In this paper, we establish Carleman estimates for the twodimensional isotropic non-stationary Lamé system with the zero Dirichlet boundaryconditions. Using this estimate, we prove the uniqueness and thestability in determining spatially varying density and two Lamécoefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomainω satisfies a non-trapping condition.

Given two measured spaces $(X,\mu)$ and $(Y,\nu)$ , and a third space Z,given two functions u(x,z) and v(x,z), we study the problem of finding twomaps $s:X\rightarrow Z$ and $t:Y\rightarrow Z$ such that the images $s(\mu)$ and $t(\nu)$ coincide, and the integral $\int_{X}u(x,s(x))d\mu-\int_{Y}v(y,t(y))d\nu$ is maximal. We give condition on u and v for whichthere is a unique solution.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e. $$\left\{\begin{array}{l} \varphi^{i}_{tt} - \varphi^{i}_{xx} = \exp(\varphi^{i+1} -\varphi^{i}) - \exp( \varphi^{i} - \varphi{i-1} ) \quad 0<x<\pi, \quad t \in \rm I\hskip-1.8pt R, i \in Z\!\!\!Z\quad (TC) \varphi^i (0,t) = \varphi^i (\pi,t) = 0 \quad\forall t, i. \end{array}\right.$$ We consider the case of “closed chains" i.e. $ \varphi^{i+N} = \varphi^i \forall i \in Z\!\!\!Z$ and some $ N \in {I\!\!N}$ and look for solutions which are peirodicin time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In the framework of transport theory, we are interested in the following optimization problem: given the distributions µ+ of working people and µ- of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of µ+ from µ- with respect to a metric which depends on the transportation network.

We consider an optimal control problem for a class of non-linearelliptic equations. A result of existence and uniqueness of the state equation is proven under weaker hypotheses than in theliterature. We also prove the existence of an optimalcontrol. Applications to some lubrication problems and numericalresults are given.

In this article we are interested in the following problem: tofind a map $u: \Omega \to \mathbb{R}^2$ that satisfies $$\left\{\begin{array}{ll}D u \in E\,\, &\mbox{{\it a.e.} in } \Omega\\ u(x)=\varphi(x) &x \in \partial \Omega\end{array}\right.$$ where Ω is an open set of $\mathbb{R}^2$ and E is acompact isotropic set of $\mathbb{R}^{2\times 2}$ . We will show anexistence theorem under suitable hypotheses on φ.

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.