We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.

The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.

The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical variational approach without and with a regularization term in order to smooth the contours of the classified image. Then we present the general topological asymptotic analysis, and we finally introduce its application to the grey-level image classification problem.

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit. From...

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...

We prove an abstract version of concentration compactness principle in Hilbert space and show its applications to a range of elliptic problems on unbounded domains.

We are interested in algorithms for constructing surfaces $\Gamma$ of possibly small measure that separate a given domain $\Omega$ into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the $p$-Laplacians, $p\to 1$, under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients.

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with $5/6$ as shear correction factor....

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with 5/6 as shear correction...

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree...

We establish the existence of a solution to the Neumann problem in the half-space with a subcritical nonlinearity on the boundary. Solutions are obtained through the constrained minimization or minimax. The existence of solutions depends on the shape of a boundary coefficient.