ECOLE POLYTECHNIQUE. Piecewise rigidity CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS A. Chambolle, A. Giacomini, M.

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1 ECOLE POLYTECHNIUE CENTRE DE MATHÉMATIUES APPLIUÉES UMR CNRS PALAISEAU CEDEX (FRANCE). Tél: Fax: Piecewise rigidity A. Chambolle, A. Giacomini, M. Ponsiglione R.I. 595 February, 2006

2 PIECEWISE RIGIDITY ANTONIN CHAMBOLLE, ALESSANDRO GIACOMINI, AND MARCELLO PONSIGLIONE Abstract. In this paper we provide a Liouville type theorem in the framework of fracture mechanics, and more precisely in the theory of SBV deformations for cracked bodies. We prove the following rigidity result: if u SBV (Ω, R N ) is a deformation of Ω whose associated crack J u has nite energy in the sense of Grith's theory (i.e., H N 1 (J u) < ), and whose approximate gradient u is almost everywhere a rotation, then u is a collection of an at most countable family of rigid motions. In other words, the cracked body does not store elastic energy if and only if all its connected components are deformed through rigid motions. In particular, global rigidity can fail only if the crack disconnects the body. Contents 1. Introduction 1 2. Notations and preliminaries 4 3. curl u is a measure for u SBV (Ω) 5 4. Some estimates for vector elds in a cube 9 5. Proof of Theorem Acknowledgments 16 References Introduction A classical rigidity result in nonlinear elasticity, due to Liouville, states that if an elastic body is deformed in such a way that its deformation gradient is pointwise a rotation, then the body is indeed subject to a rigid motion. If the body is supposed to be hyperelastic with an elastic energy density W dened on a natural reference conguration Ω, a standard assumption for W which comes from its frame indierence is that W is minimized exactly on the set of rotations SO(3). Hence the rigidity result implies that the body doesn't store elastic energy if and only if it is deformed through a rigid motion. From a mathematical viewpoint, Liouville's Theorem can be stated as follows: if Ω R N is open and connected, u C (Ω; R N ) is such that u(x) SO(N) for every x Ω, then u = a + R x for some a R and R SO(N). The assumption on the regularity of u has been fairly weakened, and now the same rigidity result is available for deformations in the class of Sobolev maps (see Yu. Reshetnyak [17]). In this case the deformation gradient is dened only almost everywhere in Ω, so that the assumption for rigidity is u(x) SO(N) for a.e. x Ω. A quantitative rigidity estimate has been provided recently by Friesecke, James and Müller [13], in order to derive nonlinear plates theories from three dimensional elasticity. They proved that if Ω is connected and with Lipschitz boundary, there exists a constant C depending only on Ω and N such that for every u W 1,2 (Ω, R N ) (1.1) min R SO(N) u R L 2 (Ω) C dist( u, SO(N)) L2 (Ω). 1

3 2 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE As a consequence, if the deformation gradient is close to rotations (in L 2 ), then it is in fact close to a unique rotation. Estimate (1.1) is indeed true in L p for every 1 < p < + (see [10]). The aim of this paper is to discuss the problem of rigidity in the framework of fracture mechanics, that is for bodies that can not only deform elastically, but also be cracked along surfaces where the deformation becomes discontinuous. The class of admissible deformations that we consider, in this setting, will be the space of special functions of bounded variation SBV (Ω; R N ) (see Section 2 for a precise denition). Given u SBV (Ω; R N ), the approximate gradient u (which exists at almost every point of Ω) takes into account the elastic part of the deformation, while the jump set J u represents a crack in the reference conguration. The set J u is rectiable, that is, it can be covered (up to a H N 1 negligible set) by a countable number of C 1 submanifolds of R N. So J u is, in some sense, a (N 1)dimensional surface. In the context of SBV deformations, we cannot expect a rigidity result as for elastic deformations, because a crack can divide the body into two parts, each of one subject to a dierent rigid deformation. We prove that this is essentially the only way rigidity can be violated, provided the crack J u has nite energy (which, in the framework of Grith's theory, means that its total (N 1)dimensional surface is nite). If the body is not suitably divided by a crack in several components, then rigidity as in the elastic case holds. In order to formulate our result, we need some notions from geometric measure theory in order to make precise the notion of a partition Ω in connection with SBV deformations. We refer to Section 2 for more details. We say that a partition (E i ) i N of Ω is a Caccioppoli partition if i N P (E i, Ω) < +, where P (E i, Ω) denotes the perimeter of E i in Ω. Given a rectiable set K Ω, we say that a Caccioppoli partition (E i ) i N of Ω is subordinated to K if (up to a H N 1 negligible set) the reduced boundary E i of E i is contained in K for every i N. We say that Ω \ K is indecomposable if the only Caccioppoli partition subordinated to K is the trivial one, i.e., E 0 = Ω. The main rigidity result of the paper is the following Liouville's type theorem for SBV - deformations. Theorem 1.1. Let u SBV (Ω, R N ) such that H N 1 (J u ) < + and u(x) SO(N) for a.e. x Ω. Then u is a collection of an at most countable family of rigid deformations, i.e., there exists a Caccioppoli partition (E i ) i N subordinated to J u such that u = i N(K i x + b i )1 Ei (x), where K i SO(N) and b i R N (and, as a consequence, J u i N E i ). In particular, if Ω \ J u is indecomposable, then u is a rigid deformation, i.e., u(x) = a + R x for some a R N and R SO(N) (hence, J u = ). Let us observe that the assumption that H N 1 (J u ) is nite is essential in this result. Indeed, it has been shown by Alberti [1, 5] that any Ndimensional L 1 vector eld can be the gradient of a suitable SBV function, so that the rigidity clearly fails if one just assumes u(x) SO(N) for a.e. x Ω. In the context of fracture mechanics, Theorem 1.2 implies the following fact. Assume that the density of the elastic energy stored in the cracked body is represented by a function W vanishing exactly on SO(N). Then a deformation u of class SBV does not store elastic energy if and only if the crack J u divides Ω in several subbodies, each of one subject to a rigid motion. If J u is not enough to create subbodies of Ω, then u is a rigid motion for the entire body (and there is no jump J u at all). In this respect, the space SBV seems to be appropriate for the study of elastic properties of cracked hyperelastic bodies.

4 PIECEWISE RIGIDITY 3 The main diculty to prove Theorem 1.1 is that the dierential constraint curl u = 0, valid for every Sobolev function, does not hold in general for SBV functions, because u is only a part of the distributional derivative of u. However we prove that if u SBV (Ω; R N ) with u L (Ω; M N N ) then curl u is a measure, which is absolutely continuous with respect to H N 1 J u. This result (up to our knowledge, new and interesting on its own), combined with the quantitative rigidity estimate (1.1) is enough to obtain our rigidity result. The set of rotations in R N can be replaced by any compact set of matrices K M N N which satisfy a L p -quantitative rigidity estimate for 1 < p < N N 1, i.e., there exists C > 0 depending on N and p such that, for every u W 1,p (Ω, R N ), (1.2) min K K u K L p (Ω) C dist( u, K) L p (Ω). Theorem 1.1 is obtained as a particular case of the following rigidity result. Theorem 1.2 (The rigidity result). Let K M N N be a compact set such that the quantitative rigidity estimate (1.2) holds for some p (1, N/(N 1)). Let u SBV (Ω, R N ) be such that H N 1 (J u ) < + and u(x) K for a.e. x Ω. Then there exists a Caccioppoli partition (E i ) i N of Ω subordinated to J u such that where K i K and b i R N u = i N(K i x + b i )1 Ei (x), (and, as a consequence, J u i N E i ). In particular if Ω \ J u is undecomposable, then u = Kx + b for some K K, b R N (hence, J u = ). In order to prove Theorem 1.2, the key point is to show that u is a piecewise constant function that can jump only on J u, i.e., u SBV (Ω, M N N ) with ( u) = 0 and J u J u : this implies that u is constant on a Caccioppoli partition subordinated to J u, and hence that u is ane on the same partition. In order to establish that u is piecewise constant with jumps on J u, we use an approximation based on a covering argument inspired by [13]. First of all we split our domain in a disjoint union of small cubes h of size h. On many of these cubes, H N 1 (J u h ) will be small, showing that curl u is close to zero in h. A Helmholtz' type estimate for L 1 vector elds with curlmeasure shows then that u is close in L p to the gradient w h of a Sobolev function, which by the quantitative rigidity estimate (1.2) is close in L p to a unique matrix K( h ) K. We show that u is approximated by the piecewise constant functions ψ h such that ψ h = w h on h. The sequence (ψ h ) h N has a uniformly bounded total variation which is controlled by curl u and so by H N 1 J u : we prove this, as in [13], by using again the quantitative rigidity estimate on the union of neighboring cubes. An application of Ambrosio's compactness theorem for SBV functions [2, 3, 4] is then enough to get the conclusion. Let us mention that a local version of Liouville Theorem on sets of nite perimeter, for Lipschitz mappings, was already given in [12]. There, Dolzmann and Müller prove that if u : Ω R N is in W 1, (Ω; R N ), det u c > 0, and u SO(N) for a.e. x E, where E is a subset of Ω with nite perimeter, then u1 E BV (Ω), and D( u1 E ) (Ω \ E) = 0. (So that the thesis of Theorem 1.1 holds inside E.) Rigidity results in the spirit of Liouville's Theorem play also an important role in order to understand possible microstructures arising in elastic bodies. The problem of microstructures can be stated mathematically in the following way: given a set of matrices K M N N, nd Lipschitz mappings u : Ω R N such that u(x) K for a.e. x Ω. K is said to be rigid if it doesn't admit nontrivial microstructures, i.e., if the only maps u W 1, (Ω) such that u(x) K for a.e. x Ω are ane.

5 4 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE An example of rigid set of matrices is provided by a famous result by Ball and James [6]: K = {K 1, K 2 } is rigid if and only if rank (K 1 K 2 ) 2. In this case, Ball and James proved that rigidity holds also in the stronger sense of approximate solutions: for every sequence (u h ) h N of equi-lipschitz functions such that dist( u h, K) 0 in measure, then either dist( u h, K 1 ) 0, or dist( u h, K 2 ) 0 in measure. Theorem 1.2 can be used to infer a similar result in the framework of the discontinuous deformations of class SBV. The quantitative rigidity estimate we need to apply our arguments has been recently provided by De Lellis and Székelyhidi [11]: they prove that if K M N N is a nite set of matrices which is rigid for approximate solutions, then the quantitative rigidity estimate (1.2) holds for any p (1, + ) provided that Ω is Lipschitz-regular. As a consequence, we can deduce the following result. Theorem 1.3. Let K := {K 1, K 2 }, with rank (K 1 K 2 ) 2. Let u SBV (Ω, R N ) such that u(x) K for a.e. x Ω. Then there exists a Caccioppoli partition (E i ) i N of Ω subordinated to J u such that u = i N(K i x + b i )1 Ei (x), where K i K and b i R N (and, as a consequence, J u i N E i ). In particular if Ω \ J u is indecomposable, then u = Kx + b for some K K and b R N (hence, J u = ). The rigidity result with respect to approximate solutions by Ball and James has been generalized to the case where K consists of three matrices without any rank-1 connection in [18], so that Theorem 1.3 still holds in this case. For completeness, let us say that if K consists of four matrices without any rank-1 connection, rigidity can fail for approximate solutions for a suitable choice of the involved matrices (see [19], [20]), while K is always rigid with respect to exact solutions (see [9]). The case N = 5 is nicely illustrated in [14] by a non-rigid ve point conguration without any rank-1 connection. The paper is organized as follows. In Section 2 we recall some facts from geometric measure theory and from the theory of SBV spaces. In Section 3, we show that the curl of a function u that satises the assumptions of the rigidity theorem is a Radon measure absolutely continuous with respect to H N 1 J u. Section 4 is devoted to the statement and proof of some Helmholtz type estimates in cubes. The proof of Theorem 1.2 is then given in Section Notations and preliminaries In this section we recall the denition of the space SBV and some facts from geometric measure theory that will be used throughout the paper. We refer to [5] for further details. The space SBV. Let Ω be an open set in R N. We say that u BV (Ω; R N ) if u L 1 (Ω; R N ), and its distributional derivative Du is a vector-valued Radon measure on Ω. We say that u SBV (Ω; R N ) if u BV (Ω; R N ) and its distributional derivative can be represented as Du(A) = u(x) dx + (u + (x) u (x)) ν x dh N 1 (x), A A J u where u denotes the approximate gradient of u, J u denotes the set of approximate jumps of u, u + and u are the traces of u on J u, ν x is the normal to J u at x, and H N 1 is the (N 1)-dimensional Hausdor measure. The symbol denotes the tensorial product of vectors: (a b) ij = a i b j for every a, b R N.

6 PIECEWISE RIGIDITY 5 Note that if u SBV (Ω; R N ), then the singular part of Du is concentrated on J u which is a countably H N 1 -rectiable set: there exists a set E with H N 1 (E) = 0 and a sequence (M i ) i N of C 1 -submanifolds of R N such that J u E i N M i. We set, for d 1, (2.1) SBV (Ω; R d ) := {u SBV (Ω; R d ) : u L (Ω; M d N ), H N 1 (J u ) < + }. and, as usual, SBV (Ω) := SBV (Ω; R) whenever d = 1. Piecewise constant functions and Caccioppoli partitions. Let Ω be an open set in R N, and let E Ω. We say that E has nite perimeter in Ω if 1 E SBV (Ω). The set of jumps of 1 E is denoted by E and is called the reduced boundary of E: the derivative of 1 E is concentrated on E, and its total variation is given by H N 1 E. The perimeter of E in Ω is given by H N 1 ( E). We say that a partition (E i ) i N of Ω is a Caccioppoli partition if i N HN 1 ( E) < +. Given a rectiable set K Ω, we say that a Caccioppoli partition (E i ) i N of Ω is subordinated to K if (up to a H N 1 negligible set) the reduced boundary E i of E i is contained in K for every i N. We say that Ω \ K is indecomposable if the only Caccioppoli partition subordinated to K is the trivial one, i.e., E 0 = Ω. Caccioppoli partitions are naturally associated to piecewise constant functions, i.e., functions u SBV (Ω; R N ) such that u = 0 a.e. on Ω. These functions are said piecewise contant in Ω because they are indeed constant on the subsets E i of a Caccioppoli partition of Ω. More precisely (see [5, Theorem 4.23]) there exists a Caccioppoli partition (E i ) i N of Ω such that (2.2) u = i N b i 1 Ei, with b i b j for i j. Notice that if K is a rectiable set in Ω such that Ω \ K is indecomposable, then a piecewise constant function u in Ω with J u K is necessarily constant on Ω. 3. curl u is a measure for u SBV (Ω) In this section, we show that the curl of a function u that satises the assumptions of the theorem is in fact a measure, estimated with H N 1 J u. Theorem 3.1. Let u SBV (Ω). Then curl u is a measure µ concentrated on J u such that µ c u H N 1 J u. In this statement, the constant c depends on the dimension N. However, we conjecture that the optimal constant is 2 2 (considering the Frobenius norm for matrices). Remark 3.2. Clearly, if u SBV (Ω; R d ) is a vector-valued function (d 2), then the result still holds (with the same constant c if the norm on tensors is still the Euclidean norm of the associated matrix). Proof. Let u SBV (Ω). We have: u L 1 (Ω), L := u < +, and H N 1 (J u ) < +. The distribution curl u is formally equal to the matrix ( i ( j u) j ( i u)) 1 i,j N and is dened by curl u, ϕ = i u(x) j (ϕ i,j ϕ j,i )(x) dx, i,j=1 Ω

7 6 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE for any ϕ Cc (Ω; M N N ). The thesis of the theorem is local, so that it is enough to prove that if Ω is a hypercube in Ω, then for any ϕ Cc (; M N N ), one has (3.1) i u(x) j (ϕ i,j ϕ j,i )(x) dx c u ϕ H N 1 (J u ). i,j=1 Without loss of generality, we may assume that = (0, 1) N. We will approximate u in with a piecewise smooth function, jumping only on facets of smaller hypercubes. This will be done using a simplied variant of the discretization/reinterpolation technique presented in [7, 8], and inspired from [16]. Step 1. Consider the set J = J u. Denote by (e i ) N i=1 the canonical basis of RN (e i = (δ i,j ) N j=1 ). One easily shows that for any i, the set J ε i := { te i + x : t [0, ε], x J} is Lebesgue-measurable. Indeed, up to a H N 1 negligible set N, J is a countable union of compact sets: hence Ji ε is the union of [ εe i, 0] + N, which has Lebesgue measure zero, and of a countable union of compact sets. We have the estimate J ε i εh N 1 (J), which can be derived in several ways, and more precisely one can show (3.2) Ji ε ε ν i (x) dh N 1 J where ν(x) = (ν 1 (x),..., ν N (x)) is the normal to J at x, dened for H N 1 -a.a. x J. For y (0, 1) N, we now also dene the discrete binary variable l y ε,i (k) := 1 Ji ε(εy +k), for any k εzn. One shows that for any i = 1,..., N ε N 1 l y ε,i (k) dy = ε 1 1 J ε i (y + k) dy = ε 1 Ji ε. (0,1) N (0,ε) n k εz N Hence using (3.2) and N i=1 ν i N, (0,1) N ε N 1 i=1 k εz N Using Fatou's lemma, we deduce lim inf ε 0 εn 1 (0,1) N i=1 k εz N k εz N l y ε,i (k) dy NH N 1 (J). l y ε,i (k) dy NH N 1 (J), so that for any δ > 0, there exists a set A of positive measure in (0, 1) N, such that (3.3) y A lim inf ε 0 εn 1 i=1 k εz N l y ε,i (k) NH N 1 (J) + δ. Step 2. Let now (t) := max{1 t, 0} (t R) and N (ξ) = N i=1 (ξ i) for all ξ R N (which is known in nite elements approximation as the 1 interpolation function). If we let v y ε (x) := k εz N u(εy + k) N ( x k ε ) y, it is well known that for a.e. y (0, 1) N, v y ε u in L 1 () (see for instance [7]).

8 PIECEWISE RIGIDITY 7 Step 3. The slicing properties of BV functions (see [5]) also ensure that for all i and H N 1 -a.e. z {x : x i = 0}, the function (0, 1) t u(z + te i ) is in SBV (0, 1), with nite jump set given by {t : z + te i J}, and whose derivative is given by t i u(z + te i ), which by assumption is bounded by L. We deduce that for a.e. y (0, 1) N, the discrete function vε y satises vε y (k + εe i ) vε y (k) Lε for any i = 1,..., N and k εz N such that k + εe i and J [εy + k, εy + k + εe i ] =, which is equivalent to l y ε,i (k) = 0. Step 4. From steps 1, 2 and 3, there exists y A such that: (3.4) lim inf ε 0 εn 1 i=1 k εz N l y ε,i (k) NH N 1 (J) + δ, v y ε u in L 1 (), and v y ε (k + εe i ) v y ε (k) Lε for any i = 1,..., N and k εz N such that k + εe i and l y ε,i (k) = 0. We choose a sequence (ε j) j 1 such that the lim inf in (3.4) is in fact a limit, and let v j := v y ε j, and l j,i := l y ε j,i. From now on, since we refer only to the grids {ε j y+ε j k : k Z N } which we use to interpolate u, we can assume (up to translation) that y = 0, so that they coincide with the grids {ε j k : k Z N }. In a small cube k + (0, ε j ) N in (k ε j Z N ), as soon as J does not intersect any edge of the cube, one has i v j L for all i = 1,..., N so that v j NL inside the cube. Given an edge [k, k + ε j e i ], if l j,i (k) = 1, then J intersects the edge. In this case, we cannot control v j in all the cubes in that share this edge, whose total number is at most 2 N 1. We let K j be the union of all such cubes: by (3.4) we have the estimate (3.5) K j 2 N 1 ε N j On the other hand we have i=1 k ε jz N H N 1 ( K j ) (N + 1)2 N 1 ε N 1 j l j,i (k) cε j. i=1 k ε jz N so that (using (3.4), with the lim inf ε 0 replaced with lim j ) l j,i (k), (3.6) lim sup H N 1 ( K j ) C(N)(N + 1)2 N 1 ( NH N 1 (J) + δ). j Let v j = v j1 \Kj. By (3.5), we still have v j u in L1 () as j. The previous discussion shows that in any, for j large enough, v j SBV ( ) with v j NL, v j is piecewise smooth and S(v j ) HN 1 ( K j ) is a subset of a nite number of facets of hypercubes. By Ambrosio's theorem [5, Theorem 4.36], we know that v j u in Lp ( ) (for any p < + ). Hence curl v j curl u as j, in the distributional sense. On the other hand, since Dv j = v j(x) dx + v j ν Kj H N 1 K j, (where ν Kj is the exterior normal to K j and v j stands here for the non-zero trace of v j exterior surface of K j ), and since curl Dv j = 0, one has on the which can be shown to be equal to curl v j = curl (v j ν Kj H N 1 K j ), ( τ v j) ν Kj H N 1 K j where a b denotes the antisymmetric tensor product a b b a. Hence its total variation, as a measure, is bounded by 2NLH N 1 ( K j ). If ϕ Cc (; M N N ) is xed, one has therefore

9 8 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE (choosing such that suppϕ ), curl v j, ϕ 2NL ϕ H N 1 ( K j ). Passing to the limit and recalling (3.6), we get curl u, ϕ 2NL ϕ (N + 1)2 N 1 ( NH N 1 (J) + δ). Sending δ to zero and recalling J = J u and L = u, we conclude that (3.1) holds with a constant c 2N(N + 1)2 N 1. This shows the thesis of the Theorem. Remark 3.3. The set J u is rectiable: for H N 1 -a.e. x J u, if ρ > 0 is small enough, J u B(x, ρ) is a C 1 hypersurface that cuts the ball B = B(x, ρ) into two disjoint Lipschitz sets, up to a set of H N 1 measure o(ρ N 1 ). Moreover, up to a change of basis, we have ν e 1 (and ν 1 1, ν i << 1 for i 2) in J u B. A similar study (see again [7, 8]) will show that in such a ball B, curl u (B) 2 2N u H N 1 (J u B). Passing to the limit ρ 0, we improve the constant c in the Theorem: c 2 2N. We expect, however, that a dierent approximation technique, possibly not based on a discretization, would help remove the N in that constant. Remark 3.4. Notice that the assumption u SBV (Ω) is essential in order to obtain that curl u is a measure absolutely continuous with respect to H N 1 J u. In general, curl u is not even a measure in Ω for u SBV (Ω). In fact it suces to consider f L 1 (Ω) such that curl f is a distribution of order one in Ω, and the function u SBV (Ω) given by Alberti's result [1] such that u = f. More explicit counterexamples can be constructed as follows. We consider functions dened on Ω R 2, so that we can identify curl u with the distribution curl u, ϕ := ( 2 u 1 ϕ 1 u 2 ϕ) dx, where ϕ C c (Ω). Ω (a) If we drop the assumption u L (Ω, R N ), we can consider Ω as the square 1 =] 1, 1[ 2 of R 2 and u SBV ( 1 ) dened as { ln(x 2 + y 2 ) if y > 0 u(x, y) := 0 if y < 0. It can be easily checked that curl u is a distribution of order one. (b) If we drop the assumption H N 1 (J u ) < +, we can reason as follows. Let ϑ be the 2-periodic function on R such that ϑ(x) = 1 x for x [ 1, 1], and let ϑ k (x) := 1 k ϑ(kx). Let 1 =] 1, 1[ 2, and let for n 1 { 1 S n := (x, y) 1 : n + 1 < y < 1 }. n We can nd k n N in such a way that k n + and { ϑ2 kn (x) if (x, y) S n u(x, y) := 0 if y < 0 belongs to SBV ( 1 ). Moreover, u(x, y) = 1 a.e. on 1, so that u L ( 1, R N ). Clearly curl u is a Radon measure on every open set A n := {(x, y) 1 : 1/2 < x < 1 1/2, n+1 < y < 3 4 } (which is compactly contained in 1), but curl u (A n ) = n 1. As a consequence curl u cannot be a measure on 1.

10 PIECEWISE RIGIDITY 9 4. Some estimates for vector fields in a cube Let us rst show the following estimate, valid for smooth vector elds. Proposition 4.1 (Helmholtz's type estimate). Let = (0, 1) N f L 1 () and let ϕ C (; R N ) be a vector eld such that div ϕ = 0 in, curl ϕ = f in, ϕ ν = 0 on. be the unit cube of R N, let Then for every 1 p < N N 1 there exists a constant C depending only on N and p such that (4.1) ϕ L p () C f L 1 (). Proof. Let us consider η = (η 1,..., η N ) Cc (; R N ), and let g = (g 1,..., g N ) H 1 (; R N ) be a solution of the equation g = η in, (4.2) g i = 0 on e i g i ν = 0 on, e i where still, {e i : i = 1,..., N} is the canonical basis of R N, and e i and e, denote the faces i of orthogonal and parallel to e i respectively. (Observe that (4.2) corresponds to nding g that minimizes the energy g 2 + 2η g, with boundary condition g ν = 0 on.) It is quite standard that such a g is smooth, and we will show later on that for every 1 < q < +, we have the estimate (4.3) g W 2,q () C η Lq (), where C depends only on N and q. If in particular q > N, by Sobolev's embedding theorem, (4.3) yields (4.4) g L () C η Lq (). Let ϕ = (ϕ 1,..., ϕ N ). We observe (also g being smooth) that (4.5) curl ϕ g dx = ( i ϕ j j ϕ i ) j g i dx i,j = j j g i ν i ϕ i j g i ν j ) dh i,j (ϕ N 1 + i,j We claim that (4.6) i,j (ϕ j j g i ν i ϕ i j g i ν j ) dh N 1 = 0. ( ϕ j 2 i,jg i + ϕ i 2 j,jg i ) dx. Indeed, for i j, in view of the boundary conditions in (4.2), we have that j g i = 0 on e i and on e j, so that (since by denition, ν = ±e i on e i ) ϕ j j g i ν i dh N 1 = ± ϕ j j g i dh N 1 = 0 e i and ϕ i j g i ν j dh N 1 = ± ϕ i j g i dh N 1 = 0. e j

11 10 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE By (4.5) and (4.6) we get curl ϕ g dx = i,j ( ϕ j i,jg 2 i + ϕ i j,jg 2 i ) dx = ϕ (div g) dx + ϕ g dx. Since div ϕ = 0 in and ϕ ν = 0 on, we have ϕ (div g) dx = 0, so that curl ϕ g dx = ϕ g dx = ϕ η dx. By (4.4) we conclude that ϕ η dx g L () curl ϕ L 1 () C curl ϕ L 1 () η L q (), hence (taking the supremum on η C c (Ω; R N ) with η L q () 1), ϕ L q () C curl ϕ L 1 () where q = q/(q 1). As q varies on (N, + ), we get that p = q N ranges over (1, N 1 ), so that (4.1) holds. In order to complete the proof, we need to justify the estimate (4.3). Consider the component g 1 of g (the proof for any other component is identical): we have g 1 = η 1 on (4.7) g 1 = 0 on e 1 g 1 ν = 0 on e 1. We proceed extending g 1 and η 1 to ĝ 1 and ˆη 1 dened on R N and two-periodic in each variable, in such a way that (4.8) ĝ 1 = ˆη 1 on R N. First of all, we extend g 1 and η 1 on the cube [0, 2] N. For 1 x 1 2 and 0 x i 1 with i = 2,..., N, we set {ĝ1 (x) := g 1 (2 x 1, x 2,..., x N ) ˆη 1 (x) := η 1 (2 x 1, x 2,..., x N ). Then for 1 x 2 2 and 0 x i 1 with i = 3,..., N we set {ĝ1 (x) := ĝ 1 (x 1, 2 x 2, x 3,..., x N ) ˆη 1 (x) := ˆη 1 (x 1, 2 x 2, x 3,..., x N ) and we proceed in the same way for the coordinates x 3, x 4,... x N. We can then extend ĝ 1 and ˆη 1 by periodicity to the entire R N. We get immediately that equation (4.8) is satised, so that in particular ĝ 1 is smooth. Moreover we have that ĝ 1 = 0 on the hyperplanes orthogonal to e 1 and passing through the points of the form (k, 0, 0,..., 0) with k Z. By L p -regularity estimates [15, Theorem 9.11] we get that there exists a constant C depending only on N and q such that ( ) ĝ 1 W 2,q () C ĝ 1 Lq ( ) + ˆη 1 Lq ( ),

12 PIECEWISE RIGIDITY 11 where is the cube centered at the origin with side of length 4. In view of the periodicity of ĝ 1 and ˆη 1, we obtain that (4.9) g 1 W 2,q () C ( g 1 Lq () + η 1 Lq ()). Let us assume by contradiction that claim (4.3) is false. Then there exist ĝ n 1 and ˆη n 1 periodic on R N such that (4.10) ĝ n 1 W 2,q () n ˆη n 1 Lq (). We can assume that ĝ1 n L q () = 1. Then by (4.9) we obtain ( ) ĝ1 n W 2,q () C 1 + ĝn 1 W 2,q (). n so that (4.11) ĝ n 1 W 2,q () C. In particular ĝ1 n is compact in W 1,α loc (RN ) for all α < there exists ĝ periodic in R N such that Nq N q ĝ n 1 ĝ strongly in W 1,α loc (RN ). In particular we get ĝ L q () = 1. By (4.7), (4.10) and (4.11) we get ĝ n 1 Lq () C n 0, (or in a suitable Hölder space). Then so that ĝ is harmonic in R N. Since ĝ is periodic, we conclude that it is constant. As ĝ1 n = 0 on e 1, we nally deduce that ĝ = 0. But this is against ĝ L q () = 1, so that claim (4.3) is proved. The following corollary will be used in the proof of Theorem 1.2. Corollary 4.2 (Helmholtz's type estimate with a curl measure). Let = (0, 1) N be the unit cube in R N. Let µ M(R N ; M N N ) and ϕ L 1 (, R N ) be respectively a Radon measure on and a vector eld such that Then for every 1 p < N N 1 we have that div ϕ = 0 in, curl ϕ = µ in, ϕ ν = 0 on, ϕ L p (,R N ) C µ (), where C depends only on N and p, and denotes the total variation. Proof. Let {ρ ε } ε>0 be smooth radial symmetric kernels. We claim that we can extend ϕ and µ to ˆϕ L 1 loc (RN ; R N ) and ˆµ M(R N ; M N N ) in such a way that (4.12) ˆµ ( ) = 0, and such that (4.13) ϕ ε := ˆϕ ρ ε, µ ε := ˆµ ρ ε

13 12 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE satisfy (4.14) div ϕ ε = 0 in ϕ ε ν = 0 on curl ϕ ε = µ ε in. If this is true, by (4.1) we will obtain that for all p < N N 1 ˆϕ ε L p () C ˆµ ε L 1 (), where C depends only on N and p. Letting ε 0, in view of (4.12) we will deduce ϕ Lp (;R N ) C µ () = C curl ϕ (). As in the proof of Proposition 4.1, we now build the extension ( ˆϕ, ˆµ) satisfying (4.12) and (4.14). First of all, we extend ϕ and µ to the cube (0, 2). For 1 x 1 2 and 0 x i 1 with i = 2,..., N, let us set ˆϕ := ϕ 1 (2 x 1, x 2,..., x N ) ϕ 2 (2 x 1, x 2,..., x N ). ϕ N (2 x 1, x 2,..., x N ) For i, j 1, and E Borel set contained in + e 1, let us set ˆµ 1j (E) := µ 1j (T 1 1 (E)), ˆµ j1 (E) := µ j1 (T 1 1 (E)), ˆµ ij (E) := µ ij (T 1 1 (E)), where T 1 (x) := (2 x 1, x 2,..., x N ). Then we proceed in the same way for the components x 2, x 3,..., x N. We obtain ˆϕ L 1 ((0, 2); R N ) and ˆµ M b ((0, 2); M N N ). We extend now ˆϕ and ˆµ to the entire R N by periodicity. We obtain ˆϕ L 1 loc(r N ; R N ) and ˆµ M b (R N ; M N N ) with ˆµ satisfying (4.12). By construction we have (4.15) { div ˆϕ = 0 curl ˆϕ = ˆµ. The rst identity in (4.15) is easily checked by appropriate integration against test functions. For the second, we need to show that for any ψ Cc (R N ; M N N ), one has (4.16) i,j=1 R N ˆϕ i j (ψ i,j ψ j,i ) dx = i,j=1 R N ψ i,j dˆµ i,j. By construction, this clearly holds if ψ has compact support in k ZN (k + ). We thus need to show that any other test function ψ can be approximated by functions with such a support, without perturbing too much both terms of the equality in (4.16). We observe that not only (4.12) holds, but also, ( ) ˆµ (k + ) = 0. k Z N

14 PIECEWISE RIGIDITY 13 Hence, if η ε is a family of cut-o functions that go locally uniformly to zero in R N \ k Z (k+ ), N one has (4.17) ψ i,j dˆµ i,j = lim (1 η ε )ψ i,j dˆµ i,j R N ε 0 R N i,j=1 for any smooth test function ψ. Let us choose a smooth, even η C c ( 1, 1) with η 1 in a neigborhood of 0, and 0 η 1, and let η ε (x) := η(x 1 /ε) for any x R N. We have i,j=1 R N ˆϕ i j ((1 η ε )ψ i,j (1 η ε )ψ j,i ) dx = i,j=1 i,j=1 R N (1 η ε ) ˆϕ i j (ψ i,j ψ j,i ) dx i=1 R N η ( x 1 ε ) ε ˆϕ i (ψ i,1 ψ 1,i ) dx. Hence, if we can show that the last sum goes to 0 as ε 0, together with (4.17), this will show that ψ in (4.16) may be replaced with a function whose support avoids {x 1 = 0}. In an obvious way, it will be identical to show that ψ may be replaced with a function whose support avoids {x i = k} for any i = 1,..., N and k Z. This will show that ψ may be replaced with a function with compact support in k ZN (k + ), in which case, as observed, (4.16) clearly holds. It therefore remains to show that (4.18) lim ε 0 i=1 R N η ( x 1 ε ) ε ˆϕ i (x)(ψ i,1 (x) ψ 1,i (x)) dx = 0. First of all, the rst term in the sum is clearly zero. Then, since ˆϕ i is even with respect to x 1 for any i 2, and since η is odd, one has for i 2 (letting x = (x 1, x ) for any x R N ) x 1 ε ) ε ˆϕ i (x)(ψ i,1 (x) ψ 1,i (x)) dx = η ( x 1 ε 0 R ε ) ˆϕ i(x) ψ i,1 (x 1, x ) ψ i,1 ( x 1, x ) dx dx 1 ε N 1 R N η ( where ψ i,1 := ψ i,1 ψ 1,i. The function x η ( x1 ε )( ψ i,1 (x 1, x ) ψ i,1 ( x 1, x ))/ε is clearly bounded (by c = 2 η 1 ψi,1 ) so that this integral is less than c ε ˆϕ 0 R N 1 i dx which goes to 0 as ε 0. Summing from i = 2 to N shows (4.18). Let us now consider the convolutions (4.13). Clearly we have, from (4.15), { div ϕε = 0 curl ϕ ε = µ ε. Moreover, since we have extended the i-th component oddly in the direction e i, it is readily checked that ϕ ε ν = 0 on. Since the claims (4.12) and (4.14) are proved, the proof is concluded. 5. Proof of Theorem 1.2 Let us rst deduce from the results in the two previous section the following rigidity estimate, which is valid for any compact K such that estimate (1.1) holds. Proposition 5.1 (The rigidity estimate). Let = (0, 1) N be the unit cube in R N and let 1 p < N/(N 1). Let u SBV (; R N ) be such that u(x) K for a.e. x. Then µ u := curl u is a measure concentrated on J u and there exists K K such that (5.1) u K Lp () C µ u (),

15 14 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE where C depends only on N and p. Proof. By Theorem 3.1 we have that µ u := curl u is a measure concentrated on J u such that µ u ch N 1 J u, where c is a constant depending only on u. Let us consider w H 1 (; R N ) solution of the minimization problem { } min v u 2 : v H 1 (; R N ), v(x) dx = 0. Let ϕ := u w. We have that ϕ L 2 (; M N N ), and by minimality, that ϕ : v dx = 0 for any v H 1 (; R N ), hence: { div ϕ = 0 in Moreover we have that ϕ ν = 0 on. curl ϕ = curl u curl w = µ u, i.e., curl ϕ M(; M N N ). By corollary 4.2 (applied to each component of ϕ), there exists a constant C depending only on p and N such that ϕ Lp () C µ u () so that (5.2) u w Lp () C µ u (). Moreover, by the rigidity estimate (1.1) we have that there exists K K such that (5.3) w K Lp () C dist( w, K) Lp () (possibly changing C, which still depends only on p and N). In view of (5.2) and (5.3), and since u(x) K for a.e. x, we deduce that u K L p () w K L p () + u w L p () C dist( w, K) Lp () + u w Lp () so that (5.1) holds. C dist( u, K) L p () + (1 + C) u w L p () (1 + C)C µ u () We are now in a position to prove Theorem 1.2. Proof of Theorem 1.2. Since u(x) K for a.e. x Ω, by Theorem 3.1 we have that µ u := curl u is a measure concentrated on J u and such that (5.4) µ u ch N 1 J u, where c = c( u ). Let us cover R N by means of disjoint cubes of side h, and let {(a i, h)} i I be the family of these cubes contained in Ω. We carry out the proof in several steps. Step 1: Piecewise constant approximation of u. By Proposition 5.1, using a rescaling argument, we have that for every i I there exists Ki h K such that (5.5) u K h i L p ((a i,h)) C hn/p h N 1 µ u ((a i, h)), where C depends only on p and N.

16 PIECEWISE RIGIDITY 15 Let us consider the piecewise constant function ψ h dened on Ω such that { K h (5.6) ψ h (x) := i if x (a i, h) 0 if x i I (a i, h) Step 2: Estimate for Dψ h. Let us estimate the total variation Dψ h of ψ h. We consider two neighbouring cubes (a i, h) and (a j, h). By applying estimate (5.1) to the rectangle R h i,j = int((a i, h) (a j, h)) (of size 2h in one direction and h in the N 1 other: the proof of Corollary 4.2 in that case is identical to the proof in the case of a cube, or, alternatively, can be easily deduced by an appropriate transformation of the cube), we have that there exists K K such that (5.7) u K L p (R h i,j ) C hn/p h N 1 µ u (R h i,j) where C depends only on N and p. Then, in view of (5.5) we get that K h i K h j K h i K + K K h j 2 1 1/p ( K h i K p + K K h j p) 1/p so that = 2 1 1/p h N/p K (K h i 1 (ai,h) + K h j 1 (aj,h)) Lp (R h i,j ) 2 1 1/p h N/p ( K u L p (R h i,j ) + u (K h i 1 (ai,h) + K h j 1 (aj,h)) L p (R h i,j ) ) 2 1 1/p h N/p ( K u L p (R h i,j ) + u K h i L p ((a i,h) + u K h j L p ((a j,h) (5.8) h N 1 K h i K h j C µ u (R h i,j) ) 2 1 1/p C + C h N 1 µ u (R h i,j) for some C depending only on N and p. We conclude that the variation of Dψ h accross the interface (a i, h) (a j, h) is estimated with the variation of the measure µ u in the union of the two cubes (a i, h) and (a j, h) and their common interface. Let now A, B be open and such that B A A Ω. By (5.8) we get that for h large enough (5.9) Dψ h (B) C µ u (A) for some C depending only on N and p. Step 3: u is piecewise constant. Since K M N N is compact, we have that ψ h is uniformly bounded in L (Ω; M N N ). In view of (5.9), and since µ u H N 1 J u, we can use the compactness in BV (see [5, Theorem 3.23]) obtaining ψ BV (Ω) such that and ψ h ψ strongly in L 1 (Ω, M N N ) (5.10) Dψ (A) CH N 1 (J u A) for every open set A Ω.

17 16 A. CHAMBOLLE, A. GIACOMINI, AND M. PONSIGLIONE Let us check that ψ = u. Since u and ψ h are uniformly bounded in L (Ω; M N N ), and since p <, by (5.5) we have that lim sup h + N N 1 u ψ h Lp (Ω) lim sup h + u ψ h Lp ((a i,h)) i I lim sup h + i I C hn/p h N 1 µ u ((a i, h)) lim sup h + C hn/p h N 1 µ u (Ω) = 0 so that ψ h u strongly in L p (Ω; M N N ), and ψ = u. By (5.10) we get that u SBV (Ω; M N N ), and that D( u) is concentrated on J u. Since H N 1 (J u ) < +, by [5, Theorem 4.23] we deduce that u is piecewise constant, i.e. there exists a Caccioppoli partition {D j } j N and matrices K j K such that (5.11) D j J u, and H N 1 ( D j ) = 2H N 1 (S( u)) 2H N 1 (J u ) j N (5.12) u = j N K j 1 Dj. Step 4: Conclusion. Let us consider the map w SBV (Ω) dened by w(x) := j N(K j x)1 Dj (x). Since w = u, and J w J u in view of (5.11), we deduce that D(u w) is supported by J u. By [5, Theorem 4.23], we conclude that there exists a Caccioppoli partition {F k } k N of Ω, and b k R N, such that and F k Ω J u, H N 1 ( F k Ω) = 2H N 1 (J u ) k N (5.13) u w = k N b k 1 Fk. Considering the Caccioppoli partition {E i } i N determined by the intersection of the families {D j } j N and {F k } k N, we deduce that there exist K i K and b i R N such that u = i x + b i )1 Ei (x) i N(K and the proof is concluded. Acknowledgments The authors wish to thank Gianni Dal Maso and Gilles Francfort for interesting and fruitful discussions. The rst and third authors are partially funded by the MULTIMAT European network MRTN-CT_

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