In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let $H=\bigl(\begin{smallmatrix} \sigma& 00 & \sigma^{-1} \end{smallmatrix}\bigr)$ for $\sigma>0$ . Let $0<\zeta_1<1<\zeta_2<\infty$ . Let $K:=SO\left(2\right)\cup SO\left(2\right)H$ .Let $u\in W^{2,1}\left(Q_{1}\left(0\right)\right)$ be a $\xCone$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<\zeta_2$ , $\mathrm{Lip}\left(u^{-1}\right)<\zeta_1^{-1}$ . 
There exists positive constants $\mathfrak{c}_1<1$ and $\mathfrak{c}_2>1$ depending only on σ, $\zeta_1$ , $\zeta_2$ such that if $\epsilon\in\left(0,\mathfrak{c}_1\right)$ and u satisfies the following inequalities \[ \int_{Q_{1}\left(0\right)} {\rm d}\left(Du\left(z\right),K\right) {\rm d}L^2 z\leq \epsilon\] \[ \int_{Q_{1}\left(0\right)} \left|D^2 u\left(z\right)\right| {\rm d}L^2 z\leq \mathfrak{c}_1,\] then there exists $J\in\left\{Id,H\right\}$ and $R\in SO\left(2\right)$ such that \[ \int_{Q_{\mathfrak{c}_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| {\rm d}L^2 z\leq \mathfrak{c}_2\epsilon^{\frac{1}{800}}.\]

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

We prove the conical differentiability of the solution to a boneremodeling contact rod model, for given data (applied loads andrigid obstacle), with respect to small perturbations of the crosssection of the rod. The proof is based on the special structure ofthe model, composed of a variational inequality coupled with anordinary differential equation with respect to time. Thisstructure enables the verification of the two followingfundamental results: the polyhedricity of a modified displacementconstraint set defined by the obstacle and the differentiabilityof the two forms associated to the variational inequality.

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.

We study the null controllability by one control force of some linear systems of parabolic type.We give sufficient conditions for the null controllabilityproperty to be true and, in an abstract setting, we prove that itis not always possible to control.

We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -Young measures and $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -varifolds. The energyfunctional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. Theinterest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classicalformulation. The second formulation moreover accounts for concentration effects.

This work is devoted to prove the exponential decay for the energyof solutions of the Korteweg-de Vries equation in a bounded intervalwith a localized damping term. Following the method in Menzala (2002)which combines energy estimates, multipliers and compactnessarguments the problem is reduced to prove the unique continuation ofweak solutions. In Menzala (2002) the case where solutions vanish on aneighborhood of both extremes of the bounded interval where equationholds was solved combining the smoothing results by T. Kato (1983)and earlier results on unique continuation of smooth solutions byJ.C. Saut and B. Scheurer (1987). In this article we address thegeneral case and prove the unique continuation property in twosteps. We first prove, using multiplier techniques, that solutionsvanishing on any subinterval are necessarily smooth. We then applythe existing results on unique continuation of smooth solutions.

In this paper we study linear conservative systems of finitedimensioncoupled with an infinite dimensional system of diffusive type. Computing the time-derivative of anappropriate energy functional along the solutions helps us toprove the well-posedness of the systemand a stability property.But in order to prove asymptotic stability we need to applya sufficient spectral condition. We also illustrate the sharpness of thiscondition by exhibiting some systems for which we do not have the asymptoticproperty.