Pressure live loads and the variational derivation of linear elasticity

The rigorous derivation of linear elasticity from finite elasticity by means of $\Gamma$-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we compute its linearization for small pressure by $\Gamma$-convergence. We allow for a weakly coercive elastic energy density and we prove strong convergence of minimizers.


Introduction
Linear elasticity is a well-known and powerful mathematical approximation of the nonlinear theory of elasticity, with extensive application to the structural analysis and the numerical treatment of elastic bodies.In engineering textbooks its derivation is classical and is based on a formal linearization of finite elasticity about a reference configuration.A rigorous mathematical derivation via Γ-convergence was developed only rather recently in the pioneering work [8], where a Dirichlet boundary value problem was considered.A similar approach was then applied to different frameworks in elasticity, such as rubber-like materials [3], multiwell models [1,2,32], elasticity with residual stress [27,28], and incompressible materials [21].Beyond elasticity we also mention the papers [9,10,25,26] for models in fracture mechanics, [12] for viscoelasticity, [23] for plasticity, and the recent contribution [11] for materials with stress driven rearrangement instabilities.
Linearization of pure traction problems has been recently studied in [14,16,17,19], again in the context of elasticity.In this setting a full Γ-convergence result has been obtained in [22] and later extended to incompressible materials in [20].As observed in [22], in the Dirichlet case the boundary conditions prescribe the rigid motion to linearize about, whereas in the purely Neumann case the linearization process occurs around suitable rotations that are preferred by the applied forces.
In all this literature the main focus is on understanding the behavior of the bulk elastic energy and the applied forces are usually assumed to be dead loads, namely their density in the reference configuration is independent of the actual deformation.This assumption is mathematically convenient, since the work done by the loadings turns out to be a continuous perturbation of the elastic energy, so that Γ-convergence of the total energy immediately follows from Γ-convergence of the elastic energy.However, restricting the analysis to dead loads is physically unsatisfactory, since the only realistic examples of dead loads are the gravitational body force and the zero surface load (see, e.g., [29,30] and [4,Section 2.7]).
In this paper we begin a study of the derivation of linear elasticity in the presence of live loads.More precisely, we consider a pure traction problem for a hyperelastic body Ω ⊂ R n subject to a (small) pressure load on its boundary.In this setting the total energy of a deformation y : Ω → R n is given by T ε (y) := Ω W (x, ∇y(x)) dx + ε Ω π(y(x)) det ∇y(x) dx, where the elastic energy density W : Ω × R n×n → [0, +∞] satisfies the usual assumptions of nonlinear elasticity (see (W1)-(W5)) and επ is the intensity of the applied pressure load, with ε > 0 a small parameter and π : R n → R a given function.For simplicity in this introduction we assume π to be continuous.As shown in [30,Proposition 5.1] (see also [15,Proposition 1.2.8]), the second term in the energy T ε is the potential of the pressure load − επ(y(x))(cof ∇y(x))n ∂Ω (x) for x ∈ ∂Ω (1.1) acting on the whole boundary of Ω, where cof F denotes the cofactor of the matrix F and n ∂Ω is the outward unit normal to ∂Ω.In the deformed configuration y(Ω) the pressure load (1.1) corresponds to the surface force −επ(z)n ∂(y(Ω)) (z) for z ∈ ∂(y(Ω)).
Since W (x, •) is frame-indifferent and minimized at the identity, it is immediate to see that for ε = 0 the minimizers of T ε are all the rigid motions of Ω.When ε is small, it is thus natural to expect minimizers to be close to rigid motions and their asymptotic behavior to be described by a linearization of the energy.In pure traction problems, as mentioned before, the applied forces select the class of rigid motions around which the linearization takes place (see [22]).Indeed, if y ε is a minimizer of T ε , then we have for every rotation R ∈ SO(n).
If we assume y ε to be of the form y ε (x) = R 0 (x + εu 0 (x)) with R 0 ∈ SO(n), then by a formal expansion we obtain for every rotation R ∈ SO(n).Here Q(x, •) is the quadratic form given by the Hessian of W (x, •) computed at the identity and e(u 0 ) is the symmetric gradient of u 0 .Dividing by ε and letting ε tend to zero, we deduce that R 0 is a so-called optimal rotation, that is, R 0 belongs to the set Assume now for simplicity that the identity matrix belongs to R (one can always reduce to this case, up to rotating the whole system).The previous argument suggests that in order to identify the limiting behavior of minimizers one needs to renormalize the energy as follows: Under suitable assumptions for π and a weak p-coercivity condition on W with 1 < p ≤ 2 (see (W5)), we compute the Γ-limit of the rescaled energies 1  ε 2 E ε and we establish a compactness result for deformations with equibounded energies.Here deformations are assumed to have zero average on Ω, as it is common in Neumann boundary value problems.More precisely, we prove the following results: Compactness: If E ε (y ε ) ≤ Cε 2 , then there exist rotations R ε ∈ SO(n) and displacements u ε ∈ W 1,p (Ω; R n ) such that and, up to subsequences, there holds Γ-convergence: Under the above notion of convergence y ε → (u 0 , R 0 ), the rescaled energies We also deduce strong convergence of (almost) minimizers: if (y ε ) is a sequence of (almost) minimizers, then we have in addition that u ε → u 0 strongly in W 1,p (Ω; R n ) and the pair (u 0 , R 0 ) is a minimizer of E 0 .
We now comment on the expression of the limiting energy E 0 , which features two terms: the usual linear elastic energy and a potential term accounting for the surface load −π(R 0 x)n ∂Ω (x) on ∂Ω.The emergence of this boundary term can be explained by the following heuristic considerations: on the one hand, a formal linearization of (1.1) leads to a pressure load of the above form; on the other hand, if y is smooth enough, the force term in (1.2) can be written as Hence, taking into account (1.3), computing the limit of the above expression on sequences (y ε ) with equibounded energies corresponds to a sort of shape derivative of the functional Ω → Ω π(x) dx (see, e.g., [18,Proposition 17.8]).However, we stress that in the present setting deformations y ε are only of Sobolev regularity and are close to rigid motions only in the sense of W 1,p (Ω; R n ), therefore the usual arguments in the context of shape derivatives do not apply.
From a mathematical viewpoint the main difference with respect to previous contributions dealing with dead loads, is that the force term in (1.2) is not a continuous perturbation of the elastic energy.Indeed, our assumptions on W imply that deformations are at most strongly convergent in W 1,p (Ω; R n ) with 1 < p ≤ 2 and this is not enough to guarantee convergence of the determinants.Moreover, the crucial step in the proof of compactness is to show that deformations satisfying the bound E ε (y ε ) ≤ Cε 2 have an elastic energy of order ε 2 .Once this is established, one can apply the rigidity estimate by Friesecke, James, and Müller [13] and deduce (1.3), together with a uniform bound for (u ε ) in W 1,p (Ω; R n ).In the case of dead loads deducing the ε 2 -bound on the elastic energy is straightforward, since the force term is linear with respect to the deformation.In our setting, instead, this is one of the main difficulties.We show that the problem can be solved under two different sets of conditions: • π Lipschitz continuous in a suitable neighborhood of Ω and nonnegative; • π Lipschitz continuous in a suitable neighborhood of Ω, with a growth condition on its negative part (see (π3)) and an additional coercivity property for W (•, F ) in terms of det F (see (W6)).We note that this additional coercivity condition on W is satisfied by a large class of elastic materials (see, e.g., [3,Remark 2.8]).
We also observe that both the nonlinear energy (1.2) and the Γ-limit E 0 are well defined if π is merely a continuous function.However, because of the low regularity of deformations, in our proofs we need π to be Lipschitz continuous, at least in a suitable neighborhood of Ω.How to extend our analysis to less regular pressure loads is an interesting question that we plan to consider in a future work.
Finally, in [22] it is proved that, in the case of dead loads, the set R of optimal rotations is a submanifold of SO(n) and, as a consequence, one can prove that the distance of the approximating rotations R ε in (1.3) from R is at most of order √ ε.In the last part of the paper we show that neither of these properties is true, in general, in the present setting.
Plan of the paper.In Section 2 we set the problem and we state the main assumptions.In Section 3 we discuss the case of a nonnegative pressure intensity π.We then extend our analysis to pressures with arbitrary sign in Section 4. Finally, in Section 5 we compute a refined Γ-limit, which takes into account how much deformations differ from being optimal rotations, and we make a comparison with the results proved in [22] in the case of dead loads.

Setting of the problem
2.1.Notation and preliminaries.Throughout the paper, the symbols C or c will be used to denote some positive constants not depending of ε, whose value may change from line to line.
Given two (extended) real numbers a and b the notation a ∨ b (respectively, a ∧ b) stands for the maximum (respectively, the minimum) between the two numbers.Given a scalar function f , we denote its positive and negative part by f + and f − , respectively, so that f = f + − f − .By B r ⊂ R n we mean the open ball with radius r > 0 centered at the origin.
Let Ω be an open set in R n .For p ∈ [1, ∞] the norms in L p (Ω) and L p (Ω; R n ) will be simply denoted by • p .The conjugate exponent of p ∈ [1, ∞] will be denoted by p ′ .The notation W 1,p (Ω; R n ) stands for the space of Sobolev functions y ∈ W 1,p (Ω; R n ) with zero average; if We denote by R n×n , R n×n sym , and R n×n skew the set of (n × n)-matrices and the subsets of symmetric and skew-symmetric matrices, respectively.The set of rotations is denoted by SO(n), namely Finally, we recall that for every F ∈ R n×n and ε > 0 there holds where ι k (F ) is a homogeneous polynomial of degree k in the entries of F .In particular, there exists a constant C > 0, depending only on n, such that For k = 1 and k = n we have that ι 1 (F ) = tr F and ι n (F ) = det F .For the definition and the properties of Γ-convergence we refer to the monograph [7].

The main assumptions.
Let Ω ⊂ R n , with n ≥ 2, be a bounded domain with Lipschitz boundary representing the reference configuration of a hyperelastic body.Up to a translation of the axes, we can assume without loss of generality that the origin is the barycenter of Ω, i.e.,

Ω
x dx = 0. (2.3) For future use we also introduce the set which is an open annulus (if 0 ∈ Ω; an open ball if 0 ∈ Ω) centered at 0 and containing Ω.
The stored energy density of the body is assumed to be a Carathéodory function W : Ω × R n×n → [0, +∞] satisfying the following conditions for almost every x ∈ Ω: (W1) W (x, F ) = +∞ if det F ≤ 0 (orientation preserving condition); (W2) W (x, RF ) = W (x, F ) for every F ∈ R n×n and R ∈ SO(n) (frame indifference); (W3) W (x, I) = 0 (the reference configuration is stress-free); (W4) W (x, •) is of class C 2 in a neighborhood of SO(n), independent of x, where the second derivatives of W are bounded, uniformly with respect to x ∈ Ω; (W5) W (x, F ) ≥ c 1 g p (dist(F ; SO(n))) for every F ∈ R n×n and for some p ∈ (1, 2], where g p is defined as and c 1 > 0 is a constant independent of x (coercivity).
Assumptions (W1)-(W3) are natural conditions in elasticity theory (see, e.g., [4,15]), assumption (W4) is the minimal regularity hypothesis needed to perform the linearization, while condition (W5) is satisfied by a large class of compressible rubber-like materials (see, e.g., [2,3,19,20,21]).We note that Moreover, condition (W5) implies the following bound: for a suitable constant c > 0, independent of x.Indeed, by (W5) the bound (2.6) is satisfied for F outside a neighborhood of SO(n).If instead dist(F ; SO(n)) is small enough, then one has which implies (2.6) by using again (W5).We assume the body to be subjected to a pressure load, whose (unscaled) intensity is a Borel measurable function π : R n → R such that (π1) π is Lipschitz continuous in an open set containing O.
Since we do not prescribe any Dirichlet boundary condition, the linearization process will naturally select, as in [22] in the case of dead loads, a particular set of rotations that are "preferred" by the force.This set is called the set of optimal rotations and in our framework it is defined as Since the map R → Ω π(Rx) dx is continuous and SO(n) is compact, the set of optimal rotations is not empty and is a compact subset of SO(n).For simplicity we assume that where I is the identity matrix.Indeed, if this is not the case, we can always replace π by π(R 0 •) and deformations y by R T 0 y, where R 0 is a given optimal rotation.Let R 0 ∈ R. By computing the first variation of the functional in (2.7) along the curve t → R 0 e tA with A ∈ R n×n skew , we deduce that any optimal rotation R 0 satisfies the following Euler-Lagrange equation: (2.9) Applying the Divergence Theorem, condition (2.9) can be rewritten as where n ∂Ω is the outward unit normal to ∂Ω.

Nonnegative pressure loads
We start our analysis by considering a pressure load with nonnegative intensity, that is, (π2) π(y) ≥ 0 for every y ∈ R n .
This includes, for instance, the relevant case of hydrostatic pressure π(y) = gρy − 3 , where g is the gravitational constant, ρ is the constant density of the fluid, and y − 3 denote the negative part of the third component of y.
For every ε ∈ (0, 1) we consider the energy where the set of admissible deformations is In other words, admissible deformations are orientation preserving and, as it is common in Neumann boundary value problems, have zero average on Ω.By (2.3) any rigid motion of the form belongs to Y p .Moreover, under the assumption (π2), the energy is well defined since the two integrands W (•, ∇y) and π(y) det ∇y are nonnegative for y ∈ Y p .
Remark 3.1.Here we do not assume deformations to be injective in any sense.However, one can easily include the requirement that admissible deformations are a.e.injective (see, e.g., [4]), without affecting the results of the paper, see also Remark 3.13.
The key ingredient in the proof of compactness is the following variant of the celebrated rigidity estimate by Friesecke, James, and Müller [13], whose proof can be found, e.g., in [3,Lemma 3.1].Similar variants of the rigidity estimates with mixed growth condition have been proved in [5,24,31].Theorem 3.2.There exists a positive constant C = C(Ω, p) > 0 with the following property: The following generalized rigidity estimate will be used in Theorem 3.14 to infer strong convergence of almost minimizers.For a proof we refer to [6, Theorem 1.1].
there exist a constant rotation R ∈ SO(n) and two functions g i ∈ L p i (Ω) such that Our arguments will strongly rely on the Lipschitz continuity of the pressure function π.This, however, holds only in a suitable set Ω ′ containing Ω (see (π1)).Since deformations y ∈ Y p may be a priori valued outside Ω ′ , it is convenient to introduce an auxiliary Lipschitz continuous function that coincides with π in Ω ′ and is bounded on the whole of R n .This is the content of the following lemma, which is clearly not necessary if π itself is Lipschitz continuous and bounded.
Lemma 3.4.Assume (π1) and (π2).Then there exists a Lipschitz continuous function π : R n → [0, +∞), with compact support, such that π coincides with π in an open neighborhood of O and π(y) ≤ π(y) for all y ∈ R n .In particular, π is bounded.Remark 3.5.Note that the set of optimal rotations (2.7) stays the same if π is replaced by π, since π and π coincide in a neighborhood of O.
Proof of Lemma 3.4.Assume 0 ∈ Ω, so that O is an open annulus centered at 0 (the case where 0 ∈ Ω can be treated similarly).Let 0 < r 1 < r 2 and 0 where It is easy to see that π is Lipschitz continuous in its domain; moreover, by construction it coincides with π in an open neighborhood of O.We now show that π ≤ π on B r 2 +δ \ B r 1 −δ .Indeed, if r 1 − δ ≤ |y| < r 1 , by the Lipschitz continuity of π we have and similarly if r 2 ≤ |y| ≤ r 2 + δ.Finally, we note that, if y ∈ ∂B r 1 −δ ∪ ∂B r 2 +δ , then We now conclude by considering π(y We are now in a position to state and prove some estimates which will be crucial to infer compactness of deformations with equibounded (rescaled) energy.Lemma 3.6.Assume (W1)-(W5), (π1), and (π2).If E ε (y ε ) ≤ Cε 2 for every ε ∈ (0, 1), then there holds 2) for the definition of y Rε ) satisfy and are uniformly bounded in W 1,p (Ω; R n ).
Proof.Let E ε be the auxiliary energy defined as in (3.1) with π replaced by the function π given by Lemma 3.4.Since π ≥ π everywhere and π ≡ π on Ω, we have that For the sake of brevity we introduce the notation , we have that y ε belongs to Y p , hence in particular det ∇y ε > 0 a.e. in Ω.By (W5) and Theorem 3.2 we infer the existence of By (2.8) we deduce that Since π ≥ 0 by construction, the integrand in the last integral above is nonpositive on Ω + ε .Thus, using the fact that π is Lipschitz continuous and bounded, and applying Hölder's inequality we deduce that (3.11) By combining (3.10) and (3.11) we deduce and, as a consequence of (3.4), (3.9), and (3.12), we obtain Using the definition (2.4) of g p this implies that By Hölder's inequality we obtain ) where the last inequality follows from the fact that t p 2 ≤ 1 + t for t ≥ 0. Again from (2.4) and recalling that p ≤ 2 we also have that By (3.15), (3.16), and the continuous embedding of Since u ε has zero average, Poincaré-Wirtinger inequality finally yields Finally, if (R ′ ε ) is a sequence of rotations whose corresponding rescaled displacements satisfy (3.5), then As an immediate corollary we obtain that the infimum of the energy E ε is of order ε 2 .Corollary 3.7.Assume (W1)-(W5), (π1), and (π2).Then Proof.Let (y ε ) be a minimizing sequence satisfying Using the fact that W is nonnegative and arguing as in the proof of Lemma 3.6, we deduce that where the last inequality follows from Lemma 3.6.This proves the first inequality in (3.17).
The other inequality in (3.17) follows trivially by the fact that the energy E ε is zero on the identity map.
Since SO(n) is a compact set, there exists R 0 ∈ SO(n) such that R ε → R 0 , up to subsequences.To prove that R 0 ∈ R we argue as in the proof of Lemma 3.6 and deduce Note that in the last integral we used that π ≡ π on O.By multiplying by ε and then letting ε → 0 we infer that This implies that R 0 ∈ R since by assumption the identity matrix is an optimal rotation.Uniqueness of R 0 is a straightforward consequence of (3.6).Uniqueness (up to an infinitesimal rigid motion) of u 0 follows by arguing as in [22,Theorem 5.1], recalling that displacements have zero average in our setting.
The following proposition will be useful in both the liminf and the limsup inequalities to characterize the asymptotic behavior of the rescaled pressure potential.Note that, besides the presence of π in place of π, the integral at the left-hand side of (3.20) differs from the rescaled pressure potential in the total energy whenever R ε is not an optimal rotation.Proposition 3.9.Let π be a function as in Lemma 3.4
Since by (2.2) we have that for k = 1, . . ., n for a.e.x ∈ G ε , we deduce that G ε ⊂ Ω − ε for ε small enough.Therefore, using the nonnegativity of π and (2.1) again, we obtain Indeed, since π = π on O, we may write Using the Lipschitz continuity of π and the definition of G ε , the first integral at the right-hand side can be bounded as follows: , where the last term goes to zero, as ε → 0. Since χ Gε converges to 1 boundedly in measure, we have that χ Gε div u ε ⇀ div u 0 weakly in L p (Ω), hence the second term in (3.23) goes to zero, as well.This proves (3.22).
We now prove that both J 2 ε and J 3 ε converge to 0, as ε → 0. By (2.2) and (3.4), since π is bounded, we obtain To bound J 3 ε we use (3.3) and deduce which vanishes by (3.19).By combining the previous inequalities we conclude that lim inf ε→0 We now claim that lim ε→0 Assuming this is true, the thesis follows by (3.21), (3.24), (3.25), and the Divergence Theorem, since ).To conclude we only need to prove (3.25).We can write the integrand in and owing to the Lipschitz continuity of π we have Hence, proving (3.25) is equivalent to show that lim ε→0 On the other hand, u ε → u 0 strongly in L 1 (Ω; R n ) by compact embedding.Thus, by (3.27) and the Generalised Dominated Convergence Theorem, (3.28) is proved if we show that the integrand (3.26) converges a.e. to ∇π(R 0 x) • R 0 u 0 (x).
To this aim, we first note that, up to subsequences, Indeed, the convergence is actually in L 1 (O; R n ).This can be easily proved by approximating ∇π with functions in C 0 (O; R n ).Now, by Rademacher Theorem (we point out that we are working with a countable sequence of rotations R ε ) for almost every x ∈ Ω we have Since u ε → u 0 a.e., up to subsequences, and π ≡ π on O, we deduce the desired convergence.This concludes the proof.
With the result of Proposition 3.9 at hand, we are now in a position to state and prove the liminf and the limsup inequalities for the energy functionals 1  ε 2 E ε .Proposition 3.10 (Liminf inequality).Assume (W1)-(W5), (π1), and (π2).For every ε ∈ (0, 1) let y ε ∈ W 1,p (Ω; R n ) be such that there exist R ε ∈ SO(n) converging to R 0 ∈ R and the corresponding displacements u ε , defined as in (3.4), The density Q(x, •) is the quadratic form given by and e(u 0 ) denotes the symmetric gradient of u 0 .
Remark 3.11.For any optimal rotation R 0 ∈ R the functional E 0 (•, R 0 ) is invariant under perturbations by infinitesimal rigid motions.Indeed, if u ′ 0 (x) = u 0 (x) + Ax with A ∈ R n×n skew , then clearly e(u ′ 0 ) = e(u 0 ) and by (2.10) Proof of Proposition 3.10.Without loss of generality we can assume lim inf so that y ε ∈ Y p and, up to subsequence, E ε (y ε ) ≤ Cε 2 .By Lemma 3.6 and Proposition 3.8 there exist a (possibly different) sequence (R ′ ε ) ⊂ SO(n) such that, up to subsequences, R ′ ε → R 0 , the corresponding displacements u ′ ε satisfy (3.5) and, up to subsequences, weakly converge to u 0 +Ax for some A ∈ R n×n skew .However, by Remark 3.11 we can assume, without loss of generality, that R ε = R ′ ε and so, u ε = u ′ ε and A = 0. Let E ε be the auxiliary energy defined as in (3.1) with π replaced by the function π given by Lemma 3.4.By the properties of π we have Arguing as in [3, Proof of Theorem 2.4] one can prove that lim inf Since condition (3.3) is satisfied by Lemma 3.6, we can apply Proposition 3.9 and we obtain lim inf Finally, assumption (2.8) guarantees that the last term in (3.30) is nonnegative.This proves the desired inequality.
We first observe that y ε ∈ Y p for ε small enough.Indeed, by (2.3) it has zero-average and by (2.1) it satisfies for a.e.x ∈ Ω. (3.32) Since by (2.2) we have that for k = 1, . . ., n for a.e.x ∈ Ω, for ε small enough we obtain and thus, det ∇y ε > 0 a.e. in Ω.By (3.31) we have that for ε small enough the set y ε (Ω) is contained in the neighborhood of O where π and π coincide.Therefore, using also that R 0 ∈ R, we can write 1 Arguing as in [3, Proof of Theorem 2.4], one can show that lim sup On the other hand, by (3.32) we have that 3) is satisfied and we can apply Proposition 3.9.By (3.33) we deduce This concludes the proof.Combining together the previous propositions, we can prove the main result of this section.It ensures that almost minimizers of the nonlinear energy strongly converge to minimizers of the limiting energy.Theorem 3.14 (Convergence of almost minimizers).Assume (W1)-(W5), (π1), and (π2).If (y ε ) is a sequence of almost minimizers for the energies E ε , that is, then there exist R ε ∈ SO(n) such that, up to passing to a subsequence, we have Proof.Let (y ε ) be a sequence of almost minimizers.By Corollary 3.7 we have that inf hence by Proposition 3.8 there exist u 0 ∈ H1 (Ω; R n ) and R 0 ∈ R such that, up to a subsequence, We now show that (u 0 , R 0 ) is a minimizer of E 0 .To this aim let (v, S) ∈ H1 (Ω; R n ) × R and let (v ε , S ε ) be a recovery sequence for (v, S), as in Proposition 3.12.Let z ε (x) := S ε (x + εv ε (x)).By Proposition 3.10 we have This implies that E 0 is minimized at (u 0 , R 0 ) and, as a consequence, (3.35) hold.
To conclude, it remains to prove that u ε → u 0 strongly in W 1,p (Ω; R n ).We adapt the argument in [3, Theorem 2.5] to our framework.We claim that the following properties hold: (a) χ Gε e(u ε ) → e(u 0 ) strongly in L 2 (Ω; R n×n sym ), where the set G ε is defined as in (3.18);(b) the sequence 1 The thesis follows from (a) and (b), by using Vitali's convergence theorem together with Korn's second inequality, see [3, proof of Theorem 2.5] for more details.
We now prove (a).By choosing (v, S) = (u 0 , R 0 ) in (3.36) we deduce . By (3.30) and assumption (2.8) we have 1 Therefore, letting ε → 0 and applying Proposition 3.9 yield lim sup On the other hand, by Taylor expansion of W around I and by the weak convergence of χ Gε e(u ε ) to e(u 0 ) in L 2 (Ω; R n×n sym ) (see property (ii) in the proof of Proposition 3.8) we obtain lim sup see, e.g., [3,8,22].Combining the previous inequalities yields We now prove claim (c).Given α > p and η > 0, we deduce by (b) that there exists M η > 0 such that, setting where we have that where the last inequality follows from Hölder's inequality, (3.38), and (3.39).
On the other hand, by (3.40) we can write Thus, by (3.38), (3.39), and (3.41) we have that for every measurable set Now, for every δ > 0 we can choose first η = η(δ) and then ω = ω(δ, η) in such a way that the right-hand side above is less than δ for every measurable set A ⊂ Ω with |A| < ω.This proves claim (c) and concludes the proof of the theorem.

Pressure loads of arbitrary sign
Here we extend the results of the previous section to pressure loads whose intensity π is not necessarily nonnegative (and still satisfies (π1)).To deal with the negative part of π we need to assume an additional bound from below for W (•, F ) in terms of det F : (W6) W (x, F ) ≥ c 2 g q (| det F − 1|) for a.e.x ∈ Ω and for every F ∈ R n×n , for some q ∈ [1, 2], where g q is defined as in (2.4) and c 2 > 0 is a constant independent of x.According to the value of q in (W6), we assume π to satisfy the following condition: We note that the growth condition in (π3) is at most linear, since p, q ∈ (1, 2] implies p/q ′ ∈ (0, 1]. In the current framework the energy E ε is defined as in (3.1) with the set of admissible deformations Y p replaced by Owing to (π3) the energy is well defined on Y p q : indeed, if y ∈ Y p q , then the composition π − • y belongs to L q ′ (Ω) and thus, π(y) det ∇y is integrable.Clearly, all rigid motions y R with R ∈ SO(n) are still admissible deformations.As observed in Remark 3.1, also in this setting the a.e.injectivity condition can be included in the definition of Y p q without altering the results of this section.
As in the previous section we need a Lipschitz continuous function that extends π outside a neighborhood of O, is below π everywhere, and satisfies the same growth condition (π3) as π.
Proof.We consider only the case q ∈ (1, 2], being the case q = 1 analogous and even simpler.Let C > 0 be a constant for which (π3) is satisfied and let Since p/q ′ ∈ (0, 1], the function h is Lipschitz continuous in the whole of R n and π ≥ −h by (π3).Hence we can apply Lemma 3.4 to the function Π := π + h.This provides us with a function Π.It is now easy to check that the function π := Π − h has all the required properties.
Under this new set of assumptions, the results of Section 3 can be modified as follows.
If, moreover, (R ′ ε ) ⊂ SO(n) is another sequence for which the rescaled displacements, defined as in (3.4), satisfy (4.3), then Proof.The choice of ε 0 will be made throughout the proof.We follow the lines of the proof of Lemma 3.6.By Lemma 4.1 inequality (3.7) still holds.By (W5) and Theorem 3.2 there exists a sequence (R ε ) ⊂ SO(n) such that and We denote by P ε the last term in the above inequality.In the following, c 2 is the constant in condition (W6).If q = 1, the function π is bounded and thus, recalling the definition (3.8) of the sets Ω ± ε , we have where we used Cauchy's inequality and (2.5).If instead q ∈ (1, 2], the function π is Lipschitz continuous and satisfies a p/q ′ -growth condition with p/q ′ ≤ 1, so that we obtain Using that | det ∇y ε − 1| ≤ 1 on Ω − ε and applying Hölder's inequality, from the previous equation we deduce where we used Young's inequality, (2.5), and the fact that q ′ ≥ 2. Combining (4.4) with the previous bounds on P ε , we obtain that in both cases q = 1 and q ∈ (1, 2].Now, if ε 0 ≤ c 2 /(4C), where C is a constant for which (4.5) is true, then by (W6) we deduce that for every ε ∈ (0, ε 0 ).Combining (4.6) and (4.5) yields Up to choosing ε 0 smaller, if needed, we obtain Inequality (4.2) now follows easily from (4.6), while (4.3) is a consequence of (4.7) and (W5).The last two statements of the lemma can be proved arguing exactly as in the proof of Lemma 3.6 and Corollary 3.7.
The proof of the following compactness result is completely analogous to that of Proposition 3.8.Proposition 4.3 (Compactness).Assume (W1)-(W6), (π1), and (π3).If E ε (y ε ) ≤ Cε 2 for ε ∈ (0, ε 0 ), then for any R ε , u ε given by Lemma 4.2, we have that, up to subsequences, as ε → 0.Moreover, R 0 is independent of the choice of R ε and u 0 is independent up to infinitesimal rigid motions of the form Ax, with A ∈ R n×n skew .The next proposition is the analog of Proposition 3.9.However, in the present setting, owing to the assumptions (W6) and (π3), we can improve the result and show convergence on the whole of Ω.
Proof.We follow the lines of the proof of Proposition 3.9.The only difference is in the analysis of the term I ε in (3.21), which now can be written as where the sets Ω ± ε and G ε are defined as in (3.8) and (3.18).Here we recall that G ε ⊂ Ω − ε for ε small enough.The first integral J1 ε can be handled exactly as in the proof of Proposition 3.9.To conclude it is enough to show that the remaining terms are infinitesimal, as ε → 0.
To deal with J4 ε we consider the two cases q = 1 and q ∈ (1, 2], separately.If q = 1, the function π is bounded and so, we have where in the last inequality we used (4.2).
If instead q ∈ (1, 2], by using the p/q ′ -growth of π we have By (4.2) and the boundedness of (u ε ) in W 1,p (Ω; R n ) we deduce It is easy to verify that 2 q + p q ′ − 1 > 0.Moreover, since G ε ⊂ Ω − ε for ε small enough, we have that |Ω + ε | → 0 by (3.19).Therefore, we conclude that in both cases J 4 ε is infinitesimal, as ε → 0. The next proposition collects both the liminf and the limsup inequalities, which thus provide a full Γ-convergence result.For every ε ∈ (0, ε 0 ) let y ε ∈ W 1,p (Ω; R n ) be such that there exist R ε ∈ SO(n) converging to R 0 ∈ R and the corresponding displacements u ε , defined as in (3.4), weakly converge in , where E 0 is the functional defined in (3.29).

On the other hand, for every
Proof.The proof is the same of Propositions 3.10 and 3.12, once we have at our disposal Proposition 4.4.The only additional remark is that the deformations y ε in the recovery sequence are admissible since det ∇y ε ∈ L ∞ (Ω) by construction (see (4.1) for the definition of Y p q ).
Combining the previous results and arguing as in Theorem 3.14, one can infer the following convergence result for almost minimizers.If (y ε ) is a sequence of almost minimizers for the energies E ε , that is, then there exist R ε ∈ SO(n) such that, up to passing to a subsequence, we have

A refined Γ-limit and a comparison with dead loads
In this section we make a comparison with the results obtained in [22], in the case of dead loads.In particular, we compute a refined version of the Γ-limit of the rescaled energies 1  ε 2 E ε .We assume (W1)-(W5) and (π1), together with either (π2) or (π3) and (W6).In addition, we require (π4) π is of class C 2 in the open set given by (π1).
Under this assumption any optimal rotation R 0 ∈ R satisfies, in addition to (2.9), the following condition: This is obtained by imposing that the second variation of the functional in (2.7) is positive semidefinite along the curve t → R 0 e tA .By the Divergence Theorem, condition (5.1) can be rewritten as (5.2) In [22] the applied body force is assumed to be a dead load of the form where g ∈ L 2 (Ω; R n ) is given.In this setting the authors proved that the set of optimal rotations, which is defined as is a submanifold of SO(n) (see [22,Proposition 4.1]).Moreover, if (y ε ) is a sequence of deformations with total energy of order ε 2 , then any sequence of rotations (R ε ) provided by the rigidity estimate converges to an optimal rotation R 0 (as in Propositions 3.8 and 4.3) and, in addition, where dist SO(n) is the intrinsic distance in SO(n), that is, dist SO(n) (R, S) := min |A| : A ∈ R n×n skew , R = Se A .Finally, the Γ-limit of the rescaled energies can be expressed as where u 0 is the limit displacement, A 0 ∈ R n×n skew is the limit of (which exists up to subsequences), and P g is the projection operator on R g (see [22,Section 5]).We note that the last term in (5.5) is the second variation of the functional in (5.3) at R 0 computed in the direction A 0 .In [22] it is then proved that A 0 = 0 for sequences (y ε ) of almost minimizers, so that the last term in (5.5) is identically equal to 0 on minimizers.
In our setting of a pressure live load, a first difference with [22] is that the set of optimal rotations may not be a manifold, as the following example shows.
Example 5.1.Let n = 2 and let Ω be the set given in polar coordinates by , and ρ < 2 otherwise} (see Figure 1).We note that (2.3) is satisfied.
This example shows that, in general, we cannot expect R to be a manifold.
Since R is not in general a manifold, the projection operator on R is not well defined.However, in the limiting process we can keep track of the distance of the approximating rotations R ε from R through a suitable sequence of skew-symmetric matrices A ε .In contrast with (5.4), the scaling of this distance may be larger than √ ε (actually, larger than k √ ε for any given k > 2), see Example 5.3.To recover compactness of (A ε ) we rescale it by |A ε | ∨ √ ε and we denote by A 0 its limit.The Γ-limit of the rescaled energies can be then expressed as where F : R× R n×n skew → [0, +∞) is the second variation of the functional in (2.7).This additional term measures the cost due to the fluctuations of the approximating rotations from the set R. Arguing as in (5.1) and (5.2), the functional F takes the form For sequences (y ε ) of almost minimizers the limit A 0 may be different from 0; however, we have F(R 0 , A 0 ) = 0.More precisely, we have the following result.Theorem 5.2.Under the assumptions of Proposition 3.8 or Proposition 4.3, we have in addition that there exist A ε ∈ R n×n skew such that R ε = S ε e Aε for some S ε ∈ R and, up to subsequences, If also (π4) is in force, then with respect to the following convergences: skew as above such that, up to a subsequence, we have Furthermore, the triplet Proof.As for the compactness statement, since R is a closed set, there exist S ε ∈ R and Since, up to subsequences, R ε → R 0 ∈ R, we have that A ε → 0. The remaining properties follow from the fact the sequence Aε |Aε|∨ √ ε is bounded by 1.We now give a sketch of the proof of the liminf inequality.By Proposition 3.10 or 4.5 we have that lim inf In fact, the last term above was always neglected in the previous computations, since it is nonnegative by (2.8).Using that S ε ∈ R, a Taylor expansion of π and of the exponential map yields where t ε (x) ∈ [0, 1], H ε is a uniformly bounded matrix, and We now note that the first integral in (5.7) is equal to 0 by (2.9); thus, by multiplying and dividing by α We observe that the left-hand side is nonnegative, hence the term within square brackets is nonnegative, as well.Since The liminf inequality follows now from the Divergence Theorem.
For the construction of the recovery sequence we proceed as in Proposition 3.12 or 4.5, but choosing R ( The first two integrals at the right-hand side can be bounded by E 0 (u 0 , R 0 ), arguing as in Proposition 3.12 or 4.5.Repeating the same computations as for the liminf inequality, one can show that the last integral converges to 1 2 F(R 0 , A 0 ).Convergence of almost minimizers and of infima can be proved exactly as in Theorems 3.14 and 4.6.Finally, we observe that the functional F is always nonnegative by (5.2) and F(R, 0) = 0 for every R ∈ R. Hence, by minimality we deduce that F(R 0 , A 0 ) = 0.This concludes the proof.
We conclude the paper with an example showing that, given any sequence λ ε → 0 such that lim for some k > 2, there may exist sequences of almost minimizers of E ε for which any approximating sequence of rotations has a distance from R of order λ ε .This provides a further difference with the case of dead loads [22].
Example 5.3.We start by considering a sequence (λ ε ) satisfying (5.9) with k = 3.We assume the pressure intensity π to be of class C 3 .Moreover, we assume that the set of optimal rotations R is finite and that for every R 0 ∈ R there exists A 0 ∈ R n×n skew such that |A 0 | = 1 and F(R 0 , A 0 ) = 0. (5.10) These properties are satisfied, for instance, in Example 5.1 if the function ϕ is strictly increasing.Indeed, in this case the function φ in (5.6) attains its minimum only at α = 0 and α = π and thus, R = {±I}.Moreover, in this example ∇π(x) = ∇π(−x) = 0 for every x ∈ ∂Ω, so that F(R 0 , A 0 ) = 0 for every R 0 ∈ R and every A 0 ∈ R n×n skew .Finally, π is of class C 3 if we assume, in addition, ϕ ∈ C 4 ([0, π/2]) with ϕ (iv) (0) = ϕ (iv) (π/2) = 0 and ψ ∈ C 3 ([1, +∞)) with bounded third derivative and ψ ′′′ (1) = 0. Now let (u 0 , R 0 ) be a minimizer of E 0 on H1 (Ω; R n ) × R and let A 0 ∈ R n×n skew satisfy (5.10).Let R ε := R 0 e λεA 0 and let (u ε ) be an approximating sequence for u 0 as in (3.31).We claim that the deformations y ε (x) := R ε (x + εu ε (x)) are a sequence of almost minimizers of E ε .Indeed, arguing as in the proof of the limsup inequality in Theorem 5.2, we have by (5.8) Therefore, the claim is proved if we show that the last term above vanishes, as ε → 0. To prove it we argue as in the proof of the liminf inequality in Theorem 5.2, now expanding up to the third order.By (2.9) and (5.10) we obtain which proves the claim owing to (5.9) with k = 3.We note that the intrinsic distance of R ε = R 0 e λεA 0 from R in SO(n) is of order λ ε .Indeed, since R is finite by assumption, we have that d SO(n) (R ε ; R) = d SO(n) (R ε , R 0 ) for ε small enough.By definition we clearly have d SO(n) (R ε , R 0 ) ≤ λ ε .On the other hand, since the intrinsic distance in SO(n) is equivalent to the Euclidean distance, we obtain d SO(n) (R ε , R 0 ) ≥ c|e λεA 0 − I| ≥ cλ ε .
Using again that |R ε − R ′ ε | ≤ Cε, it is easy to see that in fact the intrinsic distance of R ′ ε from R is of order λ ε .If, instead, the sequence (λ ε ) satisfies (5.9) for some k ≥ 4, the previous arguments can be adapted with small changes as follows.We assume, in addition, that π is of class C k and that for every R 0 ∈ R there exists A 0 ∈ R n×n skew such that |A 0 | = 1 and V j (R 0 , A 0 ) = 0 for every j = 2, . . ., k − 1, (5.13) where V j (R 0 , A 0 ) is the j-th variation of the functional in (2.7) at R 0 computed in the direction A 0 .This is fulfilled by the pressure load in Example 5.1, if ϕ and ψ have enough regularity and satisfy suitable boundary conditions.By expanding up to order k in (5.11), condition (5.13) guarantees that the sequence (y ε ), constructed as above, is still a sequence of almost minimizers.The bounds on the intrinsic distance from R can be proved as before.
Remark 5.4.If condition (5.10) is not satisfied, that is, for every R 0 ∈ R one has F(R 0 , A 0 ) = 0 ⇐⇒ A 0 = 0, the phenomenon described in the previous example can not arise.More precisely, one can show that the intrinsic distance of the approximating rotations from R is at most of order √ ε, as in (5.4).The argument is the same as in [22,Theorem 5.1], combined with (5.7).

Remark 3 . 13 .
If we include a.e.injectivity in the definition of the space Y p of admissible deformations (see Remark 3.1), the limsup inequality can be proved by means of the same recovery sequence.Indeed, by [4, Theorem 5.5-1(b)] the deformations y ε are a.e.injective owing to (3.31).

Proposition 4 . 4 .
Let π be a function as in Lemma 4.1 and let y