A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
Abstract. We obtain an integral representation for the relaxation, in the space of functions of bounded deformation, of the energy Z f (Eu(x))dx Ω
with respect to L1 -convergence. Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu and the function f satisfies linear growth and coercivity conditions.
Keywords : functions of bounded deformation, relaxation, E-quasiconvexity 1991 Mathematics Subject Classification: 35J50, 49J45, 49Q20, 73E99. 2000 Mathematics Subject Classification: 35J50, 49J45, 49Q20, 74C15, 74G65.
1. Introduction The analysis of variational problems that model interesting elastic and magnetic properties exhibited by certain materials often involves minimization and relaxation of nonconvex energies of the type Z (1.1) f (∇u)dx, Ω N
where Ω ⊂ R is an open, bounded set and the density function f satisfies certain growth and coercivity conditions. In each case the function u in (1.1) represents a specific physical entity and belongs to a space which is appropriate to describe the material phenomenon in question. In the context of perfect plasticity the function u represents the displacement field of a body occupying the reference configuration Ω with volume energy density f . In this case u belongs to the space BD of functions of bounded deformation composed of integrable vector-valued functions u for which all components Eij , i, j = 1, . . . , N , of the deformation T tensor Eu := Du+Du are bounded Radon measures. Thus, the study of equilibria leads 2 naturally to questions concerning lower semicontinuity properties and relaxation of Z (1.2) f (Eu(x))dx, Ω
where Eu represents the absolutely continuous part (with respect to the Lebesgue measure) of the symmetrized distributional derivative Eu. Our goal in this paper is to show that the relaxation of the bulk energy (1.2), in the space of functions of bounded deformation, admits an integral representation where a surface energy term is naturally produced. In a forthcoming paper we will consider more general linear operators A acting on Du for which A(Du) is a bounded Radon measure (see also [11]). We remark that a relaxation result has been proved by Braides, Defranceschi and Vitali in [6] in the case where the bulk energy density is of the form kEuk2 + (div u)2 , i.e. f (ξ) := 1
2
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
|ξ|2 + |tr ξ|2 , and the total energy also includes a surface energy term. In our work no convexity assumptions are placed on the volume density f . Precisely, for u ∈ BD(Ω) we consider the energy Z f (Eu(x))dx if u ∈ W 1,1 (Ω, RN ) , F (u, Ω) := Ω +∞ otherwise , and the localized functional Z n o F(u, V ) := inf lim inf f (Eun (x))dx | un ∈ W 1,1 (V, RN ), un → u in L1 (V, RN ) {un }
n→∞
V
defined on the set O(Ω) of all open subsets V of Ω. Assuming that f is continuous and satisfies growth and coercivity conditions of the type 1 |ξ| ≤ f (ξ) ≤ C(1 + |ξ|) C
×N for all ξ ∈ MN sym ,
where C > 0, we show that F(u, ·) is the restriction to O(Ω) of a Radon measure µ which is absolutely continuous with respect to LN + |E s u|, where Eu = EuLN + E s u, and LN stands for the N -dimensional Lebesgue measure. By the Radon-Nikodym Theorem it follows that Z Z F(u, Ω) = µa (x)dx + µs (x)d|E s u|(x) , Ω
Ω
and the question now is to identify the densities µa and µs . The main ingredient of our approach is the blow-up method introduced by Fonseca and M¨ uller [9] which reduces the identification of the energy densities of the relaxed functional to the characterization of F(v, Q) when Q is a unit cube and v is obtained as the blow-up of the function u around a point x0 . Similar ideas have been used in the context of BV -functions (see, for instance [2], [4], [7], [10]) where, in order to identify the density of the absolutely continuous part µa one chooses a point x0 of approximate differentiability of the function u satisfying Z 1 (1.3) lim |u(x) − u(x0 ) − ∇u(x0 )(x − x0 )|dx = 0. ε→0+ εN +1 Q(x0 ,ε) Using the L1 -convergence, as ε → 0+ , of the rescaled functions vε (y) :=
u(x0 + εy) − u(x0 ) ε
to the homogeneous function v0 (y) = ∇u(x0 )y, which is guaranteed by (1.3), it is possible to reduce the identification of the energy density of the absolutely continuous part µa , to a relaxation problem where the target function is homogeneous. However this reasoning cannot be applied to the BD framework where the equivalent of (1.3), replacing ∇u(x0 ) by Eu(x0 ), is, in general, false (see [1]). To overcome this difficulty we will use a Poincar´e-type inequality ku − P ukL1 (Q,RN ) ≤ C|Eu|(Q), where P is the orthogonal projection onto the kernel of the operator E (cf. [12], see also Theorem 3.1 in [1]). This, together with a compact embedding result, will ensure the L1 convergence we need to proceed. We organize the paper as follows: in Section 2 we recall the main properties of the space BD which will be used in the sequel. In Section 3 we state our main theorem, prove a version of De Giorgi’s Slicing Lemma and show some properties of the localized functional F(u, V ). Section 4 is devoted to the characterization of the absolutely continuous density µa , whereas µs is studied in Section 5.
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
3
2. The space of functions of bounded deformation Let Ω be a bounded, open subset of RN with Lipschitz boundary Γ. We denote by B(Ω), O(Ω) and O∞ (Ω) the family of Borel, open and open subsets of Ω with Lipschitz boundary, respectively. We use the standard notation, LN and HN −1 , for the Lebesgue and Hausdorff measures. We recall that the symmetric product between two vectors u and v ∈ RN , u v, is the symmetric N × N matrix defined by u v = (u ⊗ v + v ⊗ u)/2, where ⊗ indicates the tensor product. We denote by B(x, ρ) the ball in RN of centre x and radius ρ, by Q(x, ρ) the cube of centre x and sidelength ρ, while Qν (x, ρ) will be used to indicate the cube with two of its faces perpendicular to the unit vector ν. When x = 0 and ρ = 1 we shall simply write B and Q. Let Q0 := Q ∩ {x ∈ RN | xN = 0}, S N −1 := ∂B and ωN := LN (B). By C we indicate a constant whose value might change from line to line. As usual, Lq (Ω, RN ), W m,q (Ω, RN ) denote Lebesgue and Sobolev spaces, respectively, C0∞ (Ω, RN ) is ∞ the space of RN -valued smooth functions with compact support in Ω, and Cper (Q, RN ) are N the smooth and Q-periodic functions from Q into R . For u ∈ L1 (Ω, RN ) define Ωu as the set of the Lebesgue points of u, i.e. the set of points x ∈ Ω such that there exists u ˜(x) ∈ RN satisfying Z 1 lim N |u(y) − u ˜(x)|dy = 0. ρ→0+ ρ B(x,ρ) The Lebesgue discontinuity set Su of u is defined as the set of points which are not Lebesgue points, that is Su := Ω \ Ωu . By Lebesgue’s Differentiation Theorem, Su is LN -negligible and the function u ˜ : Ω → RN , which coincides with u LN -almost everywhere in Ωu , is called the Lebesgue representative of u. We say that u has one sided Lebesgue limits u+ (x) and u− (x) at x ∈ Ω with respect to a suitable direction νu (x) ∈ S N −1 , if Z 1 lim |u(y) − u± (x)|dy = 0 , ρ→0+ ρN Bρ± (x,νu (x)) where Bρ± (x, νu (x)) := {y ∈ B(x, ρ) | (y − x) · (±νu (x)) > 0}. Then the rescaled functions uρ (y) := u(x + ρy) converge in L1 (B, RN ), as ρ → 0+ , to + u (x) if y · νu (x) > 0, ux,νu (x) (y) := u− (x) if y · νu (x) < 0. When u+ (x) 6= u− (x), the triplet (u+ (x), u− (x), νu (x)) is uniquely determined up to a change of sign of νu (x) and a permutation of (u+ (x), u− (x)). If u+ (x) = u− (x) then x ∈ Ωu and the one sided Lebesgue limits coincide with the Lebesgue representative of u. The jump set Ju of u is defined as the set of points in Ω where the approximate limits u+ , u− exist and are not equal. We use [u](x) to denote the jump of u at x, i.e. [u](x) = u+ (x) − u− (x). Definition 2.1. ([13]) The space of functions of bounded deformation, BD(Ω), is the set of functions u ∈ L1 (Ω, RN ) whose symmetric part of the distributional derivative Du, Eu := Du+DuT , is a matrix-valued bounded Radon measure. We denote by LD(Ω) the subspace of 2 ×N functions u ∈ BD(Ω) for which Eu ∈ L1 (Ω, MN sym ). We recall that Korn’s Inequality holds in certain subspaces of LD(Ω), e.g. for functions ×N u ∈ LD(Ω) such that u ∈ Lp (Ω, RN ), with 1 < p < +∞, and Eu ∈ Lp (Ω, MN sym ); precisely, there exists a constant C = C(Ω) such that Z N Z X p |Di uj (x)| dx ≤ C(Ω) (|u(x)|p + |Eu(x)|p ) dx , i,j=1
Ω
Ω
and thus this subspace of LD(Ω) coincides with W 1,p (Ω, RN ). A counterexample due to Ornstein [14] shows that Korn’s Inequality does not hold for p = 1 and that the space LD(Ω) differs from the Sobolev space W 1,1 (Ω, RN ).
4
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
We recall some properties of functions of bounded deformation that will be used in the sequel. For a more detailed study of BD(Ω) we refer to [1], [5], [12], [16] [17], [18] and [19]. Notice first that the space BD(Ω) endowed with the norm kukBD(Ω) := kukL1 (Ω,RN ) + |Eu|(Ω) is a Banach space. We cannot expect smooth functions to be dense in BD(Ω) with respect to this topology but it can be shown, see for instance [18], that such a density result is true in a weaker topology (cf. Theorem 2.6). Given u, v ∈ BD(Ω) we define the distance d(u, v) := ku − vkL1 (Ω,RN ) + | |Eu|(Ω) − |Ev|(Ω)|, and we denote by intermediate topology the one determined by this distance: a sequence {un }n∈N in BD(Ω) converges to a function u ∈ BD(Ω) with respect to this topology, and we i write un → u, if un → u in L1 (Ω, RN ), ∗ Eun * Eu in the sense of measures, |Eun |(Ω) → |Eu|(Ω). If un → u in L1 (Ω, RN ) and |Eun |(Ω) ≤ C then u ∈ BD(Ω) and |Eu|(Ω) ≤ lim inf |Eun |(Ω). n→∞ Indeed, Eun converges weakly-∗ in the sense of measures to some measure µ, and by the linearity of the operator E, µ = Eu, so that u ∈ BD(Ω). The inequality follows from the lower semicontinuity of the total variation of Radon measures. Proposition 2.2. If u, v ∈ BD(Ω), and ω ⊂ Ω has Lipschitz boundary, then the function w defined by u in ω w= v in Ω \ ω belongs to BD(Ω) and Ew = Euχω + EvχΩ\ω + (v − u) νHN −1 b∂ω, where ν is the outward unit normal to ∂ω. Lemma 2.3. (Lemma 4.5 [3]) Let u ∈ BD(Ω) and let ρ ∈ C0∞ (RN ) beZa non-negative function such that supp (ρ) ⊂⊂ B(0, 1), ρ(−x) = ρ(x) for every x ∈ RN and ρ(x)dx = 1. RN
For any n ∈ N set ρn (x) := nN ρ(nx) and Z 1 un (x) := (u ∗ ρn )(x) = u(y)ρn (x − y)dy, for x ∈ y ∈ Ω | dist (y, ∂Ω) > . n Ω Then un ∈ C ∞ y ∈ Ω | dist (y, ∂Ω) > n1 , RN and i) for any non-negative Borel function h : Ω → R Z Z h(x)|Eun (x)|dx ≤ (h ∗ ρn )(x)d|Eu|(x), 1 B(x0 ,δ) B (x0 ,δ+ n ) whenever δ + n1 < dist (x0 , ∂Ω); ×N ii) for any positively homogeneous of degree one, convex function θ : MN sym → [0, +∞[ and any δ ∈]0, dist (x0 , ∂Ω)[ such that |Eu|(∂B(x0 , δ)) = 0, Z Z d Eu lim θ(Eun (x))dx = θ d|Eu|; n→∞ B(x ,δ) d|Eu| B(x0 ,δ) 0 iii) limn→∞ un (x) = u ˜(x) and limn→∞ (|un − u| ∗ ρn )(x) = 0 for every x ∈ Ω \ Su , whenever u ∈ L∞ (Ω, RN ).
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
5
Proof. i) In the same way as it was shown for the distributional derivative of BV functions, it can be proved that Eun (x) = (Eu ∗ ρn )(x) for every x ∈ Ω with dist(x, ∂Ω) > n1 . Then, by Fubini’s Theorem, we get Z Z Z h(z)|Eun (z)|dz ≤ h(z)ρn (z − y)d|Eu|(y)dz B(x0 ,δ) B(x ,δ) Ω Z Z 0 h(z)ρn (z − y)dzd|Eu|(y) ≤ 1 B (x0 ,δ+ n ) RN Z = (h ∗ ρn )(y)d|Eu|(y). 1 B (x0 ,δ+ n ) ii) Letting h ≡ 1 in part i) and since |Eu|(∂B(x0 , δ)) = 0, we obtain Z Z lim sup |Eun (x)|dx ≤ lim sup d|Eu|(x) 1 n→∞ n→∞ B(x0 ,δ) B (x0 ,δ+ n ) = |Eu|(B(x0 , δ)) Z ≤ lim inf |Eun (x)|dx. n→∞
Thus
B(x0 ,δ)
Z |Eun (x)|dx = |Eu|(B(x0 , δ)),
lim
n→∞
B(x0 ,δ)
∗
and since Eun * Eu in the sense of measures in B(x0 , δ), the result now follows by applying Reshetnyak’s Theorem (see [15]) (note that the hypotheses on θ imply that 0 ≤ θ(w) ≤ Ckwk). The proof of part iii) is carried out exactly as in [3] since it is independent of the operator E. Theorem 2.4. Assume that Ω is a bounded, open subset of RN with Lipschitz boundary Γ. There exists a linear, continuous, and surjective trace operator tr : BD(Ω) → L1 (Γ, RN ) such that tr u = u if u ∈ BD(Ω) ∩ C(Ω, RN ). This operator is also continuous with respect to the intermediate topology. Furthermore, the following Gauss-Green formula holds Z Z Z (2.1) (u Dφ)(x)dx + φ(x)dEu(x) = φ(x)(tr u ν)(x)dHN −1 Ω
Ω
Γ
for every φ ∈ C 1 (Ω). Adapting the proof of Lemma II.2.2 in [18] it is easy to show the following proposition (see also [1] and [12]). Proposition 2.5. If M is a countably (HN −1 , N − 1) rectifiable Borel subset of Ω and u ∈ BD(Ω) then EubM = (u+ − u− ) νM HN −1 bM, where νM is a unit normal to M and u± are the traces of u on the sides of M determined by νM . The following result, proved in [18], asserts that it is possible to approximate a function u ∈ BD(Ω) by a sequence of smooth functions which preserve the trace of u. Theorem 2.6. For every u ∈ BD(Ω) there exists a sequence of smooth functions {un }n∈N ⊂ i C ∞ (Ω, RN ) ∩ W 1,1 (Ω, RN ) such that un → u and tr un = tr u. If, in addition, u ∈ LD(Ω), then kEun − EukL1 (Ω,MN → 0. ×N sym )
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ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
A Poincar´e-type inequality holds, precisely Theorem 2.7. There exists a constant C = C(Ω) such that for every u ∈ BD(Ω) with tr u = 0 kukL1 (Ω,RN ) ≤ C(Ω)|Eu|(Ω).
(2.2)
For the proof see [18], Remark II.2.5. Note that the kernel of the operator E is the class R of rigid motions in RN , composed of affine maps of the form M x + b where M is a skew-symmetric N × N matrix and b ∈ RN , and therefore is closed and finite-dimensional. Hence it is possible to define the orthogonal projection P : BD(Ω) → R. This operator belongs to the class considered in the following Poincar´e-Friedrichs type inequality for BD functions (see [1], [12] and [18]). Theorem 2.8. Let Ω be a bounded, connected, open set with Lipschitz boundary, and let R : BD(Ω) → R be a continuous linear map which leaves the elements of R fixed. Then there exists a constant C(Ω, R) such that Z (2.3) |u − R(u)|dx ≤ C(Ω, R)|Eu|(Ω) for every u ∈ BD(Ω). Ω
Remark 2.9. Let C(Ω) denote the smallest constant for which (2.2), (2.3) hold. Then, as usual, a simple homothety argument shows that C(λΩ) = λC(Ω). The following embedding result proved in [18], Theorem II.2.4, will be used in Sections 4 and 5 to obtain strong L1 convergence of a sequence which is uniformly bounded in BD(Ω). Theorem 2.10. The space BD(Ω) is compactly embedded in Lq (Ω, RN ) for every 1 ≤ q < N N −1 . In particular, if {un } is bounded in BD(Ω) then there exists a subsequence {unk } such that unk → u in L1 (Ω, RN ) and u ∈ BD(Ω). The next theorem was proved in [1] (see also [12]). Theorem 2.11. For every u ∈ BD(Ω) the jump set Ju is a countably (HN −1 , N − 1) rectifiable Borel set. This result, together with Proposition 2.5, yields the following decomposition of the Radon measure Eu (see Definition 4.1 and Remark 4.2 in [1]): (2.4)
Eu = EuLN + E s u = EuLN + ([u] νu )HN −1 bJu + E c u,
where Eu is the density of the absolutely continuous part of Eu with respect to LN , E s u is the singular part, E c u is the so-called Cantor part and vanishes on Borel sets B with HN −1 (B) < +∞ (see Proposition 4.4 in [1]). Definition 2.12. (cf. Definition 4.6 in [1]) The space of special functions of bounded deformation, denoted by SBD(Ω), is the set of functions u ∈ BD(Ω) such that the measure E c u in (2.4) is zero. The next lemma will be used in Section 5 to identify the surface term of the relaxed energy F(u, Ω). Lemma 2.13. (Lemma 2.6 [10]) Let u ∈ BD(Ω). Then for HN −1 -a.e. x0 ∈ Ju Z 1 |([u] νu )(x)|dHN −1 (x) = |([u] νu )(x0 )|. (2.5) lim ε→0 εN −1 J ∩(x +εQ u 0 ν(x ) ) 0
Its proof is based on the definition of rectifiable set and on the version of Lebesgue’s Differentiation Theorem proved by Ambrosio and Dal Maso [2] (see [10] for details).
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
7
3. Statement of the main result ×N Let f : MN sym → R be a continuous function satisfying
1 ×N |ξ| ≤ f (ξ) ≤ C(1 + |ξ|) for all ξ ∈ MN sym , C and for some C > 0. Assume also that there exist Λ, L > 0, 0 ≤ β < 1 such that ∞ f (tξ) |ξ|1−β ×N ≤Λ β for all ξ ∈ MN and for all t such that t|ξ| > L , (3.2) f (ξ) − sym t t where the recession function, f ∞ , of f is defined as f (tξ) f ∞ (ξ) := lim sup . t t→∞ (3.1)
×N Definition 3.1. The function f : MN → R is said to be E-quasiconvex if, for every sym ×N ξ ∈ MN , sym Z (3.3) f (ξ) ≤ f (ξ + Eu(x))dx Q
whenever u ∈
∞ Cper (Q, RN ).
Note that the E-quasiconvexity property (3.3) is independent of the size, orientation, and centre of the cube Q. Moreover, it is equivalent to the following property: for every ×N ξ ∈ MN sym , Z f (ξ)LN (Ω) ≤
f (ξ + Ev(x))dx Ω
whenever v ∈ C0∞ (Ω, RN ). ×N Given a function f : MN sym → R we define its E-quasiconvex envelope QE f by Z ∞ (3.4) QE f (ξ) := inf f (ξ + Eu(x))dx : u ∈ Cper (Q, RN ) . Q
As it was proven for the usual quasiconvex envelope (see [8] Chapter 5) it is possible to show that QE f is the greatest E-quasiconvex function that is less than or equal to f . Moreover, the definition (3.4) does not depend on the domain, i.e. Z 1 ∞ N QE f (ξ) = inf f (ξ + Eu(x))dx : u ∈ C (Ω, R ) . 0 LN (Ω) Ω Remark 3.2. If f is upper semicontinuous and locally bounded from above, it is easy to see, using Theorem 2.6 and Fatou’s Lemma, that in the definition of E-quasiconvexity ∞ Cper (Q, RN ) may be replaced by u ∈ LDper (Q). ×N Remark 3.3. Let f : MN sym → R satisfy the growth and coercivity conditions (3.1). Then QE f also satisfies (3.1) and 1 |ξ| ≤ f ∞ (ξ) ≤ C|ξ|. C Moreover, if f is E-quasiconvex then so is f ∞ . The first statement is easily verified. As for the second one, we have for any u ∈ ∞ Cper (Q, RN ) Z f (tξ) 1 ∞ f (ξ) = lim sup ≤ lim f (tk ξ + tk Eu(x))dx k→∞ tk Q t t→∞ Z 1 + tk |ξ + Eu(x)| ≤ C lim dx − k→∞ Q tk Z C(1 + tk |ξ + Eu(x)|) 1 − lim inf − f (tk ξ + tk Eu(x))dx . k→∞ tk tk Q
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ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
By the growth condition (3.1), we may apply Fatou’s Lemma to obtain Z Z 1 ∞ f ∞ (ξ + Eu(x))dx. f (ξ) ≤ lim sup f (tk (ξ + Eu(x)))dx ≤ Q Q k→∞ tk The following lemma will be used in Section 4. Lemma 3.4. Let g : Rd → R be a continuous function satisfying |g(v)| ≤ C(1+|v|) for some constant C > 0. If un → u0 in L1 (Ω, Rd ), with u0 ∈ L∞ (Ω, Rd ), and supn kvn k∞ < +∞, then Z |g(un + vn ) − g(u0 + vn )|dx = 0.
lim
n→+∞
Ω
Proof. We divide the given integral into two terms Z J := |g(un + vn ) − g(u0 + vn )|dx , 1 Z{x∈Ω : |un (x)−u0 (x)|>1} |g(un + vn ) − g(u0 + vn )|dx. J2 := {x∈Ω : |un (x)−u0 (x)|≤1}
The growth condition on g, together with the hypotheses on un and vn , ensure that the sequence {|g(un + vn )(·) − g(u0 + vn )(·)|}n∈N is equi-integrable. Thus, since un → u0 strongly in L1 (Ω, Rd ), the measure of the set {x ∈ Ω : |un (x) − u0 (x)| > 1} can be made arbitrarily small and thus for any ε > 0 there exists an n0 such that J1 < ε/2 for every n ≥ n0 . On the other hand, using the uniform continuity of g on the closed ball of radius ku0 k∞ + 1 + supn kvn k∞ centered in 0, it follows that J2 < ε/2 for every n ≥ n0 . Let u ∈ BD(Ω) and define F (u, Ω) :=
Z
f (Eu(x))dx
if u ∈ W 1,1 (Ω, RN ),
Ω
+∞
otherwise.
We consider the functional defined for every V ∈ O(Ω) by Z 1,1 N 1 N F(u, V ) := inf lim inf f (Eun (x))dx | un ∈ W (V, R ), un → u in L (V, R ) . {un }
n→∞
V
The main result of this paper is the following integral representation for the relaxed energy F(u, Ω) of F (u, Ω) when u ∈ SBD(Ω) (see also Proposition 3.9). ×N Theorem 3.5. If f : MN sym → R is a continuous function satisfying (3.1) and (3.2), then for every u ∈ SBD(Ω) we have Z Z F(u, V ) = QE f (Eu(x))dx + (QE f )∞ (([u] νu )(x))dHN −1 (x) , V
Ju ∩V
for every open subset V of Ω. Remark 3.6. This integral representation does not provide a full description of the functional since we consider only the case where the Cantor part of the deformation tensor is zero. To extend this representation result to the whole space BD a more complete characterization of the Cantor part of the measure Eu is needed. In this section we shall establish some properties of the relaxed functional, while the remaining sections will be devoted to the characterization of the volume and surface terms, respectively. We shall use the following version of De Giorgi’s Slicing Lemma (see also [4] and [7]) which allows us to modify a sequence on the boundary without increasing the energy.
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
9
×N Proposition 3.7. Let V ⊂ RN be an open, bounded set and let f : MN → R satisfy sym the growth condition (3.1). Let {un }, {vn } be sequences of functions in BD(V ) such that ∗ un − vn → 0 in L1 (V, RN ), supn |Eun |(V ) < +∞ and |Evn | * µ, |Evn |(V ) → µ(V ). Then there exist a subsequence {vnk } and a sequence {wk } ⊂ BD(V ) such that wk = vnk near the boundary of V , wk − vnk → 0 in L1 (V, RN ), and
Z
Z f (Ewk (x))dx ≤ lim inf
lim sup k→∞
f (Eun (x))dx.
n→∞
V
V
Proof. Without loss of generality we may assume that Z Z lim inf f (Eun (x))dx = lim f (Eun (x))dx < +∞. n→∞
n→∞
V
V
Then, by (3.1), there exists a bounded Radon measure λ such that ∗
|Eun | + |Evn | * λ. Choose rk % ∞ as k → ∞ and εkj & 0 as j → ∞ such that n λ x ∈ V | dist (x, ∂V ) = n µ x ∈ V | dist (x, ∂V ) = n λ x ∈ V | dist (x, ∂V ) =
(3.5)
1 rk
o
= 0, o =0 o 1 = 0. ±εk
1 rk +εk j rk
j
Let φk,j be a smooth cut-off function such that 0 ≤ φk,j ≤ 1, φk,j (x) = 1 if x ∈ V and 1 1 dist (x, ∂V ) > r −ε k , and φk,j (x) = 0 if x ∈ V and dist (x, ∂V ) < r +εk , and define k
k
j
j
wn,k,j := φk,j un + (1 − φk,j )vn . Then wn,k,j ∈ BD(V ), wn,k,j = vn near the boundary of V , and kwn,k,j − vn kL1 (V,RN ) ≤ kun − vn kL1 (V,RN ) . On the other hand, as Ewn,k,j = φk,j Eun + (1 − φk,j )Evn + (un − vn ) Dφk,j , from the growth condition on f it follows that Z f (Ewn,k,j (x))dx V Z Z Z ≤ f (Eun (x))dx + f (Evn (x))dx + f (Ewn,k,j (x))dx V
Vk,j
Lk,j
Z ≤
f (Eun (x))dx + CLN (Vk,j ) + C|Evn | (Vk,j ) +C LN (Lk,j ) + (|Eun | + |Evn |)(Lk,j ) + kDφk,j k∞ kun − vn kL1 (V,RN ) V
where (
Lk,j Vk,j
1 1 := x∈V | < dist (x, ∂V ) < k rk + ε j rk − εkj ( ) 1 := x ∈ V | dist (x, ∂V ) < . rk + εkj
) ,
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ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
Thus, by the strong convergence of un − vn to zero in L1 (V, RN ) and (3.5), Z lim sup lim sup lim sup f (Ewn,k,j (x))dx n→∞ j→∞ k→∞ V Z ≤ lim f (Eun (x))dx + C lim sup lim sup LN V˜k,j n→∞
k→∞
V
j→∞
+C lim sup lim sup µ (Vk,j ) + C lim sup lim sup λ(Lk,j ) j→∞ j→∞ k→∞ k→∞ Z ≤ lim f (Eun (x))dx, n→∞
V
where
( V˜k,j :=
1 x ∈ V | dist (x, ∂V ) < rk − εkj
) .
By a standard diagonal argument, we may choose a subsequence nk → ∞ and a sequence εkj(k) → 0 such that wk := wnk ,k,εk satisfies the required properties. j(k)
Remark 3.8. If the sequences {un }, {vn } are more regular, e.g. if un , vn ∈ W 1,1 (V, RN ) or un , vn ∈ C ∞ (V, RN ), then so is the sequence {wk }. The localized relaxed functional has the following properties Proposition 3.9. Under hypotheses (3.1) and (3.2) (1) F(·, V ) is L1 (V, RN ) lower semicontinuous; (2) for every u ∈ BD(Ω), 1 (3.6) |Eu|(V ) ≤ F(u, V ) ≤ C(LN (V ) + |Eu|(V )); C (3) F(u, ·) is the restriction to O(Ω) of a Radon measure. We use the same notation for F(u, ·) and its extension to the Borel subsets of Ω. Proof. The proof of 1) follows from a standard diagonal argument, whereas 2) is an immediate consequence of the growth conditions on f , of the lower semicontinuity of the total variation of Radon measures and of Theorem 2.6. To prove 3), we begin by showing that for every u ∈ BD(Ω) and every U , V , W ∈ O(Ω) with W ⊂⊂ V ⊂⊂ U F(u, U ) ≤ F(u, V ) + F(u, U \ W ).
(3.7) 1,1
Indeed, let vn ∈ W (V, R ) and wn ∈ W 1,1 (U \W , RN ) be such that vn → u in L1 (V, RN ), wn → u in L1 (U \ W , RN ), Z Z F(u, V ) = lim f (Evn (x))dx and F(u, U \ W ) = lim f (Ewn (x))dx . n→∞
N
n→∞
V
U \W
Let V0 ∈ O∞ (Ω) be such that W ⊂⊂ V0 ⊂⊂ V and |Eu|(∂V0 ) = 0. Applying Proposition 3.7 and Remark 3.8 to {vn } and u in V0 , we obtain a sequence {v n } ⊂ W 1,1 (V0 , RN ) such that v n = u near the boundary of V0 , v n → u in L1 (V0 , RN ) and Z Z lim sup f (Ev n (x))dx ≤ lim inf f (Evn (x))dx. n→∞
V0
n→∞
V0
By Proposition 3.7 and Remark 3.8, now applied to {wn } and u in U \ V 0 , there exists a sequence {wn } ⊂ W 1,1 (U \ V 0 , RN ) such that wn = u in a neighbourhood of ∂V0 , wn → u in L1 (U \ V 0 , RN ) and Z Z lim sup f (Ewn (x))dx ≤ lim inf f (Ewn (x))dx. n→∞
U \V 0
n→∞
U \V 0
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
Define
zn :=
vn wn
11
in V0 , in U \ V0 .
Then zn ∈ W 1,1 (U, RN ), zn → u in L1 (U, RN ), and Z F(u, U ) ≤ lim inf f (Ezn (x))dx n→∞ ZU Z ≤ lim sup f (Ev n (x))dx + lim sup f (Ewn (x))dx n→∞ n→∞ U \V 0 V Z 0 Z ≤ lim inf f (Evn (x))dx + lim inf f (Ewn (x))dx n→∞ n→∞ V U \V 0 Z 0 Z ≤ lim inf f (Evn (x))dx + lim inf f (Ewn (x))dx n→∞
n→∞
V
U \W
= F(u, V ) + F(u, U \ W ). Now let un ∈ W (3.8)
1,1
(Ω, RN ) be such that un → u in L1 (Ω, RN ) and Z F(u, Ω) = lim f (Eun (x))dx. n→∞
Ω
1
Since {f (Eun )(·)}n∈N is bounded in L (Ω), it follows that there exists a bounded Radon ∗ measure µ on Ω such that for a subsequence (not relabelled) f (Eun )χΩ LN * µ in the sense of measures and so F(u, Ω) = µ(Ω).
(3.9)
By the definition of F(u, ·), for every U ∈ O(Ω) Z (3.10) F(u, U ) ≤ lim inf f (Eun (x))dx ≤ µ(U ). n→∞
U
Let now V ∈ O(Ω) and ε > 0 be fixed, and consider W ∈ O(Ω) with W ⊂⊂ V and µ(V \ W ) < ε. Then µ(V ) ≤ µ(W ) + ε = µ(Ω) − µ(Ω \ W ) + ε. By (3.9), applying (3.10) with U = Ω \ W , and then (3.7) for U = Ω, we get µ(V ) ≤ F(u, Ω) − µ(Ω \ W ) + ε ≤ F(u, Ω) − F(u, Ω \ W ) + ε ≤ F(u, V ) + ε. Letting ε → 0+ we obtain µ(V ) ≤ F(u, V ) ≤ µ(V ) for every V ∈ O(Ω). It remains to show that F(u, U ) ≤ µ(U ) for every U ∈ O(Ω). In order to do this, we fix again an ε > 0 and choose V, W ∈ O(Ω) such that W ⊂⊂ V ⊂⊂ U and LN (U \ W ) + |Eu|(U \ W ) < ε. By (3.6), (3.7), and (3.10), we get F(u, U ) ≤
F(u, U \ W ) + F(u, V )
≤ C(LN (U \ W ) + |Eu|(U \ W )) + F(u, V ) ≤ ε + µ(V ) ≤ ε + µ(U ). To conclude it suffices to let ε → 0+ .
In the sequel, given u ∈ BD(Ω) and V ∈ O∞ (Ω) we use the notation tr u or u|∂V to indicate the trace of u on the boundary of V . For any u ∈ BD(Ω) and V ∈ O∞ (Ω) define n o m(u, V ) := inf F(v, V ) | v|∂V = u|∂V , v ∈ BD(Ω) . In order to identify the surface term of the relaxed energy we will follow the general method for relaxation introduced by Bouchitt´e, Fonseca and Mascarenhas [7]. The idea is to compare the asymptotic behaviours of m(u, Q(x0 , ε)) and F(u, Q(x0 , ε)) as ε → 0+ , and to show, via a blow-up argument, that relaxation reduces to solving a Dirichlet problem (see Lemma 3.12).
12
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
The following three lemmas are entirely similar to Lemmas 3.1, 3.3 and 3.5 in [7]; for completeness of the presentation we include their proofs here. Lemma 3.10. There exists a positive constant C such that for any u1 , u2 ∈ BD(Ω) and any V ∈ O∞ (Ω) we have Z |m(u1 , V ) − m(u2 , V )| ≤ C |tr (u1 − u2 )(x)|dHN −1 (x). ∂V
Proof. Given δ > 0 consider the set Vδ := {x ∈ V | dist (x, ∂V ) > δ} and let v ∈ BD(Ω) be such that v = u2 on ∂V . Define vδ ∈ BD(Ω) by vδ := v in Vδ and vδ := u1 in Ω \ Vδ . Then, by the definition of m, and by virtue of the additivity and locality of F, (3.11)
m(u1 , V ) ≤ F(vδ , V ) = F(v, Vδ ) + F(vδ , V \ Vδ ).
By (3.6), F(vδ , V \ Vδ ) ≤ C LN (V \ Vδ ) + |Evδ |(V \ Vδ ) (3.12)
= C LN (V \ Vδ ) + |Eu1 |(V \ Vδ ) + |E s vδ |(∂Vδ ) .
The first two terms in (3.12) tend to zero as δ → 0, and as for the third one we have Z |E s (Dvδ )|(∂Vδ ) = |((v|∂Vδ − u1|∂Vδ ) νδ )(x)|dHN −1 (x) ∂V Z δ (3.13) → |((tr u2 − tr u1 ) ν)(x)|dHN −1 (x) as δ → 0. ∂V
Therefore from (3.11), (3.12) and (3.13) it follows that Z m(u1 , V ) ≤ F(v, V ) + C |(tr (u2 − u1 ) ν)(x)|dHN −1 (x). ∂V
Taking the infimum over v and using the fact that |(tr (u2 − u1 ) ν| ≤ C|tr (u2 − u1 )|, we obtain Z m(u1 , V ) − m(u2 , V ) ≤ C |tr (u1 − u2 )(x)|dHN −1 (x). ∂V
Interchanging the roles of u1 and u2 the proof is concluded.
As in [7], fix u ∈ BD(Ω), ν ∈ S N −1 , and define µ := LN + |E s u|. Let O∗ := {Qν (x, ε) | x ∈ Ω, ν ∈ S N −1 , ε > 0} , and for δ > 0, V ∈ O(Ω), set (∞ X mδ (u, V ) := inf m(u, Qi )
: Qi ∈ O∗ , Qi ∩ Qj = ∅, Qi ⊂ V,
i=1
) diam (Qi ) < δ, µ(V \
∪∞ i=1 Qi )
=0 .
Since δ → mδ (u, V ) is a decreasing function, we define n o m∗ (u, V ) := sup mδ (u, V ) : δ > 0 = lim mδ (u, V ). δ→0+
Lemma 3.11. (Lemma 3.3 [7]) Let u ∈ BD(Ω). Under conditions 1), 2), 3) of Proposition 3.9, for any V ∈ O(Ω) F(u, V ) = m∗ (u, V ).
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
13
Proof. Since F(u, ·) is a Radon measure and m(u, V ) ≤ F(u, V ), the inequality m∗ (u, V ) ≤ F(u, V ) is clear. To show the reverse inequality, fix δ > 0 and let (Qδi ) be an admissible sequence for mδ (u, V ) such that ∞ X
(3.14)
m(u, Qδi ) < mδ (u, V ) + δ.
i=1
Let now viδ ∈ BD(Ω) be such that viδ |∂Qδ = u|∂Qδi and i
F(viδ , Qδi ) ≤ m(u, Qδi ) + δLN (Qδi ).
(3.15) Define
∞ X
δ
v :=
viδ χQδi + uχN0δ ,
i=1
= Ω\ where Clearly v ∈ L1 (Ω, RN ). On the other hand, given φ ∈ C0∞ (Ω), using the fact that v − u = 0 in N0δ , and integrating by parts (see (2.1)) over the cubes Qδi we get * ! + ∞ X δ δ < E(v − u), φ > = E χQδi (v − u) , φ N0δ
δ ∪∞ i=1 Qi . δ
δ
i=1 ∞ D E X = E χQδi (v δ − u) , φ i=1
=
∞ Z X ∂Ω
i=1
−
φtr (χQδi (v δ − u)) νdHN −1
∞ Z X i=1
= − = −
∞ Z X
δ
v Dφdx +
δ i=1 Qi Z ∞ X
∂Qδi
i=1
+ ∞ X
φ(x)tr viδ
∞ Z X i=1
=
Ω
χQδi (v δ − u) Dφdx
∂Qδi
∞ Z X i=1
Qδi
νdH
N −1
u Dφdx +
∞ Z X Qδi
i=1
φ(x)tr viδ νdHN −1 −
φdEviδ
∞ Z X i=1
φdEu
Qδi
Eviδ bQδi , φ − Eub(Ω \ N0δ ), φ .
i=1
Hence (3.16)
δ
Ev =
∞ X
Eviδ bQδi + EubN0δ
i=1
in the sense of distributions, and due to the lower bound in (3.6) and (3.14), (3.15), the right-hand side is a bounded Radon measure on Ω, and we conclude that v δ ∈ BD(Ω). δ δ By the choice of Qδi , setting N δ := V \ ∪∞ i=1 Qi we have µ(N ) = 0, and by (3.16) δ δ δ δ |Ev |(N ) = |Eu|(N ) ≤ µ(N ) = 0. Thus, by (3.6), F(v δ , N δ ) ≤ C(LN (N δ ) + |Ev δ |(N δ )) = 0.
14
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
Since F(v δ , ·) is the restriction of a measure, by (3.14) and (3.15) it follows that F(v δ , V )
∞ X
=
F(viδ , Qδi ) + F(v δ , N δ )
i=1 ∞ X
≤
(3.17)
m(u, Qδi ) + δLN (V ) ≤ mδ (u, V ) + δ + δLN (V ).
i=1
The result is now a consequence of the lower semicontinuity of F(·, V ) provided we can show that v δ converges to u strongly in L1 (V, RN ). Indeed, if this is the case, then F(u, V ) ≤ lim inf F(v δ , V ) ≤ lim inf mδ (u, V ) = m∗ (u, V ). δ→0
viδ
δ→0
∂Qδi ,
Since = u on by Poincar´e’s Inequality (2.2) and Remark 2.9, there exists a constant C > 0 such that kviδ − ukL1 (Qδi ,RN ) ≤ Cδ|E(viδ − u)|(Qδi ). Hence kv δ − ukL1 (V,RN )
=
∞ X
δ kviδ − ukL1 (Qδi ,RN ) ≤ Cδ|E(viδ − u)|(∪∞ i=1 Qi )
i=1
≤ Cδ(|Ev δ |(V ) + |Eu|(V )). δ Since the sequence condition (3.6) and (3.17), it δ {|Ev |(V )}δ is bounded by1 the coercivity follows that v converges to u strongly in L (V, RN ).
Lemma 3.12. (Lemma 3.5 [7]) If F satisfies conditions 1)-3) of Proposition 3.9 then lim
ε→0
F(u, Qν (x0 , ε)) m(u, Qν (x0 , ε)) = lim µ a.e. x0 ∈ Ω and for all ν ∈ S N −1 . ε→0 µ(Qν (x0 , ε)) µ(Qν (x0 , ε))
The proof is the same as in [7] since it does not depend on whether we are considering the distributional derivative D or its symmetrized part E. Remark 3.13. The conclusion of this lemma still holds replacing Qν (x0 , ε) by U (x0 , ε) := x0 + εU , where U is any bounded, open, convex subset of RN containing the origin. The following result shows that in the definition of the Dirichlet functional m one may consider more regular functions. Lemma 3.14. If the function f satisfies (3.1), then for every u ∈ BD(Ω) and every V ∈ O∞ (Ω) n o m(u, V ) = m0 (u, V ) := inf F (v, V ) : v = u on ∂V, v ∈ W 1,1 (Ω, RN ) . Proof. Clearly m0 (u, V ) ≥ m(u, V ). To show the reverse inequality, fix ε > 0 and let v ∈ BD(Ω) be such that v = u on ∂V and m(u, V ) ≥ F(v, V ) − ε. Consider a sequence un ∈ W 1,1 (Ω, RN ) such that un → v in L1 (Ω, RN ), Z F(v, V ) = lim f (Eun (x))dx , n→∞
V
1,1
and, using Theorem 2.6, let vn ∈ W (Ω, RN ) satisfy vn → v in L1 (Ω, RN ), vn = v on ∂V − thus vn = u on ∂V − and |Evn |(V ) → |Ev|(V ). It suffices now to apply Proposition 3.7 to {un } and {vn }, and, in light of Remark 3.8, there exists a sequence {wk } of functions in W 1,1 (Ω, RN ) such that wk = u on ∂V and Z Z m0 (u, V ) ≤ lim sup f (Ewk (x))dx ≤ lim inf f (Eun (x))dx ≤ m(u, V ) + ε, k→∞
n→∞
V +
and we conclude by letting ε & 0 .
V
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
15
4. The volume term Proposition 4.1. If u ∈ BD(Ω) then Z F(u, Ω) ≥
QE f (Eu(x))dx, Ω
where Eu = EuLN + E s u. Proof. Let un ∈ W 1,1 (Ω, RN ) be such that un → u in L1 (Ω, RN ) and Z F(u, Ω) = lim f (Eun (x))dx < +∞. n→∞
Ω
By passing to a subsequence (not relabelled), there exists a finite non-negative Radon measure λ such that ∗ f (Eun )LN * λ in the sense of measures. Thus, it suffices to show that for LN -a.e. x0 ∈ Ω dλ (x0 ) ≥ QE f (Eu(x0 )). dLN Consider a point x0 ∈ Ω such that (4.1) (4.2) (4.3)
dλ λ(Q(x0 , ε)) (x0 ) = lim exists and is finite, dLN Z ε→0 εN 1 lim |Eu(x) − Eu(x0 )|dx = 0, ε→0 εN Q(x ,ε) 0 1 |E s u|(Q(x0 , ε)) = 0, lim ε→0 εN
where the sequence of ε & 0+ is chosen such that λ(∂Q(x0 , ε)) = 0. It is well-known that properties (4.1)-(4.3) hold for LN -almost every x0 ∈ Ω. Then Z dλ 1 (x0 ) = lim lim N f (Eun (x))dx ε→0 n→∞ ε dLN Q(x0 ,ε) Z (4.4) = lim lim f (Eun,ε (y))dy ε→0 n→∞
Q
where un (x0 + εy) − u(x0 ) . ε By (4.1), (4.4) and the coercivity hypothesis on f , we obtain un,ε (y) :=
sup |Eun,ε |(Q) < +∞
(4.5)
n,ε
and so, applying Theorem 2.8, kun,ε − P un,ε kL1 (Q,RN ) ≤ C, where P is the projection of BD(Q) onto the kernel of the operator E. Theorem 2.10, together with (4.5) and the fact that EP un,ε = 0, implies the existence of a function v ∈ L1 (Q, RN ) such that (4.6)
lim lim kun,ε − P un,ε − vkL1 (Q,RN ) = 0,
ε→0 n→∞
and this, in turn, entails (4.7)
∗
Eun,ε * Ev in the sense of distributions.
16
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
On the other hand, given φ ∈ C0 (Q), from the strong convergence in L1 of un to u and the convergence of Eun to Eu in the sense of measures, we deduce that Z Z lim lim φ(y)Eun,ε (y)dy = lim lim φ(y)Eun (x0 + εy)dy ε→0 n→∞ Q ε→0 n→∞ Q Z 1 x − x0 φ Eun (x)dx = lim lim N ε→0 n→∞ ε ε Q(x0 ,ε) Z 1 x − x0 = lim N φ dEu(x) ε→0 ε ε Q(x0 ,ε) Z 1 x − x0 = lim N φ Eu(x)dx ε→0 ε ε Q(x0 ,ε) Z 1 x − x0 + lim N φ dE s u(x) ε→0 ε ε Q(x0 ,ε) Z = lim φ(y) [Eu(x0 + εy) − Eu(x0 )] dy ε→0 Q Z + φ(y)Eu(x0 )dy Q Z 1 x − x0 + lim N φ dE s u(x) ε→0 ε ε Q(x0 ,ε) Z = φ(y)Eu(x0 )dy Q ∗
by (4.2) and (4.3). We have thus showed that Eun,ε * Eu(x0 )LN in the sense of measures, which, together with (4.7), yields (4.8)
Ev = Eu(x0 )LN .
Extracting a diagonal subsequence {um } of {un,ε − P un,ε }, according to (4.6), (4.7) and (4.8), we then have, by (4.4) Z dλ (x ) ≥ lim f (Eum (x))dx, um → v in L1 (Q, RN ). 0 m→∞ Q dLN By virtue of Proposition 3.7, without loss of generality, we may assume that um = v on ∂Q. Hence by the definition of QE f , Remark 3.2 and (4.8), we have Z dλ (x0 ) ≥ lim f (Eum (x))dx m→∞ Q dLN Z = lim inf f (Eu(x0 ) + E(um − v)(x))dx m→∞
Q
≥ QE f (Eu(x0 )). Proposition 4.2. Let u ∈ BD(Ω). Then for LN -almost every x0 ∈ Ω dF(u, ·) (x0 ) ≤ QE f (Eu(x0 )). dLN Proof. Choose a point x0 ∈ Ω such that (4.2) and (4.3) are satisfied, and dF(u, ·) F(u, Q(x0 , ε)) (x0 ) = lim exists and is finite, N ε→0 dL εN where the sequence of ε & 0+ is chosen such that |Eu|(∂Q(x0 , ε)) = 0.
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
17
∞ Let φn ∈ Cper (Q, RN ) be such that Z 1 (4.9) f (Eu(x0 ) + Eφn (x))dx ≤ QE f (Eu(x0 )) + . n Q
Extend φn to RN by periodicity and define 1 φn (mx). m
φm n (x) :=
Then Eφm n (x) = Eφn (mx) so, Z ×N (4.10) Eφm * Eφn (x)dx = 0 in L1 (Q, MN n sym ) as m → ∞ Q
and |Eφm n |(Q)
Z =
|Eφn (mx)|dx Z 1 |Eφn (y)|dy mN mQ Q
=
= |Eφn |(Q). By (4.9) and the coercivity hypothesis on f , it follows that |Eφn |(Q) is uniformly bounded. Applying Theorem 2.8, Theorem 2.10 and the argument in the previous proof, passing to a diagonal subsequence, we obtain φm(n) − P φm(n) → φ in L1 (Q, RN ) n n m(n) ∗
and Eφn * Eφ in the sense of measures, which, together with (4.10), yields Eφ = 0 and thus φ is a rigid motion. Define m(n) φn (x) := φm(n) (x) − P φn (x), n x − x0 x − x0 ε un (x) := (ρn ∗ u)(x) + ε φn −φ . ε ε Then, as n → ∞, uεn → u in L1 (Q(x0 , ε), RN ) so that uεn ∈ W 1,1 (Q(x0 , ε), RN ) is admissible for F(u, Q(x0 , ε)). Hence we obtain, taking into account that Eφ = 0, F(u, Q(x0 , ε)) dF(u, ·) (x0 ) = lim ε→0 dLN εN Z 1 x − x0 ≤ lim sup lim inf N f E(u ∗ ρn )(x) + Eφn dx n→∞ ε ε ε→0 Q(x0 ,ε) Z 1 x − x0 ≤ lim sup lim inf N f Eu(x0 ) + Eφn dx n→∞ ε ε ε→0 Q(x0 ,ε) Z 1 x − x0 + lim sup lim inf N f E(u ∗ ρn )(x) + Eφn n→∞ ε ε ε→0 Q(x0 ,ε) x − x0 −f Eu(x0 ) + Eφn dx ε =: I1 + I2 where, by changing variables and using the periodicity of φn and (4.9), Z I1 = lim inf f (Eu(x0 ) + Eφn (y))dy = QE f (Eu(x0 )). n→∞
Q
On the other hand, by Lemma 2.3 and setting u0 (x) := Eu(x0 )x, Z 1 lim sup lim inf N |E(u ∗ ρn )(x) − Eu(x0 )|dx n→∞ ε ε→0 Q(x0 ,ε)
18
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
Z 1 |E((u − u0 ) ∗ ρn )(x)|dx n→∞ εN Q(x ,ε) ε→0 0 Z 1 ≤ lim sup lim inf N d|E(u − u0 )|(x) n→∞ ε 1 ε→0 Q(x0 ,ε+ n ) Z 1 |Eu(x) − Eu(x0 )|dx ≤ lim sup lim inf N n→∞ ε 1 ε→0 Q(x0 ,ε+ n ) Z 1 + lim sup lim inf N d|E s u|(x) n→∞ ε 1 ε→0 Q(x0 ,ε+ n ) Z 1 |Eu(x) − Eu(x0 )|dx = lim sup N ε→0 ε Q(x0 ,ε) 1 + lim sup N |E s u|(Q(x0 , ε)) = 0 ε ε→0 =
lim sup lim inf
where we have used (4.2), the fact that ε & 0+ were chosen such that |Eu|(∂Q(x0 , ε)) = 0, and (4.3). This, together with Lemma 3.4 proves that I2 = 0. 5. The surface term Proposition 5.1. Let u ∈ SBD(Ω). Then, under hypotheses (3.1), (3.2), Z F(u, V ∩ Ju ) = g(x, u+ (x), u− (x), νu (x))dHN −1 (x) for all V ∈ O(Ω), V ∩Ju
m(uλ,θ,ν , Qν (x, ε)) where g(x, λ, θ, ν) := lim sup εN −1 ε→0 λ if y · ν > 0 and uλ,θ,ν (y) = θ if y · ν < 0. Proof. Since u+ , u− are Borel functions and Ju is rectifiable, writing ν := νu (x0 ), for HN −1 almost every x0 ∈ Ju Z 1 (5.1) lim |u(x) − u+ (x0 )| dx = 0, ε→0+ εN Q+ ν (x0 ,ε) Z 1 (5.2) |u(x) − u− (x0 )| dx = 0, lim ε→0+ εN Q− ν (x0 ,ε) (5.3)
dF(u, ·) F(u, Qν (x0 , ε)) (x0 ) = lim+ N −1 d|[u] νu |H bJu ε→0 |[u] νu |HN −1 bJu (Qν (x0 , ε)) exists and is finite.
Choose one such point x0 ∈ Ju . Let α := |[u] νu |HN −1 bJu and set µ := LN + α. By Lemma 3.12 the limit in (5.3) is equal to lim
ε→0+
Recall that
m(u, Qν (x0 , ε)) for α a.e. x0 ∈ Ω. α(Qν (x0 , ε))
n o m(u, V ) = inf F(v, V ) | v|∂V = u|∂V , v ∈ BD(Ω) .
By Lemma 2.13 the point x0 can be chosen to satisfy also 1 1 (5.4) lim |Eu|(Qν (x0 , ε)) = lim+ N −1 α(Qν (x0 , ε)) = |([u] νu )(x0 )|. ε→0+ εN −1 ε→0 ε Let uε : Qν → RN be defined by uε (y) := u(x0 + εy). Then, by (5.1) and (5.2) uε → ux0 ,ν strongly in L1 (Qν , RN ), where + u (x0 ) if y · ν > 0, ux0 ,ν (y) := u− (x0 ) if y · ν < 0.
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
19
Also, from Lemma 2.13 it follows that |Euε |(Qν ) =
1 |Eu|(Qν (x0 , ε)) −→ |([u] ν)(x0 )| = |Eux0 ,ν |(Qν ) , εN −1
so that uε → ux0 ,ν in the intermediate topology of BD(Qν ), and since the trace operator is continuous with respect to this topology (cf. Theorem 2.4), we deduce that Z Z 1 N −1 (5.5) lim N −1 |tr (u − ux0 ,ν )| dH = lim |tr (uε − ux0 ,ν )| dHN −1 = 0. ε→0 ε ε→0 ∂Q ∂Qν (x0 ,ε) ν We have, by (5.3), Lemma 3.10 and (5.4), (5.5), dF(u, ·) F(u, Qν (x0 , ε)) (x0 ) = lim+ dα α(Qν (x0 , ε)) ε→0 m(u, Qν (x0 , ε)) = lim ε→0+ α(Qν (x0 , ε)) m(u, Qν (x0 , ε)) − m(ux0 ,ν , Qν (x0 , ε)) + m(ux0 ,ν , Qν (x0 , ε)) = lim+ εN −1 ε→0 εN −1 · α(Qν (x0 , ε)) m(ux0 ,ν , Qν (x0 , ε)) = lim+ (|([u] ν)(x0 )|)−1 . εN −1 ε→0 Hence Z F(u, V ∩ Ju ) = V ∩Ju
dF(u, ·) (x)dα(x) = dα
Z
g(x, u+ (x), u− (x), ν(x))dHN −1 .
V ∩Ju
We will show now that the surface energy density may be more explicitly characterized. Proposition 5.2. If (3.1), (3.2) hold and if u ∈ SBD(Ω), then for HN −1 -almost every x0 ∈ Ju g(x0 , u+ (x0 ), u− (x0 ), νu (x0 )) = (QE f )∞ (([u] νu )(x0 )). Proof. By Lemma 3.14, for HN −1 -almost every x0 ∈ Ju the function g is given by nZ 1 g(x0 , u+ (x0 ), u− (x0 ), ν) = lim sup N −1 inf f (Ew(x))dx ε→0 ε Qν (x0 ,ε) o w ∈ W 1,1 (Ω, RN ), w = ux0 ,ν on ∂Qν (x0 , ε) where ν := νu (x0 ). Hence, for any k ∈ N there exists a function wk,ε , depending also on ε, such that, setting wk,ε (y) := wk,ε (x0 + εy), 1 g(x0 , u (x0 ), u (x0 ), ν) + k +
(5.6)
−
≥ lim sup
1
Z
f (Ewk,ε (x))dx εN −1 Qν (x0 ,ε) Z 1 = lim sup εf Ewk,ε (y) dy ε ε→0 Qν Z 1 Ewk,ε (y) dy, ≥ lim sup εQE f ε ε→0 Qν ε→0
20
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
and Z
1 Ewk,ε (y) dy ε Z Z 1 = εQE f Ewk,ε (y) − (QE f )∞ (Ewk,ε (y)) dy + (QE f )∞ (Ewk,ε (y))dy ε Qν Qν =: I1 + I2 .
εQE f Qν
Using hypothesis (3.2), the growth condition of (QE f )∞ , and H¨older’s inequality, it follows that Z 1 ∞ |I1 | ≤ εQE f ε Ewk,ε (y) − (QE f ) (Ewk,ε (y)) dy Qν ∩{|Ewk,ε |≤Lε} Z εQE f 1 Ewk,ε (y) − (QE f )∞ (Ewk,ε (y)) dy + ε Qν ∩{|Ewk,ε |>Lε} Z 1 ≤ εC 1 + |Ewk,ε (y)| + C|Ewk,ε (y)| dy ε Qν ∩{|Ewk,ε |≤Lε} Z + Λεβ |Ewk,ε (y)|1−β dy Qν
≤ Cε + CLε + Λε
β
1−β
Z |Ewk,ε (y)|dy
.
Qν
By the coercivity of QE f (cf. Remark 3.3) and (5.6) it follows that Z Z 1 C≥ε QE f Ewk,ε (y) dy ≥ |Ewk,ε (y)|dy, ε Qν Qν where C is independent of ε, and thus Z Z 1 (5.7) εQE f Ewk,ε (y) dy = O(ε) + (QE f )∞ (Ewk,ε (y))dy. ε Qν Qν Set
Then zk,ε obtain
u− (x0 ) + u+ (x0 ) zk,ε (y) := wk,ε (y) − ([u] ⊗ νu )(x0 )y + . 2 1,1 ∈ Wper (Qν , RN ) so using the E-quasiconvexity of (QE f )∞ (cf. Remark 3.3) we (QE f )∞ (([u] νu )(x0 )) ≤
Z
(QE f )∞ (([u] νu )(x0 ) + Ezk,ε (y))dy
Qν
Z (5.8)
=
(QE f )∞ (Ewk,ε (y))dy.
Qν
From (5.6), (5.7) and (5.8) we conclude that g(x0 , u+ (x0 ), u− (x0 ), νu (x0 )) +
1 k
Z ≥ lim sup O(ε) + ε→0
(QE f )∞ (Ewk,ε (y))dy
Qν
≥ (QE f )∞ (([u] νu )(x0 )). It suffices to let k → ∞. To show the reverse inequality, in view of Remark 3.13, consider the parallelepiped εk ε Qk,ε (x0 ) := x ∈ RN | |(x − x0 ) · νi | < , |(x − x0 ) · ν| < , i = 1, . . . , N − 1 , 2 2 where {ν1 , . . . , νN −1 , ν} is an orthonormal basis of RN , and define, for x ∈ Qk,ε (x0 ), wε (x) :=
1 u+ (x0 ) + u− (x0 ) ([u] ⊗ ν)(x0 )(x − x0 ) + . ε 2
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
21
Then wε has the same trace as ux0 ,ν when (x − x0 ) · ν = ± 2ε but not on the remaining lateral faces of the boundary of Qk,ε (x0 ). In order to obtain an admissible sequence for m(ux0 ,ν , Qk,ε (x0 )), we consider a smooth function ϕk,ε ∈ C ∞ (Qk,ε (x0 )), depending only on x · νi , i = 1, . . . , N − 1, such that 0 ≤ ϕk,ε ≤ 1, ϕk,ε (x) = 1 if x ∈ Qk−1,ε (x0 ), ϕk,ε (x) = 0 1 if |(x − x0 ) · νi | = εk 2 and kDϕk,ε k∞ = O( ε ). Thus ϕk,ε wε + (1 − ϕk,ε )ux0 ,ν has the same trace as ux0 ,ν on ∂Qk,ε (x0 ), and so, g(x0 , u+ (x0 ), u− (x0 ), ν)
m(ux0 ,ν , Qk,ε (x0 ) (kε)N −1 F(ϕk,ε wε + (1 − ϕk,ε )ux0 ,ν , Qk,ε (x0 ) ≤ lim sup lim sup . (kε)N −1 ε→0 k→∞ =
lim sup lim sup k→∞
(5.9)
ε→0
∞ By definition of E-quasiconvexity, let φn ∈ Cper (Qk,1 (0), RN ) be such that Z 1 1 k N −1 N −1 (5.10) f ([u] νu )(x0 ) + Eφn (x) dx ≤ k (QE f ) ([u] νu )(x0 ) + . ε ε n Qk,1 (0) ∞ As in the proof of Proposition 4.2, we may find a sequence φn ∈ Cper (Qk,1 (0), RN ) associated to φn , and a function φ, such that Eφ = 0, |Eφn |(Qk,1 (0)) = |Eφn |(Qk,1 (0)) and
φn → φ in L1 (Qk,1 (0), RN ). i
By Theorem 2.6, let un ∈ C ∞ (Qk,ε (x0 ), RN ) ∩ W 1,1 (Qk,ε (x0 ), RN ) be such that un → ux0 ,ν , tr un = tr ux0 ,ν on ∂Qk,ε (x0 ) and define x − x0 x − x0 uk,ε,n (x) := (ϕk,ε wε + (1 − ϕk,ε )un )(x) + ε φn −φ . ε ε Then uk,ε,n → ϕk,ε wε +(1−ϕk,ε )ux0 ,ν strongly in L1 (Qk,ε (x0 ), RN ) as n → ∞ and uk,ε,n ∈ W 1,1 (Qk,ε (x0 ), RN ). Hence uk,ε,n is admissible for F(ϕk,ε wε + (1 − ϕk,ε )ux0 ,ν , Qk,ε (x0 )) and therefore, as x − x0 Euk,ε,n (x) = ϕk,ε (x)(Ewε − Eun )(x) + Eun (x) + (wε − un ) Dϕk,ε (x) + Eφn , ε it follows from (5.9) that Z 1 lim inf f (Euk,ε,n (x))dx N −1 n→∞ ε→0 (kε) k→∞ Qk,ε (x0 ) Z x − x0 lim inf f Ewε (x) + Eφn dx n→∞ ε Qk−1,ε (x0 )
g(x0 , u+ (x0 ), u− (x0 ), ν) ≤ lim sup lim sup
1 N −1 (kε) k→∞ 1 + lim sup lim sup N −1 (kε) ε→0 k→∞ Z 1 x − x0 · lim inf C 1 + |Ewε |(x) + |Eun |(x) + |wε − un |(x) + |Eφn | dx n→∞ ε ε Lk,ε ≤ lim sup lim sup ε→0
=: I1 + I2 , where Lk,ε := Qk,ε (x0 ) \ Qk−1,ε (x0 ). By (5.10) Z 1 x − x0 1 I1 ≤ lim sup lim sup lim inf f ([u] νu )(x0 ) + Eφn dx N −1 n→∞ ε ε ε→0 (kε) k→∞ Qk,ε (x0 ) Z ε 1 = lim sup lim sup N −1 lim inf f ([u] νu )(x0 ) + Eφn (y) dy n→∞ ε ε→0 k k→∞ Qk,1 (0) 1 ε ≤ lim sup lim inf ε(QE f ) ([u] νu )(x0 ) + = (QE f )∞ (([u] νu )(x0 )). n→∞ ε n ε→0
22
ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER
On the other hand, as LN (Lk,ε ) = O(εN k N −2 ), Z ε 1 1 1 C 1 + |([u] νu )(x0 )| dx = O +O . N −1 (kε) ε k k Lk,ε By the coercivity condition in (3.1) and (5.10), |Eφn |(Qk,1 (0)) is uniformly bounded in n, thus Z 1 x − x0 lim sup lim sup |Eφn | dx (kε)N −1 n→∞ Lk,ε ε ε→0 Z εN ≤ lim sup lim sup |Eφn |(y)dy N −1 n→∞ ε→0 (kε) Qk,1 (0) Z ε = lim sup N −1 lim sup |Eφn |(y)dy = 0. n→∞ ε→0 k Qk,1 (0) From the convergence of un to ux0 ,ν in the intermediate topology and the fact that ([u] νu )(x0 )HN −1 bJux0 ,ν (∂Lk,ε ) = 0, we deduce that Z
Z
|([u] νu )(x0 )|dHN −1 (x);
|Eun |(x)dx =
lim
n→∞
Lk,ε ∩Jux0 ,ν
Lk,ε
also
Z |wε − un |(x)dx =
lim
n→∞
Z
Lk,ε
|wε − ux0 ,ν |(x)dx, Lk,ε
with 1 |([u] νu )(x0 )|HN −1 (Lk,ε ∩ Jux0 ,ν ) N −1 (kε) ε→0 1 O(εN −1 k N −2 ) = 0, = lim sup lim sup N −1 ε→0 (kε) k→∞
lim sup lim sup k→∞
and
1 lim sup lim sup N −1 (kε) ε→0 k→∞
since |(x − x0 ) · ν| < I2 = 0.
ε 2
Z Lk,ε
1 |wε − ux0 ,ν |(x)dx = 0, ε
in Lk,ε , which implies that |wε − ux0 ,ν | ≤ C. Thus we conclude that
Acknowledgements. The authors wish to thank G. Dal Maso for many useful discussions concerning the subject of this paper. The research of Ana Cristina Barroso was partially supported by FCT, PRAXIS XXI and project PRAXIS 2/2.1/MAT/125/94. The research of Irene Fonseca was partially supported by the National Science Foundation under Grants No. DMS-9500531 and DMS-9731957 and through the Center for Nonlinear Analysis (CNA). This research was done while Rodica Toader was a Postdoctoral Research Fellow at the CMAF-University of Lisbon. The financial support of CMAF-UL, the Funda¸c˜ao para a Ciˆencia e Tecnologia and PRAXIS XXI is gratefully acknowledged. References [1] Ambrosio, L., A. Coscia and G. Dal Maso. Fine properties of functions with bounded deformation. Arch. Rat. Mech. Anal. 139 (1997), 201-238. [2] Ambrosio, L. and G. Dal Maso. On the relaxation in BV (Ω; Rm ) of quasi-convex integrals. J. Funct. Anal. 109 (1992), 76-97. [3] Ambrosio, L., S. Mortola and V. M. Tortorelli. Functionals with linear growth defined on vector valued BV functions. J. Math. Pures et Appl. 70 (1991), 269-323.
A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION
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[4] Barroso, A. C., G. Bouchitt´ e, G. Buttazzo and I. Fonseca. Relaxation of bulk and interfacial energies. Arch. Rat. Mech. Anal. 135 (1996), 107-173. [5] Bellettini, G., A. Coscia and G. Dal Maso. Special functions of bounded deformation. Math. Z. 228 (1998), 337-351. [6] Braides, A., A. Defranceschi and E. Vitali. A relaxation approach to Hencky’s plasticity. Appl. Math. Optimization 35 (1997), 45-68. [7] Bouchitt´ e, G., I. Fonseca and L. Mascarenhas. A global method for relaxation. Arch. Rat. Mech. Anal. 145 (1998), 51-98. [8] Dacorogna, B. Direct Methods in the Calculus of Variations. Springer, 1989. [9] Fonseca, I. and S. M¨ uller. Quasiconvex integrands and lower semicontinuity in L1 . SIAM J. Math. Anal. 23 (1992), 1081-1098. [10] Fonseca, I. and S. M¨ uller. Relaxation of quasiconvex functionals in BV (Ω; Rp ). Arch. Rat. Mech. Anal. 123 (1993), 1-49. [11] Fonseca, I. and S. M¨ uller. A-quasiconvexity, lower semicontinuity and Young measures. To appear in SIAM J. Math. Anal. [12] Kohn, R. V. New Estimates for Deformations in Terms of Their Strains, Ph.D. Thesis, Princeton University, 1979. [13] Matthies, H., G. Strang and E. Christiansen. The Saddle Point of a Differential Program. Energy Methods in Finite Element Analysis, Wiley, New York, 1979. [14] Ornstein, D. A non-inequality for differential operators in the L1 -norm. Arch. Rat. Mech. Anal. 11 (1962), 40-49. [15] Reshetnyak, Y. G. Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968), 1039-1045 (translation of Sibirsk. Mat. Z. 9 (1968), 1386-1394). [16] Suquet, P.M. Existence et r´ egularit´ e des solutions des ´ equations de la plasticit´ e parfaite. C. R. Acad. Sci. Paris S´ er. A 286 (1978), 1201-1204. [17] Suquet, P.M. Un espace fonctionel pour les ´ equations de la plasticit´ e. Ann. Fac. Sci. Toulouse 1 (1979), 77-87. [18] Temam, R. Probl` emes Math´ ematiques en Plasticit´ e. Gauthier-Villars, 1983. [19] Temam, R. and G. Strang. Functions of bounded deformation. Arch. Rat. Mech. Anal. 75 (1980), 7-21. (Ana Cristina Barroso) CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal ´tica, Faculdade de Cie ˆncias, Universidade de Lisboa, 1749-016 Lisboa, Departamento de Matema Portugal E-mail address, A. C. Barroso:
[email protected] (Irene Fonseca) Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA E-mail address, I. Fonseca:
[email protected] (Rodica Toader) CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal E-mail address, R. Toader:
[email protected]
