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We study stochastic homogenization by Γ-convergence of nonconvex integrals of the calculus of variations in the space of functions of bounded deformation.

We establish a convergence theorem for a class of nonlinear reaction-diffusion equations when the diffusion term is the subdifferential of a convex functional in a class of functionals of the calculus of variations equipped with the Mosco-convergence. The reaction term, which is not globally Lipschitz with respect to the state variable, gives rise to bounded solutions, and cover a wide variety of models. As a consequence we prove a homogenization theorem for this class under a stochastic homogenization framework.

In this paper we establish an unbounded version of the integral representation theorem by Buttazzo and Dal Maso [see Buttazzo Dal Maso (Nonlinear Anal 9(6):515–532, 1985) and also Bouchitte et al. (Arch Ration Mech Anal 165(3), 187–242, 2002)]. More precisely, we prove an integral representation theorem (with a formula for the integrand) for functionals defined on $$W^{1,p}$$ with $$p>N$$ (N being the dimension) that do not satisfy a standard p-growth condition from above and can take infinite values. Applications to $$\Gamma $$ -convergence, relaxation and homogenization are also developed.

International audience; We establish a convergence theorem for a class of two components nonlinear reaction-diffusion systems. Each diffusion term is the subdifferential of a convex functional of the calculus of variations whose class is equipped with the Mosco-convergence. The reaction terms are structured in such a way that the systems admit bounded solutions, which are positive in the modeling of ecosystems. As a consequence, under a stochastic homogenization framework, we prove two homogenization theorems for this class. We illustrate the results with the stochastic homogenization of a prey-predator model with saturation effect.

A necessary condition called $$H_\mu ^{1,p}$$-quasiconvexity on p-coercive integrands is introduced for the lower semicontinuity with respect to the strong convergence of $$L^p_\mu (X;\mathbb {R}^m)$$ of integral functionals defined on Cheeger–Sobolev spaces. Under polynomial growth conditions it turns out that this condition is necessary and sufficient.

International audience; We prove subadditive theorems à la Akcoglu-Krengel on measure spaces with acting groups. Applications to homogenization of nonconvex integrals in Cheeger-Sobolev spaces are also developed.

We establish a continuity theorem for non-local functionals indexed by Young measures, that we use to deal with homogenization of stochastic non-diffusive reaction differential equations. Non-local effects induced by homogenization of such stochastic differential equations are studied.

International audience; Stability under a variational convergence of nonlinear time delays reaction-diffusion equations is discussed. Problems considered cover various models of population dynamics or diseases in heterogeneous environments where delays terms may depend on the space variable. As a consequence a stochastic homogenization theorem is established and applied to vector disease and logistic models. The results illustrate the interplay between the growth rates and the time delays which are mixed in the homogenized model.

We study Γ-convergence of nonconvex variational integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces. Applications to relaxation and homogenization are given.

International audience; We study relaxation of nonconvex integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces when the integrand does not have polynomial growth and can take infinite values.

International audience; We study periodic homogenization by $\Gamma $-convergence of integral functionals with integrands $W(x,\xi )$ having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to infinity as the volume of matter tends to zero, i.e., $W(x,\xi )=\sum _{j\in J}\mathbf{1}_{V_j}(x)H_j(\xi )$ where $\lbrace V_j\rbrace _{j\in J}$ is a finite family of open disjoint subsets of $\mathbb{R}^N$, with $|\partial V_j|=0$ for all $j\in J$ and $|\mathbb{R}^N\setminus \bigcup _{j\in J}V_j|=0$, and, for each $j\in J$, $H_j(\xi )\rightarrow \infty $ as $\det \xi \rightarrow 0$. In fact, our results apply to integrands of type $W(x,\xi )=a(x)H(\xi )$ when $H(\xi )\rightarrow \infty $ as $\det \xi \rightarrow 0$ and $a\in L^\infty (\mathbb{R}^N;[0,\infty [)$ is $1$-periodic and is either continuous almost everywhere or not continuous. When $a$ is not continuous, we obtain a density homogenization formula which is a priori different from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.

In [Anza Hafsa, O., Mandallena, J.-P., The nonlinear membrane energy: variational derivation under the constraint “ det ∇ u ≠ 0 ”, J. Math. Pures Appl. 86 (2006) 100–115] we gave a variational definition of the nonlinear membrane energy under the constraint “ det ∇ u ≠ 0 ”. In this paper we obtain the nonlinear membrane energy under the more realistic constraint “ det ∇ u > 0 ”.

Acerbi, Buttazzo and Percivale gave a variational definition of the nonlinear string energy under the constraint “ det ∇ u > 0 ” (see [E. Acerbi, G. Buttazzo, D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity 25 (1991) 137–148]). In the same spirit, we obtain the nonlinear membrane energy under the simpler constraint “ det ∇ u ≠ 0 ” 1 .

We prove a new interchange theorem of infimum and integral. Its distinguishing feature is, on the one hand, to establish a general framework to deal with interchange problems for nonconvex integrands and nondecomposable sets, and, on the other hand, to link the theorems of Rockafellar and Hiai-Umegaki with the one of Bouchitte-Valadier. We give an application to relaxation of nonconvex geometric integrals of Calculus of Variations.

We study homogenization by $\Gamma$-convergence of periodic nonconvex integrals when the integrand has quasiconvex growth with convex effective domain., Comment: 26 pages

International audience; We study homogenization by Г-convergence, with respect to the L1-strong convergence, of periodic multiple integrals in W1,∞ when the integrand can take in finite values outside of a convex bounded open set of matrices.

We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper semicontinuity which plays the role of convexity, and allows to prove a radial representation result for nonconvex functions. An application to the relaxation of multiple integrals with constraints on the gradient is given., Comment: 16 pages

We study homogenization by Gamma-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside of a convex set of matrices., Comment: 24 pages, submitted

International audience; We study the integral representation of relaxed functionals in the multi-dimensional calculus of variations, for integrands which are finite in a convex bounded set with nonempty interior and infinite elsewhere.

Relaxation theorems which apply to one, two and three-dimensional nonlinear elasticity are proved. We take into account the fact an infinite amount of energy is required to compress a finite line, surface or volume into zero line, surface or volume. However, we do not prevent orientation reversal.

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ 2 , and let \(W:\mathcal{M}^{3\times 2}\to[0,+\infty]\) be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type: $$\inf \left\{\int_{\Omega} W\big(\nabla u(x)\big)dx-\int_{\Omega} \big\langle f(x),u(x)\big\rangle dx:u\in W^{1,p}_{\ast}(\Omega;\mathcal{R}^3)\right\}$$ , where \(W^{1,p}_{\ast}(\Omega;\mathcal{R}^3):= \{u\in W^{1,p}(\Omega;\mathcal{R}^3):u(x)=(x,0)\hbox{ on }\partial \Omega\}\) with \(p >1,\langle\cdot,\cdot\rangle \) is the scalar product in ℝ 3 and\(f\in L^{q}(\Omega;\mathcal{R}^3), \hbox{ with } 1/p+1/q=1\)is the external loading per unit surface. We take into account the fact that an infinite amount of energy is required to compress a finite surface of the plate into zero surface, i.e., $$W\big(\xi_1\mid\xi_2\big)\to+\infty\ \hbox{ as }\ \big|\xi_1 \land \xi_2\big|\to 0.$$

Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ -limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.