functional in Calculus of Variations whose computation is not reducible to one dimensional ...... [1] R.Adams: Sobolev spaces, Academic Press, New York, 1975.
Second Order Variational Problems with Free Discontinuity and Free Gradient Discontinuity Michele Carriero1 , Antonio Leaci2 , Franco Tomarelli3
Contents 1. Special Bounded Hessian functions (SBH) (8). 2. The class GSBV 2 (20). 3. Slicing properties and second derivatives in GSBV 2 (22). 4. Compactness in GSBV 2 and in SBH (24). 5. Lower semicontinuity (31). 6. Existence of minimizers (44). 7. Examples (45). 1 Dipartimento
di Matematica ”Ennio De Giorgi”– Via Provinciale per Arnesano – 73100 – Lecce – Italia 2 Dipartimento di Matematica ”Ennio De Giorgi”– Via Provinciale per Arnesano – 73100 – Lecce – Italia 3 Dipartimento di Matematica ”Francesco Brioschi”– Politecnico – Piazza Leonardo da Vinci 32 – 20133 – Milano – Italia
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Introduction In recent years many functionals depending on higher derivatives have been studied [12],[13],[14],[15], [16],[18],[5],[32],[23]: many related examples of variational principles for image segmentation have been focused [9],[38],[36], [34],[37],[15], [16], [17],[20],[21] and several models of elastic plastic plates [12], [13],[46] and elastic energies with damage at small scale were introduced and studied ([39], [40],[41], [24],[31],[33] and the contributed paper [22] in this book). In this contribution we unify general properties of functionals depending on first and second derivatives together with free discontinuities and free gradient discontinuities. Our framework has an interest in itself, nevertheless we emphasize that the scheme includes many relevant applications to image processing and continuum mechanics. More precisely we focus the functional F(v, α, β, γ, σ, A) defined below and shortly denoted by F(v) or F(v, A) when the parameters α, β, γ, σ and/or the localization in A are prescribed. For any scalar function v defined in Ω and Borel set A ⊂ Ω we study: (0.1) Z
|∇2 v|2 + σ|∇v|2 + G(x, v) dx + A Z + αHn−1 (Sv ∩ A) + βHn−1 ((S∇v \ Sv ) ∩ A) + γ |[∇v]| dH n−1 .
F(v, α, β, γ, σ, A) =
S∇v ∩A
The following structural hypotheses are assumed in the definition (0.1) of the functional F and in the whole paper. (0.2)
α, σ, γ ≥ 0, β > 0, A ⊂ Ω ⊂ Rn open sets, n ≥ 1, G Carath´eodory.
We focus the two following main issues. Problem 0.1. Minimize F(v, 0, β, γ, σ, Ω) among v ∈ SBH(Ω) under the assumptions (0.2) and (0.3)
(0.3)
α = 0, β > 0, γ > 0 , σ ≥ 0.
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Definition and properties related to the space SBH(Ω) of Special Bounded Hessian Functions are collected in Section 1. Problem 0.2. Minimize F(v, α, β, γ, σ, Ω) among v ∈ GSBV 2 (Ω) under the assumptions (0.2) and (0.4)
(0.4)
0 < β ≤ α ≤ 2β ,
γ, σ ≥ 0 .
The set GSBV 2 (Ω), which arises in a natural way when considering the finite energy set for Problem 0.2, is the subset of generalized special bounded variation functions whose approximate gradient belongs to the same class. Definition and properties related to the class GSBV 2 are given in Section 2. Notice that, when α = 0 , then the minimization of F is not well-posed in GSBV 2 (Ω) ; while the functional space SBH(Ω) is the natural setting: on this subject see also Remark 5.7 Some explicit examples of forcing term G allowed by our framework are given in Section 7. Here we list the main results: existence of solutions for both Problems 0.1, 0.2 corresponding to weak form of nonhomogeneous Dirichlet boundary value problem or homogeneous Neumann boundary value problem; they are proved by the direct method in Calculus of Variations. For relevant explicit problems and a more general setting see Theorems 7.1, 7.2, 7.3, 7.4, 7.5, 7.6. Theorem 0.3. Assume (0.2),(0.4), ∃w ∈ GSBV 2 (Ω) s.t. F(w, α, β, γ, σ, Ω) is finite and ( spt G(·, s) ⊂ Ω ∀s (0.5) ∃c3 ∈ R, c4 ∈ L1 (Rn ) : c3 > 0, c3 |s|2 − c4 ≤ G(x, s) ∀s, a.e. x. Then there is a minimizer of F(v, α, β, γ, σ, Ω) among v in GSBV 2 (Ω) with finite energy. Theorem 0.4. Assume (0.2),(0.4),(0.5), ∃w ∈ GSBV 2 (Rn ) s.t. F(w, α, β, γ, σ, Rn ) is finite. Then there is a minimizer of F(v, α, β, γ, σ, Rn ) among v ∈ GSBV 2 (Rn ) such that v = w in Rn \ Ω with finite energy.
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Theorem 0.5. Assume (0.2),(0.3) and w ∈ SBH(Rn ) s.t. F(w, 0, β, γ, σ, Rn ) is finite and
(0.6)
Ω bounded with |Ω| = |Ω| , spt G(·, s) ⊂ Ω ∀s , ∃c1 > 0, c2 ∈ L1 (Rn ) : G(x, s) ≤ c1 |s| + c2 (x) a.e.x, ∀s, if n = 1, 2, n G(x, s) ≤ c1 |s| n−2 + c2 (x) a.e. x, ∀s, if n ≥ 3, γ ∃λ : 0 < λ < and ∃ c0 ∈ L1 (Rn ) s.t. Cn (Ω) G(x, s) ≥ c0 (x) − λ s , a.e. x, ∀s,
where Cn (Ω) and Kn (Ω) are constants such that 1 2 if n = 1 2 |Ω| 1 Cn (Ω) = if n = 2 4 |Ω| 2/n |Ω| Kn (Ω) if n > 2 n kvkL n−2 ≤ Kn (Ω) |D2 v|T (Ω) (Ω)
∀v ∈ SBH(Rn ) s.t. spt v ⊂ Ω , n > 2 .
Then there is a minimizer of F(v, 0, β, γ, σ, Rn ) among v ∈ SBH(Rn ) such that v = w in Rn \ Ω with finite energy. Theorem finite, (0.7)
0.6. Assume (0.2),(0.3) and ∃w ∈ SBH(Ω) s.t. F(w, 0, β, γ, σ, Ω) is
Ω is R regular (Definition 1.5), spt G(·, s) ⊂ Ω ∀s , ∃c1 > 0, c2 ∈ L1 (Ω) : G(x, s) ≤ c1 |s| + c2 (x) a.e.x, ∀s if n = 1, 2 n G(x, s) ≤ c1 |s| n−2 + c2 (x) a.e.x, ∀s if n ≥ 3, γ ∃λ : 0 < λ < and ∃ c0 ∈ L1 (Ω) s.t. Cn (Ω) G(x, s) ≥ c0 (x) − λ s , a.e.x, ∀s,
where, referring to the constant Γn (Ω) in Theorem 1.12, Cn (Ω) is given by |Ω| Γ (Ω) if n = 1, 2 n Cn (Ω) = |Ω|2/n Γn (Ω) if n > 2 and Z
Z G(x, v(x) + t + r · x) =
(0.8) Ω
Ω
G(x, v(x)) ∀t ∈ R, r ∈ Rn , v ∈ SBH(Ω).
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Then there is a minimizer of F(v, 0, β, γ, σ, Ω) among v in SBH(Ω) with finite energy. Regularity results (existence of strong solutions with essentially closed singular set) were proved by the authors [13, 16] in some relevant cases of Problems 0.1 and 0.2. The typical difficulty of second order functionals in GSBV 2 (Ω) amounts to the fact that, though truncating the competing functions is allowed in the finite energy space, nevertheless the truncation jeopardizes the estimates on distributional hessian D2 u since functional (0.1) does not provide any information on the L1 norm of the jump of ∇u (the absolutely continuous part of the distributional gradient Du) when γ = 0, in contrast with Lipschitz behavior of truncation in the space BH(Ω) of functions whose hessian matrix is a Radon measure (see [43],[44]). Moreover, since discontinuous competing functions u are admissible in Problem 0.2, when σ = 0 and α > 0 there is lack of compactness for sub-levels of the functional F even with respect to the a.e. convergence, as shown by the Example 2.5 In search for a finite energy space of F with σ = 0, α > 0 where appropriate compactness and lower semicontinuity properties of minimizing sequences hold, an additional difficulty is that finiteness of (0.1) does not provide any estimate on Du or even on ∇u : more precisely |∇u| may be essentially unbounded even if the energy (0.1) is arbitrarily small, as shown by the Example 2.5. In addition the fact that the competing functions u are allowed to be discontinuous immediately drives the analysis outside the framework of the space of functions with Special Bounded Hessian SBH(Ω) ([12]) where second order energies for elastic-perfectly plastic plates and rigid-plastic slabs have been successfully studied [11], [12], [13], [46]. One significant issue in Problems 0.1 and 0.2, is that F is an example of functional in Calculus of Variations whose computation is not reducible to one dimensional slices, as like as in the case of perimeter. This fact produces some technical obstacles, nevertheless F can be estimated through lower dimensional sections: in this way we deal with mixed derivatives in Section 5. The proofs rely on the direct method of the Calculus of Variations and
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are split in the following steps: coerciveness (Theorems 1.15, 1.16), slicing properties of first and second derivatives (Section 3), an interpolation inequality (Theorem 4.3), a compactness theorem for the sublevels of the functional F (Theorems 4.4, 4.5) and a lower semicontinuity property with respect to a.e. convergence (Theorems 5.13, 5.15). The case α = 0, σ = 0 with the choice G(x, v) = −g(x)v (of type dead load acting on the transverse displacement v), includes the elastic plastic plate with Neumann boundary conditions ([12],[13],[14],[39], [40],[41]) : Z (0.9) F(v, 0, β, 0, 0, Ω) = |∇2 v|2 − g v dx + βHn−1 (S∇v ) , Ω
On this subject see also Theorems 7.3, 7.4 about functional (7.3) where the Kirchhoff-Love elastic energy of the plate coupled with free plastic yield is studied. The case α > 0, σ > 0 is new, and includes a case interesting in image segmentation ([9],[36]): see functional (7.2) and related Theorem 7.2 . The case α > 0, σ = 0 includes the weak formulation of Blake & Zisserman functional ([15],[16],[5]), with the choice G(x, v) = |v(x) − g(x)|2 (of kind penalization of quadratic correction), leads to
(0.10)
F(v, Z α, β, 0, 0, Ω) = = |∇2 v|2 + |v − g|2 dx + αHn−1 (Sv ) + βHn−1 (S∇v \ Sv ) Ω
which is the weak formulation of the Blake & Zisserman functional for image segmentation ([9],[34]) (called the thin plate surface under tension): a relaxed version of the following strong counterpart F defined as follows Z F (K0 , K1 , u) = |D2 u|2 + |u − g|2 dy (0.11) Ω\(K0 ∪K1 ) + αH n−1 (K0 ∩ Ω) + βH n−1 ((K1 \ K0 ) ∩ Ω) to be minimized over closed sets K0 , K1 and u ∈ C 0 (Ω\K0 )∩C 2 (Ω\(K0 ∪K1 )) in order to achieve an optimal segmentation K0 ∪ K1 of the noisy image with g intensity level on the picture in Ω (where K0 is the set of jump points for u and K1 \ K0 is the set of crease points). About functional F in (0.11) among various results proved in [16] we recall the following statement.
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Theorem 0.7. If n = 2, 0 < β ≤ α ≤ 2β , and g ∈ L4loc (Ω) ∩ L2 (Ω) , then there is at least one triplet among K0 , K1 ⊂ R2 Borel sets with K0 ∪ K1 closed and u ∈ C 2 (Ω \ (K0 ∪ K1 )) approximately continuous on Ω \ K0 minimizing functional (0.11) and having finite energy F. Moreover the sets K0 ∩ Ω and K1 ∩ Ω are (H1 , 1) rectifiable. The assumption 0 < β ≤ α ≤ 2β , is necessary for the l.s.c. of F ([15]) in the case α > 0 ; actually in such case the terms dependent on α and β are l.s.c. as a whole and cannot be split. On the other hand the term dependent on γ is l.s.c. by itself (see Remarks 5.5, 5.9, 5.14), for any α ≥ 0. R We emphasize that the contribution of |[∇v]|dH n−1 in functional F has to be taken into account on the whole set S∇v and not on S∇v \ Sv to prevent the lack of lower semicontinuity even in the one dimensional case, as shown by the example in Remark 5.8. Eventually we notice that in dimension 1 the weak solutions are also strong solutions for Problem 0.1 and Problem 0.2, since strong and weak formulation of functional F coincide when n = 1 and is given by: Z 1 F (vh , α, β, γ, σ, A) = (|¨ vh (x)|2 + σ|v˙ h (x)|2 + G(x, vh (x))) dx + A X + α #(Svh ∩ A) + β #((Sv˙ h \ Svh ) ∩ A) + γ |[v˙ h ]| . Sv˙ h ∩A
1. Special Bounded Hessian functions (SBH) In this section we recall some properties of the functional setting suitable for functional F when α = 0 : say the class SBH. Given a connected open subset Ω ⊆ Rn (n ≥ 1) we define the class of real valued functions with special bounded hessian SBH(Ω) and we point out some of its properties. For a given set U ⊂ Rn we denote by U , ∂U its topological closure and boundary; moreover we denote by H n−1 (U ) its (n − 1)–dimensional Hausdorff measure (in particular H0 (U ) is the counting measure also denoted by #(U )) and by Ln (U ) (or shortly |U |) its Lebesgue outer measure. We indicate by
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Bρ (x) the open ball {y ∈ Rn ; |y − x| < ρ}, and we set Bρ = Bρ (0). If Ω , Ω0 are open subsets in Rn , by Ω ⊂⊂ Ω0 we mean that Ω is compact and Ω ⊂ Ω0 . We introduce the following notations: a ∧ b = min{a, b}, a ∨ b = max{a, b} for every a, b ∈ R; Mk,n stands for k × n matrices (k ≥ 1) and I for the P identity in Mn,n ; given two vectors a = {ai }, b = {bi }, we set a · b = i ai bi , (a ⊗ b)ij = ai bj . e k = Rk ∪ {∞} Let v : Ω → Rk be a Borel function; for x ∈ Ω and z ∈ R (the one point compactification of Rk ) we say, following [27], that z is the approximate limit of v at x, and we write z = ap lim v(y), y→x
if R g(z) = lim
g(v(y))dy
Bρ (x)
ρ→0
|Bρ |
e k ). for every g ∈ C 0 (R The set Sv = {x ∈ Ω; ap lim v(y) y→x
does not exist }
is a Borel set, of negligible Lebesgue measure (see e.g. [30], 2.9.13); for brevity’s e k the function sake we denote by v˜ : Ω \ Sv → R v˜(x) = ap lim v(y). y→x
Let x ∈ Ω \ Sv be such that v˜(x) ∈ Rk ; we say that v is approximately differentiable at x if there exists a k × n matrix ∇v(x) such that ap lim
y→x
|v(y) − v˜(x) − ∇v(x)(y − x)| = 0. |y − x|
If v is a smooth function then ∇v is the jacobian matrix. In the following with the notation |∇v| we mean the euclidean norm of ∇v. If p ∈ [1, +∞], we denote by Lp (Ω, Mk,n ) and by W 1,p (Ω, Mk,n ) the Lebesgue and Sobolev spaces of functions with values in Mk,n , endowed with the usual norms k · kLp and k · kW 1,p respectively. We denote by M(Ω, Mk,n ) the space of
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the bounded measures on Ω with values in Mk,n and by | · |T the total variation of a measure of M(Ω, Mk,n ), i.e. Z X X |µ|T = sup ϕij dµij ; ϕij ∈ C00 (Ω), ϕ2ij ≤ 1 in Ω . Ω ij
ij
If A is any open set then |µ|T (A) is defined in the same way with ϕij ∈ C00 (A) and we define a Borel measure |µ| setting for every Borel set B ⊂ Ω |µ|T (B) = inf |µ|T (A) ; B ⊂ A, A open . We recall the definition of the space of functions of bounded variation in Ω with values in Rk : BV (Ω, Rk ) = {v ∈ L1 (Ω, Rk ); Dv ∈ M(Ω, Mk,n )} where Dv = {Dj vi }i=1,..k,j=1,..n denotes the distributional derivatives of v. For every v ∈ BV (Ω, Rk ) the following properties hold: 1) ve(x) ∈ Rk for H n−1 –almost all x ∈ Ω \ Sv (see [47], 5.9.6); 2) Sv is countably (H n−1 ; n − 1) rectifiable (see [47], 5.9.6); 3) ∇v exists a.e. on Ω and coincides with the Radon–Nikodym derivative of Dv with respect to the Lebesgue measure (see [30], 4.5.9(26)); 4) for H n−1 almost all x ∈ Sv there exist ν = νv (x) ∈ ∂B1 , v + (x), v − (x) ∈ Rk (outer and inner trace, respectively, of v at x in the direction ν) such that (see [47], 5.14.3) Z −n lim ρ |v(y) − v + (x)| dy = 0, ρ→0
lim ρ−n
ρ→0
{y∈Bρ (x);y·ν>0}
Z
|v(y) − v − (x)| dy = 0,
{y∈Bρ (x);y·ν 1) be a bounded open set with the exterior cone property. Then (1.3)
BH(Ω) ⊂ W 1,q (Ω)
with continuous embedding if q ≤ In particular (1.4)
n n−1 ;
the embedding is compact if q
2; for any q ≥ 1 (compactly for finite q) if n = 2. If Ω ⊂ R is a bounded interval then BH(Ω) is compactly embedded in C 0,1 (Ω).
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Since we want to consider both the case of smooth domains and polygonal ones, we introduce the following definition. Definition 1.5. We say that a set Ω ⊂ Rn (n ≥ 1) satisfies the property R if it is a bounded connected open set and Ω is strongly Lipschitz and ∂Ω is the union of finitely many C 2 curves, if n = 2, Ω is C 2 uniformly regular, if n > 2 (see e.g. [1], 4.5, 4.6). Theorem 1.6. Let Ω ⊂ R2 be an open set. If v ∈ BH(Ω) has compact support in Ω then (1.5)
kvkL∞ (Ω) ≤
1 2 |D v|T (Ω) . 4
If Ω satisfies property R then BH(Ω) ⊂ C 0 (Ω).
(1.6)
(See Theorem 3.3 and Remark 3.2 in [28]). The following extension theorem holds (see Theorem 2.2, Remark 2.1 in [28]). Theorem 1.7. Let Ω ⊂ Rn with property R and let x ∈ Ω. Then there is a constant Mn = Mn (Ω) > 0 and a linear continuous map Π : BH(Ω) → BH(Rn ) such that Π v = v a.e. in Ω
spt(Πv) ⊂ Bt (¯ x) where t = 2 diamΩ
kΠvkBH(Rn ) ≤ Mn kvkBH(Ω) , Π W 2,1 (Ω) ⊂ W 2,1 (Rn ) . Remark 1.8 - If Ω does not satisfy property R both theorems 1.6, 1.7 may p fail: take, for instance, v(x, y) = log x2 + y 2 and Ω = {(x, y) : 0 < x < 1, |y| < x2 }. Then v Ω belongs to BH(Ω) but it is unbounded. Of course v B 6∈ BH(B1 ). 1
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In the space BH(Ω) the following trace theorem holds (see [28], Appendix). Theorem 1.9. Let Ω ⊂ Rn with the property R. Two bounded linear maps exist γ0 : BH(Ω) → W 1,1 (∂Ω)
γ1 : BH(Ω) → L1 (∂Ω)
such that γ0 (v) = v
, ∂Ω
γ1 (v) =
∂v ∂N ∂Ω
2
for every v ∈ C (Ω), where N is the outward normal to ∂Ω. Moreover γ1 is onto.
Remark 1.10 - We notice explicitely that C ∞ (Ω) is neither dense in BH(Ω) nor in SBH(Ω) with respect to the strong topology; nevertheless, if Ω is strongly Lipschitz, the density holds true with respect to the intermediate topology associated to the distance Z Z d2 (u, v) = ku − vkL1 (Ω) + |D2 u| − |D2 v| , Ω
Ω
as in the case of BV (Ω) and SBV (Ω) with Z Z d1 (u, v) = ku − vkL1 (Ω) + |Du| − |Dv| Ω
Ω
(see [45], III, 2.8 and I, 1.3, where the result is obtained by mollification of the trivial extension).
From now on we denote by P1 (Ω) the space of the affine functions. We define the linear map p : BH(Ω) → P1 (Ω) by (1.7)
(pv)(x) = vΩ + (∇v)Ω · (x − xΩ )
∀v ∈ BH(Ω)
where vΩ = |Ω|−1
Z v dx, Ω
(∇v)Ω = |Ω|−1
Z ∇v dx, Ω
xΩ = |Ω|−1
Z x dx. Ω
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Obviously (1.8)
pv = v p(v − pv) = 0
(1.9)
∀v ∈ P1 , ∀v ∈ BH(Ω).
Now we can prove a Poincar´e type inequality. Theorem 1.11. Assume Ω ⊂ Rn with the property R. Then there are con√ stants Kn = Kn (Ω) > 0, Sn = 1 + nKn (1 + Kn ) such that Z (1.10) kvkL1 (Ω) ≤ Kn |Dv|T (Ω) ∀v ∈ BV (Ω) with v dx = 0, Ω
kv − pvkBH(Ω) ≤ Sn |D2 v|T (Ω)
(1.11)
∀v ∈ BH(Ω).
Proof – Take vh ∈ C ∞ (Ω), such that, referring to remark 1.10, Z d1 (vh , v) → 0, vh dx = 0, Ω
then the well known Poincar´e inequality in W 1,1 (Ω) gives the existence of a constant Kn = Kn (Ω) such that kvh kL1 (Ω) ≤ Kn kDvh kL1 (Ω) = Kn |Dvh |T (Ω)
∀h ∈ N
and since |Dvh |T (Ω) → |Dv|T (Ω) , we get (1.10). Then, by choosing vh ∈ C ∞ (Ω) such that d2 (vh , v) → 0 and by using (1.9), (1.10), we get kvh − pvh kBH(Ω) = kvh − pvh kW 2,1 (Ω) = kvh − pvh kL1 (Ω) + kD (vh − pvh ) kL1 (Ω,Rn ) + kD2 vh kL1 (Ω,Mn,n ) ≤ (1 + Kn ) kD(vh − pvh )kL1 (Ω,Rn ) + kD2 vh kL1 (Ω,Mn,n ) ≤ (1 +
√
nKn (1 + Kn )) D2 vh L1 (Ω,Mn,n )
= Sn D2 vh T (Ω) . Taking the limit as h → +∞, inequality (1.11) follows.
q.e.d.
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By summarizing Theorems 1.4, 1.6, 1.7 and 1.11, and by direct computation in the case n = 1, we get Theorem 1.12. For any set Ω ⊂ Rn with property R, there is a constant Γn (Ω) > 0, such that for every v ∈ BH(Ω) n kv − pvkL n−2 ≤ Γn (Ω) |D2 v|T (Ω) (Ω)
kv − pvkL∞ (Ω) ≤ Γn (Ω)|D2 v|T (Ω)
if n > 2, if n = 1, 2.
More precisely Γ2 (Ω) = 14 M2 (Ω)S2 (Ω) for n = 2 and Γ1 (Ω) = |Ω| for n = 1. Moreover, for any open set Ω ⊂ R we have kvkL∞ (R) ≤
1 |Ω||v 00 |T (Ω) 2
∀v ∈ SBH(R) : spt v ⊂ Ω .
In the next theorem we point out the structure of the singular part of the hessian matrix of a function v ∈ SBH(Ω). On this subject we refer also to [7] and [2]. Theorem 1.13. Let Ω ⊂ Rn be an open set and v ∈ SBH(Ω). Then ∂v 2 s n−1 1) (D v) = [Dv] ⊗ ν dH SDv = ν ⊗ ν dH n−1 SDv , ∂ν
2)
3)
2 s (D v)
Z T (Ω)
=
|[Dv]| dH n−1 =
SDv
2 s (D v)
T (Ω)
∂v n−1 , ∂ν dH SDv
Z
= |∆s v|T (Ω) ,
∂v where ν = νv , ∂ν = ν · Dv, (D2 v)s and ∆s v denote respectively the singular part of the distributional hessian and laplacian of v with respect to Ln .
Proof – By the definition of SBH(Ω) we have Dv ∈ SBV (Ω, Rm ) so that the first equalities in 1) and in 2) immediately follow. Even we get ∂v [Dv] = ν. ∂ν
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Since the singular part of the hessian matrix is rank one and symmetric we get n Z Z X 2 s ∂v n−1 (D v) = dH = [D v] ν dH n−1 = |∆s v|T (Ω) i i ∂ν T (Ω) SDv SDv i=1
and the proof is achieved.
q.e.d.
We state two basic results on SBV functions, that will be applied to the gradients of a minimizing sequence for the functional (0.1). Theorem 1.14 is a compactness property and Theorem 1.17 is related to semicontinuity. Theorems 1.15 and 1.16 are coerciveness results. Theorem 1.14. Let Ω ⊂ Rn be an open set with property R. Let φ : [0, +∞[→ [0, +∞] be a convex, non decreasing function satisfying the condition φ(t) lim = + ∞, t→+∞ t let a, b be strictly positive constants and let {zh }h∈N be a sequence of functions in SBV (Ω, Rk ) such that Z zh dx = 0 ∀h ∈ N, Ω
(Z φ(|∇zh |) dx +
sup h∈N
Ω
)
Z
+
a + b|zh − zh
−
| dH n−1
< +∞.
Szh
Then there is a function z ∈ SBV (Ω, Rk ) and a subsequence {zhi }i∈N such that 1) zhi → z strongly in L1 (Ω, Rk ); 2) ∇zhi * ∇z weakly in L1 (Ω, Mk,n ); 3) Dzhi − ∇zhi dx = (zhi + − zhi − ) ⊗ ν dH n−1 Szhi converges weakly∗ in M(Ω, Mk,n ) to Dz − ∇zdx = (z + − z − ) ⊗ ν dH n−1 Sz ; 4)
R Ω
z dx = 0.
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Proof – We can assume φ(s) ≥ cs − d for every s ∈ [0, +∞[, so that there exists a constant C 0 such that |Dzh |T (Ω) ≤ C 0 and, by Theorem 1.11, kzh kL1 ≤ C 0 . Hence there exists a subsequence, still denoted by zh , and a function z ∈ BV (Ω, Rk ) such that zh → z in L1 (Ω, Rk ). Let t > 0 and denote by zht the vector valued function whose components are (zht )j = ((zh )j ∧ t) ∨ (−t) (j = 1, . . . , k). Then for every t we have zht → z t in L1 (Ω, Rk ) and Z t t n−1 sup kzh kL∞ + φ(|∇zh |)dx + H Szht < +∞, h
Ω
so by Theorem 2.1 of [4] there exists a subsequence such that zht i → z t ∈ SBV (Ω, Rk ) in L1 (Ω, Rk ). Since z ∈ BV (Ω, Rk ) and z t ∈ SBV (Ω, Rk ) for every t > 0, then we obtain z ∈ SBV (Ω, Rk ). The second assertion is proved in [4], Theorem 2.2. The third assertion follows by difference and the fourth one is trivial. q.e.d.
Theorem 1.15. Assume (0.2),(0.3)(0.6) and fix w ∈ SBH(Rn ) such that the energy F(w, 0, β, γ, σ, Rn ) is finite. Then (γ − λCn (Ω)) > 0 and
(1.12)
F(v, 0, β, γ, σ, Rn ) ≥ (γ − λCn (Ω)) |D2 v|T (Ω) + δn (Ω, w) ∀v ∈ SBH(Rn ) s.t. spt(v − w) ⊂ Ω
where γ2 δn (Ω, w) = F w, Rn \ Ω + kc0 kL1 (Ω) − |Ω| − λτn (Ω, w) − λCn |D2 w|T (Ω) , 4 τ1 = τ2 = |Ω| kwkL∞ (Ω) ,
τn = |Ω|2/n kwkLn/(n−2) (Ω) if n > 2 .
19
Proof – Since α = 0 Z F(v, Rn ) =
(1.13)
Z |∇2 v|2 + σ|∇v|2 dx + G(x, v) dx + Rn Z Rn + βHn−1 ((S∇v \ Sv )) + γ |[∇v]| dH n−1 = S∇v Z Z G(x, v) dx + = |∇2 v|2 + σ|∇v|2 dx + Z Ω Rn + βHn−1 ((S∇v \ Sv )) + γ |[∇v] || dH n−1 ≥ S ∇v R R ≥ F w, Rn \ Ω + k∇2 vk2L2 (Ω) +γ S∇v ∩Ω |[∇v]| + kc0 kL1 (Ω) −λ Ω v
If n = 1, by Young inequality and applying the last inequality in Theorem 1.12 to (v − w), we get R R k∇2 vk2L2 (Ω)+ γ S∇v ∩Ω |[∇v]| + kc0 kL1 (Ω) −λ Ω v dx = R P ˙ + kc0 kL1 (Ω) −λ Ω v dx ≥ = k¨ v k2L2 (Ω) +γ Sv˙ ∩Ω |[v]| P ˙ − λ |Ω| kv − wkL∞ (Ω) + ≥ γ k¨ v kL1 (Ω) + Sv˙ ∩Ω |[v]| (1.14) 2 + kc0 kL1 (Ω) − γ4 |Ω| − λ|Ω|kwkL∞ (Ω) ≥ ≥ γ − 12 λ|Ω|2 |v 00 |T (Ω) + δ1 (Ω, w) − F w, R\ Ω = = (γ − λC1 (Ω)) |v 00 |T (Ω) + δ1 (Ω, w) − F w, R\ Ω . If n = 2, by (1.5) of Theorem 1.6, we get R R k∇2 vk2L2 (Ω)+ γ S∇v ∩Ω |[∇v]| + kc0 kL1 (Ω) −λ Ω v dx ≥ R ≥ γ k∇2 vkL1 (Ω) + S∇v ∩Ω |[∇v]|dH n−1 − λ |Ω| kv − wkL∞ (Ω) + 2 (1.15) + kc0 kL1 (Ω) − γ4 |Ω| − λ|Ω| kwkL∞ (Ω) ≥ ≥ γ − λ|Ω| |D2 v|T (Ω) + δ2 (Ω, w) − F w, R2 \ Ω = 4 = (γ − λC2 (Ω)) |D2 v|T (Ω) + δ2 (Ω, w) − F w, R2 \ Ω . If n > 2, by Theorem 1.4 we get R R k∇2 vk2L2 (Ω)+ γ S∇v ∩Ω |[∇v]| + kc0 kL1 (Ω) −λ Ω v dx ≥ R n ≥ γ k∇2 vkL1 (Ω) + S∇v ∩Ω |[∇v]| − λ|Ω|2/n kv − wkL n−2 + (Ω) 2 γ (1.16) n + kc0 kL1 (Ω) − 4 |Ω| − λ|Ω|2/n kwkL n−2 ≥ (Ω) ≥ γ − λ |Ω|2/n Kn (Ω) |D2 v|T (Ω) + δn (Ω, w) − F w, Rn \ Ω = = (γ − λCn (Ω)) |D2 v|T (Ω) + δn (Ω, w) − F w, Rn \ Ω .
20
q.e.d. Theorem 1.16. Assume (0.2),(0.3)(0.7),(0.8). Then (γ − λCn (Ω)) > 0 and 2 F(v, 0, β, γ, σ, Ω)) ≥ (γ − λCn (Ω)) |D2 v|T (Ω) + kc0 kL1 (Ω) − γ4 |Ω| (1.17) ∀v ∈ SBH(Ω) . Proof – Since F(v, Ω) = F(v−p(v), Ω), the proof goes as like as the one of the previous Theorem, except the fact that, by handling Theorem 1.12, we estimate kv −p(v)kL∞ (Ω) (and kv −p(v)kLn/(n−2) (Ω) ) instead of the corresponding norms of v − w. q.e.d. Theorem 1.17. Let {zh }h∈N be a sequence in SBV (Ω, Rk ). Assume that {zh }h∈N converges in measure to z and that {∇z h }h∈N is weakly compact in L1 (A, Mk,n ) for every open set A ⊂⊂ Ω. Moreover let θ : [0, +∞[→ [1, +∞[ be a concave, non decreasing function. Then Z Z θ(|z + − z − |) dH n−1 ≤ lim inf θ(|zh + − zh − |) dH n−1 . Sz
h→+∞
Proof – See Theorem 3.7 in [4].
Szh
q.e.d.
Lemma 1.18. Hypotheses (0.2) and, either (0.6) or (0.7),(0.8) together, entail the sequential weak l.s.c. in BH(Ω) of the following functional Z v −→ G(x, v(x)) dx . Ω
Proof – It follows by Fatou Lemma.
q.e.d.
2. The class GSBV 2 In this section we recall some properties of the functional setting suitable for functional F when α > 0 : say the class GSBV 2 . Definition 2.1. For Ω ⊂ Rn open set GBV (Ω) = v : Ω → R Borel function; −k ∨ v ∧ k ∈ BVloc (Ω) ∀k ∈ N , GSBV (Ω) = v : Ω → R Borel function; −k ∨ v ∧ k ∈ SBVloc (Ω) ∀k ∈ N .
21
Lemma 2.2. Let v ∈ GSBV (Ω). Then 1) Sv is countably (H n−1 , n − 1) rectifiable; 2) ∇v exists a.e. in Ω. Proof – See [4], proposition 1.3 and 1.4.
q.e.d.
Definition 2.1 has a very simple interpretation in dimension one, as it is shown in the following lemma (whose proof is trivial): Lemma 2.3 below shows that in dimension one strong formulation and weak formulation of Problem 0.2 coincide and it will be used in the proof of the interpolation theorem. Rb Lemma 2.3. Let a, b ∈ R and v ∈ GSBV (a, b). If a |v(t)|dt ˙ + H0 (Sv ) < +∞ Rb 2 then v ∈ SBV (a, b). If a |v(t)| ˙ dt < +∞ and H0 (Sv ) = 0 then v ∈ H 1,2 (a, b). Now we may define the following function spaces related to several boundedness properties of second derivatives, to avoid confusion with different notations. Definition 2.4. For any open bounded Ω ⊂ Rn we set SBH(Ω) = v ∈ W 1,1 (Ω), Dv ∈ [SBV (Ω)]n SBV 2 (Ω) = v ∈ SBV (Ω), ∇v ∈ [SBV (Ω)]n GSBV 2 (Ω) = v ∈ GSBV (Ω), ∇v ∈ [GSBV (Ω)]n For functions in SBH(Ω), SBV 2 (Ω) or GSBV 2 (Ω) we use the notation ∇2ij v = ∇j (∇i v), ∇2i v = ∇i (∇i v) and, in the one dimensional case, v¨ = (v)˙. ˙ Notice 2 2 that Dv = ∇v in SBH(Ω), Dv 6= ∇v in SBV (Ω), GSBV (Ω). Moreover we set S∇v = ∪ni=1 S∇i v . We notice that in dimension n > 1 the sublevels { v ∈ SBV 2 (Ω), F(v) ≤ c } of the functional F with the choice G(x, s) = s2 , γ = σ = 0, are not compact with respect to a.e. convergence as shown by the following example.
22
Example 2.5 - We choice G(x, v) = |v|2 , y = (y1 , y2 ) ∈ R2 and h ∈ N, h ≥ 4 set rh = 2−h−1 and
vh (y1 , y2 ) =
k 4 y1 − k1 k
if y ∈ Brk ( k1 , 0) , k = 4, . . . , h
0
if y 6∈ ∪hk=4 Brk ( k1 , 0) .
We have, for σ = γ = 0, 2
F(vh , R ) =
h h X π X 1 1 |vh |2 dy + αH1 Svh = + απ ≤ c < +∞. 2 64 k 2k R2
Z
k=4
2
k=4
2
The sequence converges in L (R ) to the function v0 given by k 4 1 if y ∈ Brk ( k1 , 0) , k = 4, . . . k y1 − k v0 (y) = v0 (y1 , y2 ) = S∞ 0 if y 6∈ k=4 Brk ( k1 , 0) . 2 We see that ∇v0 is not L1loc (R2 ), hence v0 6∈ SBVloc (R2 ) and even v0 6∈ SBVloc (R2 ).
Lemma 2.6. Hypotheses (0.2) and (0.5) entail the sequential lower semicontinuity with respect to strong Lp convergence p ∈ [1, 2) of the map Z v −→ G(x, v(x)) dx . Ω
Proof – It follows from Fatou Lemma.
q.e.d.
3. Slicing properties and second derivatives in GSBV 2 In this section, for any fixed direction in Rn , we define the fibers of Ω along this direction and recall some properties related to the decomposition of derivatives for the traces on the fibers of functions defined on the whole Ω.
23
Let ν ∈ ∂B1 and define the orthogonal projection on the hyperplane perpendicular to ν : πν : R n → R n πν (y) = y − (y · ν)ν. For every subset E ⊂ Rn we define, for every x ∈ πν (E), Exν = {t ∈ R; x + tν ∈ E}, for every v ∈ GSBV (Ω) we define for H n−1 -a.e. x ∈ πν (E), vxν (t) = v(x + tν)
(defined for a.e. t ∈ Exν ).
We prove some slicing properties in the case ν = ei and Ω = (0, 1)n ; for brevity’s sake in this case we set πi = πei and we omit the label ν in the previous definitions. Theorem 3.1. Let Ω = (0, 1)n and v ∈ GBV (Ω). Then for H n−1 -a.e. x ∈ πi (Ω) (i = 1, . . . , n) (i) vx ∈ GBV (0, 1), (ii) v˙ x (t) = ∇i v(x + tei ) for a.e. t ∈ (0, 1), (iii) Svx = (Sv )x . Moreover (iv) v ∈ GSBV (Ω) iff vx ∈ GSBV (0, 1) for H n−1 -a.e. x ∈ πi (Ω), ∀i, (v) if v ∈ SBV (Ω, R ) then vx ∈ SBV (0, 1) for H n−1 -a.e. x ∈ πi (Ω), ∀i. Proof – See Theorem 2 of [15].
q.e.d.
Theorem 3.2. Let Ω = (0, 1)n and v ∈ GSBV 2 (Ω). Then for H n−1 -a.e. x ∈ πi (Ω) (i = 1, . . . , n) (i) v¨x (t) = ∇2i v(x + tei ) a.e. t ∈ (0, 1), (ii) Sv˙ x ⊆ (S∇v )x , (iii) (v˙ x )+ (t) = (∇i v)+ (x + tei ), (v˙ x )− (t) = (∇i v)− (x + tei ) ∀t ∈ (0, 1). Proof – See Theorem 3 of [15].
q.e.d.
24
Theorem 3.3. For every H n−1 measurable set E ⊂ Rn , (H n−1 , n − 1) rectifiable and for every ν ∈ ∂B1 , Z Z H0 (Ex ) dH n−1 (x) = |ν · νE (y)|dH n−1 (y) ≤ H n−1 (E), πν (E)
E
where νE (y) is an approximate unit normal to E at y. Proof – See [30], 3.2.22.
q.e.d.
Remark 3.4 - The definition of GSBV 2 (Ω) and the properties proved in Theorems 3.1, 3.2, 3.3 are invariant under translations and rotations of the coordinates (see [30], 3.1.4), hence the results obtained hold for sections of v in any direction.
4. Compactness in GSBV 2 and in SBH In this section we show compactness properties of sublevels of the functional. Many proofs could be simplified in the case σ > 0, anyway all along this section we assume the more general case σ ≥ 0. At first we show a compactness property, under assumption (0.4), valid only for 1 dimensional open sets which are finite union of intervals; then we deduce a compactness property valid for any bounded open set in any dimension n (hence also for any 1 dimensional open set). The case of assumption (0.3) is much easy. Theorem 4.1. Assume A ⊂ R be an open set which is a finite union of open intervals and there are c3 > 0, c4 ∈ L1 (A) s.t. G(x, s) > c3 |s|2 − c4 , a.e. x, fix the parameters such that , α > 0, β > 0, σ ≥ 0, γ ≥ 0. Assume that, for any h ∈ N, vh ∈ W 2 (A) the set of piece-wise H 2 functions possibly with a finite number of jumps, and sup F 1 (vh , α, β, γ, σ, A) < ∞ h∈N
25
where (4.1) Z
(|¨ vh (x)|2 + σ|v˙ h (x)|2 + G(x, vh (x))) dx + X ∩ A) + β #((Sv˙ h \ Svh ) ∩ A) + γ |[v˙ h ]| .
F 1 (vh , α, β, γ, σ, A) =
A
+ α #(Svh
Sv˙ h ∩A
Then there are vhk and v∞ ∈ W 2 (A) such that vhk −→ v∞ in Lp (A), p ∈ [1, 2), v˙ hk −→ v˙ ∞ a.e. in A. v¨hk * v¨∞ weakly in L2 (A).
(4.2)
Proof – The finiteness of the functional entails #(Svh ∪ Sv˙ h ) ≤ K < ∞. If K = 0 the statement is trivial. It is not restrictive to assume that the cardinality is definitively constant and equal to K. This leads to consider the K sequences {yki }k , i = 1, ..., K of singular points. Since A = ∪M AM with AM = A ∩ (−M, +M ), by Bolzano-Weierstrass Theorem, up to successive extraction of subsequences, without relabelling, we find a finite number of open intervals I i ⊂ AM , i = 1, ..., m(M ) ≤ K s.t. yki ∈ I i , 1 ≤ i ≤ m(M ). By summarizing we find a finite number (K(∞) = supM m(M ) ≤ K) of significative intervals which contain all the singular points. Then it is not restrictive to examine just one open interval and assume A = (a, b). Choose a minimal partition xjh , j = 1, ..., s of the interval (a, b) such that n o s j j+1 (4.3) xjh = Svh ∪ Sv˙ h x0h = a, xs+1 = b, x < x , ∪ h h h j=1
Then there are subsequences {xjh }h , j = 1, ..., s, such that, without relabelling, xjh −→ xi i = 1, ..., r ≤ s, j = ri , ..., ri+1 − 1, i i+1 1 ≤ ri ≤ ri+1 ≤ s, x 0 such that the intervals [xi − δ, xi + δ] , i = 1, ..., r are pair-wise disjoint. By the interpolation inequality (Lemma 4.10 in [1]) ! Z Z l
l
|v| ˙ 2 dt ≤ 162 max(l2 , l−2 )
(4.5) 0
|¨ v |2 + |v|2 dt
0
∀v ∈ H 2 (0, l)
26
for all s.t. 0 < < δ, by choosing l = mini (xi+1 − xi − 2δ) in (4.5), we get the existence of v ∈ H 2 (xi + , xi+1 − ) s.t. up to subsequences (I) vh −→ v strongly in H 1 (xi + , xi+1 − ) and uniformly in (xi + , xi+1 − ) (II) v˙ h −→ v 0 uniformly in (xi + , xi+1 − ) (III) v¨h −→ v 00 weakly in L2 (xi + , xi+1 − ). Moreover, by the l.s.c. of the L2 norm we get Z xi+1 − Z xi+1 − 00 2 |v | dt ≤ lim inf |¨ vh |2 dt ≤ sup F 1 (vh , A) ≤ C < +∞. h
xi +
xi +
h
2 Hence by a diagonal argument we can define v∞ ∈ Hloc (xi , xi+1 ) s.t. vhk → i i+1 1 i i+1 v∞ point-wise in (x , x ) and strongly in Hloc (x , x )
(IV )
v˙ h −→ v˙ = v 0 point-wise in (xi , xi+1 )
i = 0, ..., r
00 and v¨hk * v∞ weakly in L2loc (xi , xi+1 ). By Fatou Lemma v∞ ∈ L2 (xi , xi+1 ) and by the previous inequality as ε → 0 Z xi+1 00 2 |v∞ | dt ≤ C < +∞. xi
00 * v∞ 2 i
weakly in L2 (xi , xi+1 ) and, by the interpolation inequality Hence v¨hk (4.5), v∞ ∈ H (x , xi+1 ). By the arbitrariness of i, v∞ ∈ W 2 (A) and the thesis follows. q.e.d. In order to face the multidimensional case we introduce suitable technical tools. Truncations are allowed in SBH and GSBV 2 . Anyway the truncation has the effect of creating an increase of energy in the length of S∇v and in the jump of ∇v, moreover this amount cannot be estimated for general choices of parameters and data. To overcome the increase of the singular part of the energy due to truncations we introduce the sequence of functions (ϕk ) as follows in order to mimic the truncation with the composition with them. We set 2 0 ϕk ∈ C (R), 0 ≤ ϕk ≤ 1, (4.6) ϕk (t) = t ∀t ∈ [−k + 1, k − 1] |ϕk (t)| = k ∀|t| > k + 1.
27
Remark 4.2 - The property v belongs to GSBV (Ω) is clearly equivalent to requiring that ϕk ◦v ∈ SBVloc (Ω) ∀k ∈ N (see [4], section 1). We want a weak estimate of the first gradient: to overcome the lack of a suitable interpolation inequality we exploit the inequality in n dimensional cubes proved in [15] which allows a control of the first gradient even when γ = σ = 0 and without assuming any information on the topology of the complement of the singular set.
Theorem 4.3. ([15],Th.6) Let Q be a cube with edges of lenght l and parallel to the axes, v ∈ GSBV 2 (Q) and k ∈ N. Then for every i = 1, . . . , n the following interpolation inequality holds true Z |∇i (ϕk ◦v)|dy ≤ 2k H
n−1
Q
(Sv ∪ S∇v ) + l
n−1
2 n + l 2 +1 3
Z
2
2
12
|∇ v| dy
.
Q
The previous interpolation inequality is essential to deal with the case α > 0 which is faced in GSBV 2 while it is not necessary in the case α = 0 which is studied in SBH.
Theorem 4.4. Assume Ω ⊂ Rn be a bounded open set, n ≥ 1, α > 0, β > 0, σ ≥ 0, γ ≥ 0, (0.2),(0.4),(0.5) and vh in GSBV 2 (Ω) be such that sup F (vh , α, β, γ, σ, Ω) < ∞. h
Then there are v∞ ∈ GSBV 2 (Ω) ∩ L2 (Ω) and a subsequence vhm such that vhm → v∞ a.e. in Ω and w-L2 (Ω) vhm → v∞ strongly in Lp (Ω) for every p ∈ [1, 2) ∇vhm → ∇v∞ a.e. in Ω ∇2 vhm * ∇2 v∞ w-[L2 (Ω)]n×n .
28
Proof – Since suph kvh kL2 < ∞, there exist a subsequence that we still denote by (vh ) and v∞ ∈ L2 (Ω) such that
(4.7)
lim vh = v∞ h
w − L2 (Ω).
Fix k ∈ N and let ϕk ∈ C 2 (R) as in (4.6). Then we have kϕk ◦vh kL1 ≤ kvh kL2 |Ω|1/2
∀h ∈ N
and, by the interpolation Theorem 4.3, for every i = 1, . . . , n and for every cube Q ⊂ Ω with edges parallel to the axes of lenght l, we have Z sup |∇i (ϕk ◦vh )|dy ≤ c(k, l) < +∞. h
Q
By the definition of SBV (Ω) and by the previous inequalities, for every Ω0 ⊂⊂ Ω we obtain
(4.8)
R R sup kϕk ◦vh kBV (Ω0 ) = suph Ω0 |D(ϕk ◦vh )| + Ω0 |ϕk ◦vh |dy ≤ h R suph Ω0 (|∇(ϕk ◦vh )| + |ϕk ◦vh |) dy + 2kH n−1 (Svh ) < +∞,
hence, by the compactness theorem in BVloc (Ω), by the equiboundedness in L (Ω) with respect to h of (ϕk ◦vh ) and by a diagonalization argument, there exist a subsequence (that we do not relabel) and wk ∈ BVloc (Ω) such that ∞
(4.9)
a.e. and in Lp (Ω)
lim ϕk ◦vh = wk h
1 ≤ p < +∞, ∀k ∈ N.
Set Ek = {|wk | ≤ k − 1}, then by (4.7) lim vh χEk = v∞ χEk h
w − L2 (Ω),
by the properties of ϕk and by (4.9) lim vh χEk = lim (ϕk ◦vh )χEk = wk χEk h
h
a.e. in Ω,
29
hence, since |ϕk ◦vh | ≤ k a.e. in Ω, we get wk = v∞
a.e. in Ek ∀k ∈ N.
Since Ek = {|v∞ | ≤ k − 1} and v∞ ∈ L2 (Ω), we have limk |Ω \ Ek | = 0 and then
(4.10)
lim vh = v∞ h
a.e. in Ω,
hence for every i and for H n−1 -a.e. x ∈ πi (Ω) we have
(vh )x → (v∞ )x
(4.11)
a.e. in Ωx .
By (4.7), (4.10) and the boundedness of Ω we get vh → v∞ strongly in Lp (Ω) for p ∈ [1, 2). By (4.8), (4.9) and (4.10), ϕk ◦v∞ = wk ∈ BVloc (Ω) ∀k, hence v∞ ∈ GBV (Ω). By using Fatou’s Lemma, Fubini’s Theorem, and Theorems 3.1, 3.2, 3.3, 4.1, 4.3 and Remark 4.2, for every i, we get Z Z 2 0 lim inf |(¨ vh )x | dt + H (S(vh )x ∪ S(v˙ h )x ) dx ≤ h
πi (Ω)
Ωx
Z
Z
2
0
|(¨ vh )x | dt + H (S(vh )x
lim inf h
πi (Ω)
Z lim inf h
∪ S(v˙ h )x ) dx ≤
Ωx
|∇2i vh |2 dx + H n−1 (Svh ∪ S∇i vh ) < +∞
Ω
by the assumption, so that for H n−1 -a.e. x ∈ πi (Ω) we have Z (4.12)
lim inf h
|(¨ vh )x |2 + |(vh )x |2 dt + H0 (S(vh )x ∪ S(v˙ h )x ) < +∞,
Ωx
hence for H n−1 -a.e. x ∈ πi (Ω), by Theorem 3.3, (4.11) and the limit uniqueness, we get (v∞ )x ∈ GSBV 2 (Ωx ). By the same argument for every i = 1, . . . , n, and by Theorem 3.1 we obtain v∞ ∈ GSBV (Ω) and also ∇v∞ (x) ∈ Rn for a.e. x ∈ Ω. The property just stated allows us to use the same argument
30
in the proof of Theorem 2.1 in [3] for the sequence (∇i vh ), hence there exists fi ∈ GSBV (Ω) such that, up to a subsequence, ∇i vh → fi ∇2ij vh * ∇j fi
a.e. in Ω w − L2 (Ω).
By (4.11) and the slicing argument above we know that ∇i vh → ∇i v∞
a.e. in Ω
hence ∇i v∞ = fi ∈ GSBV (Ω) and then v∞ ∈ GSBV 2 (Ω).
q.e.d.
Eventually we give two compactness properties valid under the assumption (0.3). The proof is a straightforward consequence of Theorem 1.14, and RellichKondratiev Theorem when σ > 0. Theorem 4.5. Assume Ω ⊂ Rn be a bounded open set, (0.2), (0.3) and (0.6). Let vh in SBH(Rn ) with spt vh ⊂ Ω and sup F (vh , 0, β, γ, σ, Rn ) < ∞. h
Then there are v∞ ∈ SBH(Rn ) and a subsequence vhm such that vhm → v∞ a.e. in Rn n vhm → v∞ in Lq (Rn ) for every q ∈ [1, n−2 ) if n ≥ 3 (any q if n = 1, 2) 1 n Dvhm → Dv∞ in L (R ) ∇2 vhm * ∇2 v∞ w-[L2 (Rn )]n×n . Proof – By Theorem 1.15 we get a bound for |D2 vh |T (Rn ) . By Theorem 1.14, applied to Dvh , and by Theorem 1.4, applied in a ball containing Ω, the thesis follows. q.e.d. Theorem 4.6. Assume Ω ⊂ Rn with the property R, (0.2), (0.3), (0.7), (0.8). Let vh in SBH(Ω) be such that sup F (vh , 0, β, γ, σ, Ω) < ∞. h
Then there are v∞ ∈ SBH(Ω) and a subsequence vhm such that
31
vhm − pvhm → v∞ vhm − pvhm → v∞
a.e. in Ω, in Lq (Ω) n for every q ∈ [1, n−2 ) if n ≥ 3 (any q if n = 1, 2), 1 D(vhm − pvhm ) → Dv∞ in L (Ω), ∇2 vhm * ∇2 v∞ w-[L2 (Ω)]n×n . Proof – By Theorem 1.16 we get a bound for |D2 vh |T (Ω) . By Theorem 1.14, applied to D(vh − pvh ), and by Theorems 1.11 and 1.4, the thesis follows. q.e.d. Remark 4.7 - So far we have not proved that F(v∞ ) is finite even in the case γ > 0, where v∞ is defined in the thesis of Theorems 4.4, 4.5 and 4.6. The finiteness of F(v∞ ) will follow from the semicontinuity properties in the next section.
5. Lower semicontinuity We recall that the commutation of operators ∇i ∇j is true in SBH but fails in GSBV 2 (see [32]). In particular the functional (0.1) cannot be sliced, due to the presence of mixed derivatives in ∇2 v . Hence in such case the difficult task is showing the lower semi continuity of length terms, integral jump contribution and the square of mixed derivatives ∇ij v. Several steps are required in order to achieve this l.s.c. property. The idea is to estimate the part of the functional which cannot be sliced, by mean of a covering method and exploiting the 1-dimensional lower semicontinuity in every tile of the covering. Before stating the lower semicontinuity theorem, we prove some lemmas. The first one allows us to deal with the singular part of the mixed derivatives which cannot be estimated by simply considering any fixed one dimensional fiber.
Definition 5.1. For every (H n−1 , n − 1) rectifiable set Σ ⊂ Ω we denote by νΣ (y) the approximate unit normal to Σ at y, defined for H n−1 a.e. y ∈ Σ.
32
We remark that, up to the orientation, for every v ∈ GSBV 2 (Ω) νΣ = νSv νΣ = νS∇v
if Σ ⊂ Sv if Σ ⊂ S∇v .
For every fixed x ∈ Σ and ν ∈ ∂B1 , we can make a conventional choice of νΣ (x) satisfying νΣ (x) · ν ≥ 0. Such choice induces a unique orientation in every portion of Σ where νΣ · ν 6= 0.
Lemma 5.2. Let v ∈ GSBV 2 (Ω). Fix i ∈ {1, . . . , n}. Choose ν, η ∈ ∂B1 , and Σ ⊂ S∇i v \ Sv be a Borel set such that ν 6= η,
(i)
(ii)
ν · νΣ (y) >
3 , 4
ν·η >
η · νΣ (y) >
3 , 4
3 , 4 H n−1 a.e. y ∈ Σ,
{ν, η, ei } linearly dependent.
(iii)
Then there are two disjoint Borel sets Σ0 , Σ1 such that H n−1 (Σ\(Σ0 ∪Σ1 )) = 0 and ∀y ∈ Σ0 ∃x ∈ πν (Σ0 ), ∃t ∈ S(v˙ xν ) : y = x + tν ∀y ∈ Σ1 ∃z ∈ πη (Σ1 ), ∃τ ∈ S(v˙ zη ) : y = z + τ η. Proof – Due to the assumptions (i) and (ii) the orientations induced on νΣ by ν and η coincide on Σ and this will be understood in the following. By assumption (ii) and by Theorem 3.3, if N ⊂ Σ then H n−1 (N ) = 0
iff
H n−1 (πν (N )) = 0
iff
H n−1 (πη (N )) = 0.
Then by Theorem 3.1(ii) and Theorem 3.3 for a.e. y ∈ Σ, setting x = πν (y) and y = x + tν we have t ∈ S(v˙ xν ) iff (∇v · ν)+ (y) 6= (∇v · ν)− (y),
33
hence we can choose Σ0 = {y ∈ Σ; (∇v · ν)+ (y) 6= (∇v · ν)− (y) }. On the other hand, if we set Σ1 = {y ∈ Σ; (∇v · ν)+ (y) = (∇v · ν)− (y) }, since Σ ⊂ S∇i v \ Sv then (∇i v)+ (y) = (∇v · ei )+ (y) 6= (∇v · ei )− (y) = (∇i v)− (y)
H n−1 a.e. y ∈ Σ,
hence by the assumptions (i) and (iii) we get (∇v · η)+ (y) 6= (∇v · η)− (y)
H n−1 a.e. y ∈ Σ1 .
The thesis follows since, by setting z = πη (y), y = z + τ η, we have τ ∈ S(v˙ zη ) iff (∇v · η)+ (y) 6= (∇v · η)− (y). q.e.d. Now we show the lower semicontinuity in the one dimensional case; then we will deduce the lower semicontinuity for n ≥ 1. We would like to exploit the one dimensional compactness of Theorem 4.1 − ε), say the interval but the convergence (I)(II)(III) hold only in (xih + ε, xi+1 h endpoints may depend on h. Nevertheless, taking into account of possible collapse of several sequences, the convergence denoted by (I)(II)(III) in its proof actually holds up to the endpoints of the interval (as in the case of (IV)), say in (xi , xi+1 ) in a suitable sense: by handling translations as stated in the following lemma. Lemma 5.3. (Translations) With the notation defined in the proof of Theorem 4.1, we assume that lim xjh = xi , j = ri , ..., ri+1 − 1 and xi is not limit of other h
i i sequences. By setting E+ = [xi , xi + δ], E− = [xi − δ, xi ], we deduce that convergence denoted by (I)(II)(III)(IV) in the proof of Theorem 4.1 hold in intervals of length δ > 0 for both
(5.1)
vh (x − xi + xrhi ) −→ v(x) r −1 vh (x − xi + xhi+1 ) −→ v(x)
i x in E− i x in E+
34
Hence r −1 v(xi+ ) = lim vh (xhi+1 )+ h
v(xi− ) = lim vh ((xrhi )− ) h r −1 v(x ˙ i+ ) = lim v˙ h (xhi+1 )+
(5.2)
h
v(x ˙ i− ) = lim v˙ h ((xrhi )− ) h
Proof – It is a consequence of continuity of translations in L2 applied to v, ˙ v¨ and the uniform continuity of H 2 functions in 1 dimension. q.e.d. We now exploit the fact that, if k¨ vh kL2 is kept uniformly bounded, then the contribution to the variation of v˙ h due to the absolutely continuous part of vh00 is vanishing as the length vanishes. More precisely we have:
(5.3)
|w(x ˙ + ) − w(x)| ˙ ≤
√
ε kwk ¨ L2 (x,x+δ) ∀x ∈ A, ∀ ∈ (0, δ], ∀ w ∈ H 2 (x, x + δ)
Theorem 5.4. (1D Lower semicontinuity in SBV 2 ) Assume A ⊂ R is an open interval, (0.2),(0.4) hold true. Let v∞ , vh , v˙ ∞ , v˙ h ∈ SBV (A) (h ∈ N), such that sup F 1 (vh , α, β, γ, σ, A) < ∞ and vh → v∞
a.e. in A
h∈N
where
(5.4)
Z
(|¨ vh (x)|2 + σ|v˙ h |2 + G(x, vh (x))) dx + X ∩ A) + β #((Sv˙ h \ Svh ) ∩ A) + γ |[v˙ h ]| .
F 1 (vh , α, β, γ, σ, A) =
A
+ α #(Svh
Sv˙ h ∩A
Then F 1 (v∞ , α, β, γ, σ, A) ≤ lim inf F 1 (vh , α, β, γ, σ, A). h
35
Z
(|¨ vh (x)|2 +
Proof – The absolutely continuous part of the functional A
σ|v˙ h |2 + G(x, vh (x))) dx is sequentially l.s.c. (see Lemma 2.6), so we have only to show that
α#(Sv∞ ) + β#(Sv˙ ∞ \ Sv∞ ) + γ
X
|[v˙ ∞ ]| ≤
Sv˙ ∞
(5.5)
≤ lim inf α#(Svh ) + β #(Sv˙ h \ Svh ) + γ h
X
|[v˙ h ]| .
Sv˙ h
By a selection argument identical to the one at the beginning of the proof of Theorem 5.1 it is not restrictive to assume, by localization, that there is at most one point in Sv∞ ∪ Sv˙ ∞ . ˜ , Moreover, if there is no such point, then v∞ ∈ H 2 (A) and for h > h 2 vh ∈ H , and the l.s.c. in A is straightforward. Else there is exactly one point x ∈ Sv∞ ∪ Sv˙ ∞ : in this case, up to subsequences, vh 6∈ H 2 (A), ∀h (otherwise kvh kH 2 is bounded which is a contradiction). Hence there are only two cases to be examined. A) If there is exactly one point x ∈ Sv∞ then, either there is an approximating sequence of jumps (and possibly something more in Svh ∪ Sv˙ h ) or two approximating sequences of creases (and possibly something more in Svh ∪Sv˙ h ). This because one jump cannot be approximated by only one sequence of creases, without blow-up of the energy. In fact F 1 (vh ) bounded together the absence of jumps and #Sv˙ h = 1 leads to a continuous limit v∞ . B) If there is exactly one point x ∈ (Sv˙ ∞ \ Sv∞ ) then such x can be the limit of one or more (finite number, uniformly bounded in h) points in Svh ∪ Sv˙ h . In any of the above cases, by β ≤ α ≤ 2β we get (5.6) α#(Sv∞ ) + β#(Sv˙ ∞ \ Sv∞ ) ≤ lim inf h
α#(Svh ) + β #(Sv˙ h \ Svh )
In both cases A) B) it is not restrictive to assume that vh has a fixed number M of points xjh , in Svh ∪ Sv˙ h h ∈ N, j = 1, ..., M, say there are neither jump nor crease points of vh in A \ ∪j {xjh } and xjh → x, for all j, then v∞ ∈ H 2 (A \ {x}). By Lemma 5.3, (5.3), triangular inequality, by taking into account that
36
#(Sv∞ ∪ Sv˙ ∞ ) = 1 and that M is uniformly bounded in h , we get: X γ |[v˙ ∞ ]| = γ|[v˙ ∞ (x)]| = Sv˙ ∞ 1 = γ |v˙ ∞ (x+ ) − v˙ ∞ (x− )| = γ lim |v˙ h (xM h + ) − v˙ h (xh− )| ≤ h
≤ γ lim h
M X
|v˙ h (xjh+ )− v˙ h (xjh− )|+
M −1 X
j=1
(5.7) ≤ γ lim inf h
! v˙ h (xj+1 ) − v˙ h (xj ) ≤ h − h+
j=1
M −1q X X M [v˙ h (xj )] + xj+1 − xj k¨ vh k h
h
j=1
h
!
L2 (xjh ,xj+1 ) h
≤
j=1
! M q X j M 1 [v˙ h (x )] + (M − 1) x − x k¨ ≤ γ lim inf h h h vh kL2 (a,b) = h j=1 X = γ lim inf |[v˙ h ]| . h
Sv˙ h
Inequalities (5.6) and (5.7) together give (5.5).
q.e.d.
Remark 5.5 - We emphasize that, when γ > 0, the last three terms in (5.4) cannot be arbitrarily split if we want to mantain the l.s.c. with respect to the a.e. convergence. Nevertheless both α #(Sv ∩ A) + β #((Sv˙ \ Sv ) ∩ A)
(5.8) (5.9)
γ
X
|[v]| ˙
Sv˙ ∩A
are l.s.c. in SBV 2 (a, b). The two terms in (5.8) are not separately l.s.c. ([15]). R The absolutely continuous terms Z(the main term A |¨ v |2 dx , and lower order Z |v| ˙ 2 dx and
perturbations, σ 2
A
l.s.c.in SBV (a, b).
G(x, v(x)) dx ) are each one separately A
37
Theorem 5.6. (1D Lower semicontinuity in SBH) Assume A ⊂ R is an open interval, (0.2),(0.3) hold true . Fix the parameters such that α = 0, β > 0, γ > 0, σ ≥ 0 so that, referring to (5.4), F 1 becomes Z F 1 (v, 0, β, γ, σ, A) = (|¨ v |2 + σ|v| ˙ 2 + G(x, v(x))) dx + A X (5.10) +β #((Sv˙ ) ∩ A) + γ |[v]| ˙ . Sv˙ ∩A
Let vh , v∞ ∈ SBH(A), h ∈ N, such that sup F 1 (vh , 0, β, γ, σ, A) < ∞ and vh → v∞
a.e. in A
h∈N
Then F 1 (v∞ , 0, β, γ, σ, A) ≤ lim inf F 1 (vh , 0, β, γ, σ, A). h
Proof – The statement can be proved as like as the previous one: there are fewer and simpler cases than before. Notice that one can also deduce the thesis by [4] as in [12] since the term with σ does not affect the argument. q.e.d. Remark 5.7 - When α = 0 we are forced to restrict the set of admissible functions to SBH since the minimization of F 1 (·, 0, β, γ, σ, A) is not well posed in SBV 2 due to lack of compactness. From another viewpoint we notice that minimizing F 1 (·, ∞, β, γ, σ, A) over GSBV 2 (Ω) with γ > 0, leads to finite energy class of admissible functions contained in SBH(A), hence it is equivalent to the minimization of F(·, 0, β, γ, σ, A) over SBH(A). Remark 5.8 - We emphasize that, when γ > 0, for both functionals (5.4) and (5.10), it is essential to take into account the contribution of [v] ˙ on the whole set Sv˙ and not on Sv˙ \ Sv since the functional Z Fe1 (v) = (|¨ v (x)|2 + σ|v| ˙ 2 + G(x, v(x))) dx + A
38
X
+ α #(Sv ∩ A) + β #((Sv˙ \ Sv ) ∩ A) + γ
|[v]| ˙
(Sv˙ \Sv )∩A
is not lower semicontinuous, as shown by the counterexample A = (−1, 1)
and vh (x) =
1 sign(x) + M |x|, h
x ∈ A,
which entails vh → v∞ = M |x|, hence Fe1 (v∞ ) ≤ lim inf Fe1 (vh ) h
if and only if β + 2γM ≤ α for every M , which contradicts γ > 0. Remark 5.9 - We emphasize that the last two terms in (5.10) actually are separately l.s.c. with respect to the a.e. convergence. Now we face the l.s.c in the n dimensional case, hence we are forced to consider the wider class GSBV 2 . In the 1d case the finiteness of energy automatically restricted such class to SBV 2 . Lemma 5.10. Assume 0 < β ≤ α ≤ 2β, γ ≥ 0, σ ≥ 0. Let 0 < < 14 , ν ∈ ∂B1 , i ∈ {1, . . . , n}, v∞ , vh ∈ GSBV 2 (Ω) (h ∈ N), A ⊂ Ω open set and let Σ ⊂ A ∩ (S∇i v∞ \ Sv∞ ) be a Borel set such that ν · νΣ (y) > 1 −
2
vh → v∞
H n−1 -a.e. y ∈ Σ a.e. in Ω
sup F(vh , A) < +∞. h∈N
Then Z (1 − ) βH n−1 (Σ) + γ |[∇v]|dH n−1 ≤ lim inf F(vh , α, β, γ, σ, A). Σ
h
where F(vh , α, β, γ, σ, A) =
R A
(|∇2 vh |2 + σ|∇vh |2 + G(x, vh )) dx +
αH n−1 (Svh ∩ A) + βH n−1 ((S∇vh \ Svh ) ∩ A)) + γ
R S∇vh ∩A
|[∇vh ]|dH n−1
39
Proof – Let η ∈ ∂B1 be such that 0 < |ν − η| < dependent, hence
2
and {ν, η, ei } linearly
η · νΣ (y) > 1 − H n−1 -a.e. y ∈ Σ. Choose Σ0 , Σ1 ⊂ Σ disjoint Borel sets as in Lemma 5.2 and Kj ⊂ Σj (j = 0, 1) compact sets. Let A0 , A1 be disjoint open sets such that Kj ⊂ Aj ⊂ A (j = 0, 1). We can also assume that A0 (resp. A1 ) is a finite union of cubes with edges parallel or orthogonal to ν (resp. η). Then by |[∇v]| = |[∇i v]| on S∇i v , by Theorems 3.3 and 5.4, Fatou’s Theorem, Fubini’s Theorem and again Theorem 3.3 we get for any K0 , K1 as above Z
(1 − ε) βH n−1 (K0 ) + βH n−1 (K1 ) + γ |[∇v]|dH n−1 = K ∪K 0 1 Z ∂v n−1 νΣ · νΣ dH ≤ = (1 − ε) β+γ ∂νΣ 1 Z K0 ∪K Z ∂v ∂v νΣ · ν dH n−1 + νΣ · η dH n−1 = β + γ ≤ β + γ ∂ν ∂η K K 0 1 Z Z βH0 ((K0 )νx )+γ (vxν )˙ dH n−1 (x)+ βH0 ((K1 )ηz )+γ (vzη )˙ dH n−1 (z) ≤ πν (K0 )
πη (K1 )
Z
lim inf F 1 ((vh )νx , (A0 )νx ) dH n−1 (x) + h πν (K0 ) Z + lim inf F 1 ((vh )ηz , (A1 )ηz ) dH n−1 (z) ≤ h π (K ) η 1 Z lim inf F 1 ((vh )νx , (A0 )νx ) dH n−1 (x) + h πν (K0 ) Z + F1 ((vh )ηz , (A1 )ηz ) dH n−1 (z) ≤ πη (K1 )
lim inf (F(vh , A0 ) + F(vh , A1 )) ≤ lim inf F(vh , A). h
h
By the arbitrariness of the compact sets K0 , K1 and by the regularity of H the thesis follows. q.e.d. n−1
Lemma 5.11. Assume 0 < β ≤ α ≤ 2β, γ ≥ 0, σ ≥ 0. Let 0 < < 14 , ν ∈ ∂B1 , i ∈ {1, . . . , n}, v∞ , vh ∈ GSBV 2 (Ω) (h ∈ N), A ⊂ Ω open set and let Σ ⊂ A ∩ (S∇i v∞ ∩ Sv∞ ) be a Borel set such that ν · νΣ (y) > 1 − H n−1 -a.e. y ∈ Σ 2
40
vh → v∞
a.e. in Ω
sup F(vh , A) < +∞. h∈N
Then Z (1 − ) αH n−1 (Σ) + γ |[∇v]|dH n−1 ≤ lim inf F(vh , α, β, γ, σ, A). h
Σ
Proof – Let η ∈ ∂B1 be such that 0 < |ν − η| < dependent, hence
2
and {ν, η, ei } linearly
η · νΣ (y) > 1 − H n−1 -a.e. y ∈ Σ. Choose Σ0 , Σ1 ⊂ Σ disjoint Borel sets as in Lemma 5.2 and Kj ⊂ Σj (j = 0, 1) compact sets. Let A0 , A1 be disjoint open sets such that Kj ⊂ Aj ⊂ A (j = 0, 1). We can also assume that A0 (resp. A1 ) is a finite union of cubes with edges parallel or orthogonal to ν (resp. η). Then by Theorems 3.3 and 5.4, Fatou’s Theorem, Fubini’s Theorem and again Theorem 3.3 we get for any K0 , K1 as above Z
(1 − ε) αH n−1 (K0 ) + αH n−1 (K1 ) + γ |[∇v]|dH n−1 = K0 ∪K1 Z ∂v n−1 = (1 − ε) α+γ ≤ νΣ · νΣ dH ∂ν Σ K ∪K 0 1 Z Z ∂v ∂v νΣ · ν dH n−1 + νΣ · η dH n−1 = α + γ ≤ α + γ ∂ν ∂η K K 0 1 Z αH0 ((K0 )νx ) + γ (vxν )˙ dH n−1 (x) + πν (K0 ) Z + αH0 ((K1 )ηz ) + γ (vzη )˙ dH n−1 (z) ≤ πη (K1 ) Z 1 ν ν lim inf F ((vh )x , (A0 )x ) dH n−1 (x) + h πν (K0 ) Z + lim inf F 1 ((vh )ηz , (A1 )ηz ) dH n−1 (z) ≤ h πη (K1 ) Z lim inf F 1 ((vh )νx , (A0 )νx ) dH n−1 (x) + h πν (K0 ) Z + F1 ((vh )ηz , (A1 )ηz ) dH n−1 (z) ≤ πη (K1 )
lim inf (F(vh , A0 ) + F(vh , A1 )) ≤ lim inf F(vh , A). h
h
41
By the arbitrariness of the compact sets K0 , K1 and by the regularity of H n−1 the thesis follows. q.e.d. Lemma 5.12. Assume 0 < β ≤ α ≤ 2β, γ ≥ 0, σ ≥ 0. Let 0 < < 14 , ν ∈ ∂B1 , v∞ , vh ∈ GSBV 2 (Ω) (h ∈ N), U ⊂ Ω open set and let T ⊂ U ∩ (Sv∞ \ S∇v∞ ) be a Borel set such that ν · νT (y) > 1 −
2
H n−1 -a.e. y ∈ T
vh → v∞
a.e. in Ω
sup F(vh , α, β, γ, σ, U ) < +∞. h∈N
Then (1 − )αH n−1 (T ) ≤ lim inf F(vh , α, β, γ, σ, U ). h
Proof – We can argue as in the proof of the previous lemma in a simpler way since it is not necessary to split T into two sets due to the fact that we can use Theorem 3.1(iii) instead of Theorem 3.2(ii) and Lemma 5.2. Fix a compact set K ⊂ T . We can choose an open set A which is a finite union of cubes with edges parallel or orthogonal to νT such that K ⊂ A ⊂ U . Then by Theorem 3.3 and 5.4, Fatou’s Theorem, Fubini’s Theorem and again Theorem 3.3 we get Z Z (1 − ε) α H n−1 (K) = (1 − ε)
Z
H0 ((K)νx ) dH n−1 (x) ≤
α πν (K) Z h
πν (K)
F
lim inf πν (K)
α νT · νT dH n−1 ≤
ZK
1
((vh )νx , (A)νx ) dH n−1 (x)
α νT · ν dH n−1 =
K
lim inf F 1 ((vh )νx , (A)νx ) dH n−1 (x) ≤ h
≤ lim inf F(vh , A). h
By the arbitrariness of the compact set K and by the regularity of H n−1 the thesis follows. q.e.d. We use the previous results to show the lower semicontinuity of F.
42
Theorem 5.13. Assume Ω ⊂ Rn bounded open set, (0.2),(0.4),(0.5) hold true. Let v∞ , vh ∈ GSBV 2 (Ω) (h ∈ N), such that sup F(vh , α, β, γ, σ, Ω) < ∞ and vh → v∞
a.e. in Ω.
h∈N
Then F(v∞ , α, β, γ, σ, Ω) ≤ lim inf F(vh , α, β, γ, σ, Ω). h
Proof – Fix 0 < < 14 . Let {νk } be a finite set in ∂B1 such that for every ν ∈ ∂B1 there exists k such that |ν − νk | < 4 . We can find a finite family of Borel sets {Σj ; j = 1, . . . , p + q + r} such that Σj ⊂ (Sv∞ \ S∇v∞ ) for j = 1, . . . , p, Σj ⊂ (Sv∞ ∩ S∇v∞ ) for j = p + 1, . . . , p + q, Σj ⊂ (S∇v∞ \ Sv∞ ) for j = p + q + 1, . . . , p + q + r, Σj ∩ Σi = ∅ for j 6= i,
H n−1 ((Sv∞ ∪ S∇v∞ ) \ ∪p+q+r Σj ) = 0 j=1
H n−1 − a.e. y ∈ Σj . 2 there exist disjoint compact sets Kj ⊂ Σj such that
∀j ∃kj : νSv∞ ∪S∇v∞ (y) · νkj > 1 − By the regularity of H n−1
H n−1 (Sv∞ ∪ S∇v∞ ) \ ∪p+q+r Kj < j=1 and there exist pairwise disjoint open sets {Aj ; j = 0, 1, . . . , p + q + r} such that Kj ⊂ Aj for j = 1, . . . , p + q + r, and Z (|∇2 v∞ |2 + σ|∇v∞ |2 + G(x, v∞ (x)))dx < . Ω\A0
From Lemmas 5.10, 5.11, 5.12 we get αH n−1 (Sv∞ \ S∇v∞ ) + αH n−1 (Sv∞ ∩ S∇v∞ ) + βH n−1 (S∇v∞ \ Sv∞ )+ Z Z n−1 +γ |[∇v∞ ]| dH +γ |[∇v∞ ]| dH n−1 ≤ S∇v∞ \Sv∞
Sv∞ ∩S∇v∞ p
p+q
j=1
j=p+1
ε(α + β + γ) + αH n−1 ( ∪ Kj ) + αH n−1 ( ∪
Kj ) + βH n−1 (
p+q+r
∪
j=p+q+1
Kj )+
43
+γ
p+q+r X Z j=p+1
ε(α + β + γ) +
|[∇v∞ ]| dH n−1 ≤
Kj p+q+r X 1 lim inf F(vh , Aj ). h 1 − ε j=1
By the compactness Theorem 4.4, by Lemma 2.6 and by the lower semicontinuity of quadratic forms with respect to weak convergence in L2 Z |∇2 v∞ |2 + σ |∇v∞ |2 + G(x, v∞ (x)) dx ≤ Ω Z ≤ ε + lim inf |∇2 vh |2 + σ |∇vh |2 + G(x, vh (x)) dx. h
A0
Since {Aj } are pairwise disjoint, summing the previous inequalities and taking into account the arbitrariness of the thesis follows. q.e.d. Remark 5.14 - We emphasize that, when γ > 0, the last three terms in F cannot be arbitrarily split if we want to keep the l.s.c. with respect to the a.e. convergence (which holds true for their sum). Nevertheless both
αH n−1 (Sv ) + βH n−1 (S∇v \ Sv )
(5.11) and
Z (5.12)
|[∇v]|
γ S∇v ∩A
are l.s.c.in GSBV 2 (Ω) with respect Z to the a.e. convergence. Z 2 The absolutely continuous terms |∇v| dx and G(x, v(x))) dx are lower A
A
order perturbations, each one separately l.s.c. in GSBV 2 (A), by Lemma 2.6. Theorem 5.15. Assume Ω ⊂ Rn bounded open set, (0.2),(0.3) and, either (0.6) or (0.7),(0.8) together. Let v∞ , vh ∈ SBH(Ω) (h ∈ N), such that sup F(vh , 0, β, γ, σ, Ω) < ∞
and
vh → v∞
a.e. in Ω.
h∈N
Then F(v∞ , 0, β, γ, σ, Ω) ≤ lim inf F(vh , 0, β, γ, σ, Ω). h
44
Proof – By the compactness Theorems 4.5 and 4.6 and by Lemma 1.18 it follows that the absolutely continuous part of F is l.s.c., hence the thesis follows from Theorem 1.17. q.e.d.
6. Existence of minimizers Here we show the proofs of main Theorems. Proof of Theorem 0.3 – By (0.5) and the properties of w there is a minimizing sequence vh s.t. F(vh , Ω) ∈ R. By Theorem 4.4 we find v ∈ GSBV 2 (Ω) s.t. vh → v a.e.x ∈ Ω. By Theorem 5.13 and (0.5) we get −∞ < −kc4 kL1 ≤ F(v, Ω) ≤ inf F(vh , Ω) < +∞ . q.e.d. Proof of Theorem 0.4 – By (0.5) and the properties of w there is a minimizing sequence vh s.t. F(vh , Rn ) ∈ R . By Theorem 4.4 we find v ∈ GSBV 2 (Rn ) s.t. vh → v a.e.x ∈ Rn . By Theorem 5.13 and (0.5) we get −∞ < −kc4 kL1 ≤ F(v, Rn ) ≤ inf F(vh , Rn ) < ∞ . q.e.d. Proof of Theorem 0.5 – By (0.6) and the properties of w there is a minimizing sequence vh s.t. F(vh , Rn ) ∈ R. By Theorem 4.5 we find v ∈ SBH(Rn ) s.t. vh → v a.e.x ∈ Rn . By Theorem 5.15 and (0.6) we get F(v, Rn ) ≤ inf F(vh , Rn ). By (0.6) and Theorem 1.15 we get −∞ < (γ − λCn (Ω)) kD2 vkT (Rn ) + δn (Ω, w) ≤ F(v, Rn ) . q.e.d. Proof of Theorem 0.6 – By (0.7), (0.8) and the existence of w there is a minimizing sequence vh s.t. F(vh , Ω) ∈ R. By Theorem 4.6 we find v ∈ SBH(Ω) s.t. vh − pvh → v a.e.x ∈ Ω.
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By Theorem 5.15 and (0.7),(0.8) we get F(v, Ω) ≤ inf F(vh , Ω). By (0.7), (0.8) and Theorem 1.16 we get −∞ < (γ − λCn (Ω)) kD2 vkT (Ω) + kc0 kL1 −
γ2 ≤ F(v, Ω) . 4 q.e.d.
Remark 6.1 - The thesis of Theorem 0.6 holds true when assumption (0.8) is substituted by the following one ∀t ∈ R, r ∈ Rn , ∃µ 6= 0 s.t. R R G x, v(x) − µ(t + r · x) = Ω G(x, v(x)) ∀t ∈ R, r ∈ Rn , v ∈ SBH(Ω) , Ω as one can see by inspection of the last proof and referring to the theory of sequential recession functionals (see Def. 2.1 and Th.3.4 in [8])
7. Examples All along the paper we exploited only weak L2 lower semicontinuity and R R positive definiteness in L2 of the terms |∇2 v|2 and |∇v|2 appearing in the functional (0.1). Hence the following statement for a more general functional holds true as an easy corollary of the previous analysis. Theorem 7.1. Consider the functional Z F(v, α, β, γ, σ, A) = Q(∇2 v) + σ Q(∇v) + G(x, v) dx+ A Z (7.1) +αHn−1 (Sv ∩ A) +βHn−1 ((S∇v \ Sv ) ∩ A)+ γ |[∇v]| dH n−1 S∇v ∩A
where both Q and Q are real valued positive definite quadratic forms. Then the thesis of Theorems 0.3, 0.4, 0.5, 0.6 hold true also when functional (7.1) is substituted to functional (0.1). An important example of functional related to image analysis ([9], [36]) is the following, obtained by choosing α > 0, G(x, s) = |g(x) − s|2 , in the
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functional (7.1).
(7.2)
Z
e 2 + σ|∇v|2 + |v − g|2 dx+ |∇2 v|2 + η|∆v| A Z n−1 n−1 +αH (Sv ∩ A) +βH ((S∇v \ Sv ) ∩ A)+ γ |[∇v]| dH n−1 G(v, α, β, γ, σ, A) =
S∇v ∩A
e stands for the trace of ∇2 v. Notice that the map v → Trace∇2 v where ∆v is sequentially weakly continuous with respect to the L2 convergence of ∇2 v, R e 2 dx is sequentially weakly lower semicontinuous with respect to hence η|∆v| 2 the L convergence of ∇2 v. Theorem 7.2. Let (0.2),(0.4),(0.5), η ≥ 0, g ∈ L2 (Ω). Then functional G in (7.2) achieves a finite minimum over GSBV 2 (Ω). Another interesting example arises in continuum mechanics. The following choices in functional (7.1): n = 2, α = 0, σ = 0, G(x, s) = −g(x)s and a quadratic form Q dependent on the Lam´e coefficients λ and µ, lead to the total energy of the Kirchhoff-Love plate with damage at small scale ([39],[40],[35]): Z Z λ |∇2 v|2 + |∆a v|2 dx − gv dx+ λ + 2µ A A Z 1 +βH (S∇v ∩ A) + γ |[∇v]| dH1 .
P(v, 0, β, γ, 0, A) = (7.3)
2 µ 3
S∇v ∩A
In this context A is the natural state of the unloaded elastic-plastic plate ([40]); the first integral expresses the elastic deformation energy; the second one the potential energy due to the transverse dead load g; the last two terms together express the energy associated to damage along free plastic-yield lines. Theorem 7.3. Let µ > 0, 2µ + 3λ > 0, g ∈ M(R2 ), (0.3) and there exists w ∈ SBH(R2 ) s.t. P(w, 0, β, γ, 0, R2 ) is finite and
(7.4)
|g|T (Ω) < 4γ
(safe load condition for clamped plate ) .
Then there is a minimizer u, with finite energy, of P(v, 0, β, γ, 0, R2 ) among v ∈ SBH(R2 ) such that v = w in R2 \ Ω .
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Theorem 7.4. Let µ > 0, 2µ + 3λ > 0, Ω is R regular (Def. 1.5), g ∈ M(Ω), (0.3), there is w ∈ SBH(Ω) s.t. P(w, 0, β, γ, 0, Ω) is finite and Z (7.5)
g v dx = 0
∀ affine displacement v
Ω
(compatibility condition for free plate)
(7.6)
|g|T (Ω)
0, β > 0, γ > 0, Ω ⊂ R open bounded interval, g ∈ M(Ω), and w ∈ SBH(R), assume
(7.7)
|g|T (Ω)
0, β > 0, γ > 0, Ω ⊂ R open bounded interval, g ∈ M(Ω), assume Z g v dx = 0 ∀ affine displacement v (7.8) Ω (compatibility condition for free beam)
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(7.9)
|g|T (Ω)
