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On considère le problème : $$\left\{\begin{array}{c}-\mathrm{div}a(x,Du)=f\mathrm{dans}\Omega ,\hfill \\ u=0\mathrm{sur}\partial \Omega ,\hfill \end{array}\right.$$ où $\Omega$ est un ouvert borné de ${𝐑}^N$, où $a(x,\xi )$ est une fonction de Carathéodory, monotone en $\xi$, coercive, qui définit un opérateur dans $W_0^{1,p}\left(\Omega \right)$ (avec $1\<p\le N$), et où $f$ appartient à $L^1\left(\Omega \right)$ ou est une mesure bornée sur $\Omega$. On introduit une nouvelle définition de la solution de ce problème, la notion de solution renormalisée (ou entropique), et on montre l’existence d’une telle solution et sa continuité par rapport à $f$. Quand $f$ appartient à $L^1\left(\Omega \right)$, on montre en outre que cette solution...

We state and prove a chain rule formula for the composition $T\left(u\right)$ of a vector-valued function $u\in W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ by a globally Lipschitz-continuous, piecewise $C^1$ function $T$. We also prove that the map $u\to T\left(u\right)$ is continuous from $W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ into $W^{1,r}\left(\mathrm{\Omega }\right)$ for the strong topologies of these spaces.

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is where $\lambda >0$, $k>0$; $f(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $f(x)\ge 0$ (and so $u\ge 0$). If , we prove the existence of a solution for both the "+" and the "-" signs, while if $k\ge 1$, we prove the existence of a solution for the "+" sign only.

For conductivity problems in dimension N = 2, we prove a variant of a classical result: if a sequence $A^{ϵ}$ of matrices H-converges to $A^0$ (or in other terms if $A^{ϵ}$ converges to $A^0$ in the sense of homogenization) and if $det{A}^{ϵ}$ tends to $c^0$ a.e., then one has $det{A}^0=c^0$.

The limit behavior of the solutions of Signorini’s type-like problems in periodically perforated domains with period $\epsilon$ is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as $\epsilon \to 0$. In the critical case, it is shown that Signorini’s problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from the...

We prove the existence of solutions to $-\text{div}a\left(x,\text{grad}u\right)=f$, together with appropriate boundary conditions, whenever $a\left(x,e\right)$ is a maximal monotone graph in $e$, for every fixed $x$. We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a ${45}^{\circ }$ rotation, for every fixed $x$; in other words, the graph $d\in a\left(x,e\right)$ is defined through $d-e=\phi \left(x,d+e\right)$, where $\phi$ is a Carathéodory contraction in ${\mathbb{R}}^N$. This definition is shown to be equivalent to the fact that $a(x,\cdot )$ is pointwise monotone...

The limit behavior of the solutions of Signorini's type-like problems in periodically perforated domains with period is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from...

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose...

We consider the linearized elasticity system in a multidomain of ${𝐑}^3$. This multidomain is the union of a horizontal plate with fixed cross section and small thickness , and of a vertical beam with fixed height and small cross section of radius $r^{\epsilon }$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When and $r^{\epsilon }$ tend to zero simultaneously, with $r^{\epsilon }\gg {\epsilon }^2$, we identify the limit problem. This limit problem involves six junction conditions.