107. [21] Duff, G. F. “Mixed problems for linear systems of first order.” Canadian Journal of Mathematics 10 (1958): 127–60. [22] Earley, D. M. and V. Moncrief. “The global existence of Yang–Mills-Higgs fields in 4-dimensional Minkwoski space I. Local Existence and Smoothness Properties, II. Com171–91. [23] Friedrich, H. “On the global existence and the asymptotic behavior of solutions to the Einstein–Maxwell–Yang–Mills equations.” Journal of Differential Geometry 34, no. 2 (1991): 275–345. [24] Friedrich, H. “On the hyperbolicity of Einstein’s and Gauge Field Equations.” Communications in Mathematical Physics 100, no. 4 (1985): 525–43. [25] Hawking, S. W. and G. F. R. Ellis. The Large Scale Structure of Space–time. Cambridge: Cambridge University Press, 1973. ` ` ´ [26] Houpa, D. E. and M. Dossa. “Problemes de Goursat pour les systemes semi-lineaires hyper´ ´ ´ I 341, no. 1 boliques.” Comptes Rendus Mathematique Academie des Sciences Paris, Series (2005): 15–20. [27] Kannar, J. “On the existence of C ∞ solution to the asymptotic characteristic initial value problem in General Relativity.” Proceedings of the Royal Society of London A 452, no. 1947 (1996): 945–52. [28] Klainerman, S. and F. Nicolo. “The evolution problem in General Relativity.” Progress in ¨ Mathematical Physics. Basel: Birkhauser, 2004. [29] Lindblad, H. and I. Rodnianski. “Global existence for the Einstein vacuum equations in wave coordinates.” Communications in Mathematical Physics 256, no.1 (2005): 43–110. ´ [30] Loizelet, J. “Solutions globales des equations d’Einstein–Maxwell avec Jauge harmonique et ´ ´ jauge de Lorenz.” Comptes Rendus Mathematique Academie des Sciences Paris, Ser. I 340, no. 7 (2005): 479–82. [31] Malec, E. and P. Koc. “Trapped surfaces in Monopole-like Cauchy data of Einstein–Yang– Mills–Higgs equations.” ICTP, IC/89/225, 5662/89, September 1989. [32] Moser, J. “A rapidly convergent iteration method and non-linear differential equations.” Annali della Scuola Normale Superiore di Pisa 20, no. 2 (1966): 265–315. [33] Moser, J. “On pointwise estimate for parabolic differential equations.” Communications on Pure and Applied Mathematics 24, no. 5 (1971): 727–40. ¨ ¨ [34] Muller zum Hagen, H. and F. H. Jurgen Seifert. “On characteristic initial-value and mixed problems.” General Relativity and Gravitation 8, no. 4 (1977): 259–301.


pletion of proof.” Communications in Mathematical Physics 83, no. 2 (82, 193–212), (1982):


231

¨ [35] Muller zum Hagen, H. “Characteristic initial value problem for hyperbolic systems of second ´ Physique Theorie ´ order differential equations.” Annales de l’Institut Henri Poincare, 53, no. 2 (1990): 159–216. [36] Penrose, R. “Null hypersurface initial data for classical fields of arbitrary spin and for General Relativity.” Aerospace Research Laboratory 63-56, 1963; reprinted in General Relativity and Gravitation 12, no.3 (1980): 225–64. [37] Rendall, A. D. “Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations.” Proceedings of the Royal Society of London A 427, 1872 (1990): 221–39. [38] Rendall, A. D. “The characteristic initial value problem for the Einstein equations, non linear ematical Series 253, Harlow: Longman Sci. Tech., 154–63, 1992. [39] Sachs, R. K. “On the characteristic initial value problem in Gravitational Theory.” Journal of Mathematical Physics 3, no. 5 (1962): 908–14. [40] Stewart, J. M. “Classical General Relativity.” edited by W. B. Bonnor, J. N. Islam, and M. A. H. MacCallum. Cambridge: Cambridge University Press, 1984. [41] Stewart, J. M. and H. Friedrich. “Numerical relativity I. The initial value problem.” Proceedings of the Royal Society of London A 384, 1787 (1982): 427–54. [42] Volkov, M. S. and D. V. Gal’tsov. “Gravitating non-abelian solitons and black holes with Yang– Mills fields.” Physics Report 319, nos. 1–2, (1999): 1–83.


hyperbolic equations and field theory (Lake Como, 1991).” Pitman, Research Notes in Math-