For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.

In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also present a regularity result on linear elliptic equation where a coefficient has critical growth.

Consideraremos o problema elípitco $-\\Delta u + \\alpha\\chi_Du = \\lambda u$ em $\\Omega$, onde $\\Omega$ é um domínio de R^n com fronteira regular, e $D\\subset \\Omega$ é um subconjunto fechado de medida de Lebesgue fixada. A motivação para este problema vem da Mecânica, onde esta equação é encontrada no estudo de vibrações de uma membrana composta. Seja $\\lambda_1(D)$ o primeiro autovalor do problema, como função do conjunto D. Nesse trabalho mostraremos que $\\lambda_1$ é um autovalor simples, e estudaremos o problema de minimizar $\\lambda_1$ ao variarmos D no conjunto de todos os subconjuntos de medida fixada de $\\Omega$. Mais especificamente, determinaremos fórmulas para a variação primeira e segunda de $\\lambda_1$. / We will consider the elliptic problem $-\\Delta u + \\alpha\\chi_Du = \\lambda u in $\\Omega$, where $\\Omega$ is a domain in R^n with regular boundary, and $D \\subset\\Omega$ is a closed subset with prescribed Lebesgue measure. The motivation for this problem comes from Mechanics, where this equation models the vibrations of a composite membrane. Let $\\lambda_1(D)$ be the first eigenvalue of the problem, which is seen as a function of the set D. In this work, we will show that $\\lambda_1$ is a simple eigenvalue, and we will study the problem of minimizing $\\lambda_1(D)$ when D varies in the family of all closed subsets of $\\Omega$ with a given Lebesgue measure. More precisely, we will determine formulas for the first and the second variation of $\\lambda_1$.

I propose two Galerkin methods based on the transfer-of-approximation property for static and dynamic acoustic boundary value problems in seismic applications. For problems with heterogeneous coefficients, the polynomial finite element spaces are no longer optimal unless special meshing techniques are employed. The transfer-of-approximation property provides a general framework to construct the optimal approximation subspace on regular grids. The transfer-of-approximation finite element method is theoretically attractive for that it works for both scalar and vectorial elliptic problems. However the numerical cost is prohibitive. To compute each transfer-of-approximation finite element basis, a problem as hard as the original one has to be solved. Furthermore due to the difficulty of basis localization, the resulting stiffness and mass matrices are dense. The 2D harmonic coordinate finite element method (HCFEM) achieves optimal second-order convergence for static and dynamic acoustic boundary value problems with variable coefficients at the cost of solving two auxiliary elliptic boundary value problems. Unlike the conventional FEM, no special domain partitions, adapted to discontinuity surfaces in coe cients, are required in HCFEM to obtain the optimal convergence rate. The resulting sti ness and mass matrices are constructed in a systematic procedure, and have the same sparsity pattern as those in the standard finite element method. Mass-lumping in HCFEM maintains the optimal order of convergence, due to the smoothness property of acoustic solutions in harmonic coordinates, and overcomes the numerical obstacle of inverting the mass matrix every time update, results in an efficient, explicit time step.

Consideraremos o problema elípitco $-\\Delta u + \\alpha\\chi_Du = \\lambda u$ em $\\Omega$, onde $\\Omega$ é um domínio de R^n com fronteira regular, e $D\\subset \\Omega$ é um subconjunto fechado de medida de Lebesgue fixada. A motivação para este problema vem da Mecânica, onde esta equação é encontrada no estudo de vibrações de uma membrana composta. Seja $\\lambda_1(D)$ o primeiro autovalor do problema, como função do conjunto D. Nesse trabalho mostraremos que $\\lambda_1$ é um autovalor simples, e estudaremos o problema de minimizar $\\lambda_1$ ao variarmos D no conjunto de todos os subconjuntos de medida fixada de $\\Omega$. Mais especificamente, determinaremos fórmulas para a variação primeira e segunda de $\\lambda_1$. / We will consider the elliptic problem $-\\Delta u + \\alpha\\chi_Du = \\lambda u in $\\Omega$, where $\\Omega$ is a domain in R^n with regular boundary, and $D \\subset\\Omega$ is a closed subset with prescribed Lebesgue measure. The motivation for this problem comes from Mechanics, where this equation models the vibrations of a composite membrane. Let $\\lambda_1(D)$ be the first eigenvalue of the problem, which is seen as a function of the set D. In this work, we will show that $\\lambda_1$ is a simple eigenvalue, and we will study the problem of minimizing $\\lambda_1(D)$ when D varies in the family of all closed subsets of $\\Omega$ with a given Lebesgue measure. More precisely, we will determine formulas for the first and the second variation of $\\lambda_1$.

The thesis presents the results of our research on symmetry for some semilinear elliptic problems and on existence of solution for quasilinear problems involving singularities. The text is composed of two parts, each of which begins with a specific introduction. The first part is devoted to symmetry and symmetry-breaking results. We study a class of partial differential equations involving radial weights on balls, annuli or $R^N$ --where these weights are unbounded--. We show in particular that on unbounded domains, focusing on symmetric functions permits to recover compactness, which implies existence of solutions. Then, we stress the fact that symmetry-breaking occurs on bounded domains, depending both on the weights and on the nonlinearity of the equation. We also show that for the considered class of problems, the multibumps-solution phenomenon appears on the annulus as well as on the ball. The second part of the thesis is devoted to partial and ordinary differential equations with singularities. Using concentration-compactness tools, we show that a rather large class of functionals is lower semi-continuous, leading to the existence of a ground state solution. We also focus on the unicity of solutions for such a class of problems.

This thesis is concerned with the analysis of the finite element method and the discontinuous Galerkin method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity, if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained with numerical integration. The theoretical results are verified by numerical experiments. 1

On s’intéresse dans cette thèse à montrer que la solution approchée, par la méthode des volumes finis, converge vers la solution renormalisée de problèmes elliptiques ou paraboliques à donnée L1. Dans la première partie nous étudions une équation de convection-diffusion ellliptique à donnée L1. En adaptant la stratégie développée pour les solutions renormaliséesà la méthode des volumes finis, nous montrons que la solution approchée converge vers l’unique solution renormalisée.Dans la deuxième partie nous nous intéressons à un problème parabolique nonlinéaire à donnée L1. En utilisant une version discrète de résultats de compacité classiques, nous montrons que les résultats obtenues dans le cas elliptique restentvrais dans le cas parabolique. Dans la troisième partie nous montrons des résultats similaires pour une équationparabolique doublement non-linéaire à donnée L1. Le caractère doublement nonlinéaire de l’équation crée des difficultés supplémentaires par rapport à la partie précédente, notamment car la règle de dérivation en chaîne ne s’applique pas dansle cas discret. Enfin, dans la quatrième partie, nous utilisons les résultats établis précédemment pour étudier un système de type thermoviscoélasticité. Nous montrons que la solution approchée, obtenue par un schéma éléments finis-volumes finis, converge vers une solution faible-renormalisée du système. / In this thesis we are interested in proving that the approximate solution, obtained by the finite volume method, converges to the unique renormalized solution of elliptic and parabolic equations with L1 data. In the first part we study an elliptic convection-diffusion equation with L1 data. Mixing the strategy developed for renormalized solution and the finite volume method,we prove that the approximate solution converges to the unique renormalized solution. In the second part we investigate a nonlinear parabolic equation with L1 data. Using a discrete version of classical compactness results, we show that the results obtaines previously in the elliptic case hold true in the parabolic case. In the third part we prove similar results for a doubly nonlinear parabolic equation with L1 data. The doubly nonlinear character of the equation makes new difficulties with respect to the previous part, especially since the chain rule formula does not apply in the discrete case. Finaly, in the fourth part we use the results established previously to investigate a system of thermoviscoelasticity kind. We show that the approximate solution,obtaines by finite element-finite volume scheme, converges to a weak-renormalized solution of the system.