Smoothness properties for the optimal mixture of two isotropic materials the compliance and eigenvalue problems

The present paper is devoted to obtaining some smoothness results for the solution of two classical control problems relative to the optimal mixture of two isotropic materials. In the first one, the goal is to maximize the energy. In the second one, we want to minimize the first eigenvalue of the corresponding elliptic operator. At least for the first problem it is well known that it does not have a solution in general. Thus, we deal with a relaxed formulation. One of the applications of our results is in fact the nonexistence of a solution for the unrelaxed problem. In this sense, we improve a classical nonexistence result by Murat and Tartar for the maximization of the energy which was obtained assuming the solution smooth. We also get a counterexample to the existence of a solution for the eigenvalue problem which, to our knowledge, was an open problem.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains

In a paper of 1982, D. Cioranescu and F. Murat considered the problem satisfied by the limit u of the sequence un solution of 0 −∆un = f in Ωn, un = 0 on ∂Ωn, where Ωn is a sequence of open sets which are contained in a fixed bounded open set Ω. In order to make this, they imposed several hypotheses about the sequence Ωn. Their results were later extended to the p-Laplacian operator by N. Labani and C. Picard. In the present paper, we prove that these hypotheses may be reduced to the following one: There exists a sequence zn ∈ W1,p(Ω) which is zero in Ω \ Ωn and which converges weakly to 1 in W1,p(Ω). Indeed, G. Dal Maso and U. Mosco have solved the above homogenization problem in the general case in which we do not make any hypothesis about Ωn using Γconvergence methods and recently, G. Dal Maso and A. Garroni have also solved this general problem by a method close to the one used by D. Cioranescu and F. Murat.In un lavoro del 1982, D. Cioranescu e F. Murat hanno considerato il problema soddisfatto dal limite u di una successione un di soluzioni di 0 −∆un = f in Ωn, un = 0 su ∂Ωn, dove Ωn è una successione di insiemi aperti che sono contenuti in un fissato insieme aperto limitato Ω. Tale studio richiede di imporre numerose ipotesi sulla successione Ωn. I risultati di D. Cioranescu e F. Murat sono stati estesi in seguito da N. Labani e C. Picard al caso del p-Laplaciano. Nel presente lavoro, noi dimostriamo che le ipotesi su Ωn possono essere ridotte a un’unica ipotesi, la seguente: esiste una successione zn ∈ W1,p(Ω) che vale zero su Ω \ Ωn e che converge a 1 debolmente in W1,p(Ω). Il problema di omogeneizzazione nel caso generale in cui non si fa alcuna ipotesi sulla successione Ωn è stato risolto da G. Dal Maso e U. Mosco con metodi di Γ-convergenza e recentemente G. Dal Maso e A. Garroni hanno risolto il problema generale con metodi prossimi a quelli usati da D. Cioranescu e F. Murat

Exponential decay for the solutions of nonlinear elliptic systems posed in unbounded cylinders☆

We study the asymptotic behavior at infinity of the solutions of a nonlinear elliptic system posed in a cylinder of infinite length. The problem is written in a variational formulation, where we ask the derivative of the solutions to be in Lp. We show that an exponential decay at infinity for the second member implies exponential decay for the derivative of the solutions. We also give an application of this result to the study of boundary layers problems

Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains

In this paper, we study the homogenization of a Dirichlet problem in perforated domains for an operator which is the perturbation of the Laplace operator by a general nonlinear term with quadratic growth in the gradient. We show that a new term, which does not depend on the gradient, but which is nonlinear, appears in the limit problem. We also give a corrector result

Asymptotic behavior of nonlinear elliptic systems on varying domains

We consider a monotone operator of the form Au = −div(a(x, Du)), with Ω ⊆ Rn and a : Ω×MM×N → MM×N , acting on W1,p 0 (Ω, RM). For every sequence (Ωh) of open subsets of Ω and for every f ∈ W−1,p0 (Ω, RM), 1/p+ 1/p0 = 1, we study the asymptotic behavior, as h → +∞, of the solutions uh ∈ W1 0 (Ωh, RM) of the systems Auh = f in W−1,p0 (Ωh, RM), and we determine the general form of the limit problem

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies Fn, n ∈ N, defined in L2(Ω), for a bounded open subset Ω of R2. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fn is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence Fn, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity

Homogenization of stiff plates and two-dimensional high-viscosity Stokes equations

The paper deals with the homogenization of rigid heterogeneous plates. Assuming that the coefficients are equi-bounded in L1, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourthorder equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process. Finally, we show that the L1-boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L1-boundedness condition.Ministerio de Economía y Competitivida

Lack of compactness in two-scale convergence

This article deals with the links between compensated compactness and two-scale convergence. More precisely, we ask the following question: Is the div-curl compactness assumption sufficient to pass to the limit in a product of two sequences which two-scale converge with respect to the pair of variables (x, x/ε)? We reply in the negative. Indeed, the div-curl assumption allows us to control oscillations which are faster than 1/ε but not the slower ones

Semilinear problems with right-hand sides singular at u = 0 which change sign

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u = 0u=0. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u = 0u=0, while no restriction on its growth at u = 0u=0 is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/ |u|1/∣u∣, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1/ |u|\right.^{\gamma }\right. with 0 < \gamma < 10<γ<1