In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough "mass " is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem. Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all of which produce new results. Our method allows the construction of solutions with prescribed symmetry, and it represents a step further in the classification of positive solutions for certain nonlinear elliptic problems.

Οι ασθενείς λύσεις μιας διαφορικής εξίσωσης βρίσκονται σε μια ένα προς ένα αντιστοιχία με τα κρίσιμα σημεία ενός κατάλληλου συναρτησιοειδούς. Οι χώροι Sobolev που θα αναφερθούμε εδώ, για την ύπαρξη των ασθενών λύσεων, είναι υπόχωροι Χ του Η1=Η1(M,g). Οι άλλοι χώροι Ηqk αναφέρονται διότι θα χρησιμοποιηθούν για το πέρασμα των ασθενών λύσεων σε C2 κλασικές λύσεις. / --

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / In this paper we provide a universal solution for continuity module in the direction of the viscosity of fully nonlinear elliptic equations considering properties of the function f integrable in different situations. Established inner estimate for the solutions of these equations based on some conditions the norm of the function f. To obtain regularity in solutions of these inhomogeneous equations and coefficients of variables we use a method of compactness, which consists essentially of approximating solutions of inhomogeneous equations for a solution of a homogeneous equation in order to "inherit" the regularity that those equations possess. / Neste trabalho fornecemos mÃdulo de continuidade universal para soluÃÃes, no sentido da viscosidade,de equaÃÃes elÃpticas totalmente nÃo lineares, considerando propriedades de integrabilidade da funÃÃo f em diferentes situaÃÃes. Estabelecemos estimativa interior para as soluÃÃes dessas equaÃÃes baseadas em algumas condiÃÃes da norma da funÃÃo f. Para se obter regularidade nas soluÃÃes dessas equacÃes nÃo homogÃneas e de coeficientes variÃveis usamos um mÃtodo de compacidade, o qual consiste, essencialmente, em aproximar soluÃÃes de equaÃÃes nÃo homogÃneas por uma soluÃÃo de uma equaÃÃo homogÃnea com o objetivo de âherdarâ a regularidade que essas equaÃÃes possuem.

Nous nous intéressons à des résultats d'approximation et de régularité pour des solutions de viscosité d'équations elliptiques et paraboliques non-linéaires. Dans le chapitre 1, nous proposons, pour une classe générale d'équations elliptiques et paraboliques non-linéaires munies de conditions de Neumann inhomogènes, une interprétation de contrôle déterministe par des jeux répétés à deux personnes qui consiste à représenter la solution comme la limite de la suite des scores associés aux jeux. La condition de Neumann intervient par une pénalisation adaptée près de la frontière. En s'inspirant d'une approche abstraite proposée par Barles et Souganidis, nous prouvons la convergence en établissant des propriétés de monotonie, stabilité et consistance. Le chapitre 2 est consacré à des résultats de régularité sur les solutions d'équations paraboliques non-linéaires associés à un opérateur uniformément elliptique. Nous donnons une estimation de la mesure de Lebesgue de l'ensemble des points possédant un développement de Taylor quadratique global avec un contrôle sur la taille du terme cubique. Sous une hypothèse supplémentaire sur la régularité de la non-linéarité, nous en déduisons un résultat de régularité partielle höldérienne des solutions. Dans les chapitres 3 et 4, nous proposons une méthode générale pour obtenir des taux algébriques de convergence de solutions de schémas d'approximation vers la solution de viscosité sous l'hypothèse d'uniforme ellipticité de l'opérateur. Nous donnons un taux de convergence pour des schémas elliptiques obtenus par principe de programmation dynamique et nous prouvons un taux pour des schémas paraboliques par différences finies et implicites en temps. / In this thesis we study some approximation and regularity results for viscosity solutions of fully nonlinear elliptic and parabolic equations. In the first chapter, we consider a broad class of fully nonlinear elliptic and parabolic equations with inhomogeneous Neumann boundary conditions. We provide a deterministic control interpretation through two-person repeated games which represents the solution as the limit of the sequence of the scores associated to the games. The Neumann condition is modeled by a suitable penalization near the boundary. Inspiring by an abstract method of Barles and Souganidis, we prove the convergence of the score to the solution of the equation by establishing monotonicity, stability and consistency. The second chapter presents some regularity results about viscosity solutions of parabolic equations associated to a uniformly elliptic operator. First we obtain a Lebesgue measure estimate on the points having a quadratic Taylor expansion with a controlled cubic term. Under an additional assumption on the regularity of the nonlinearity, we deduce a partial regularity result about the Hölder regularity of these solutions. In the third and fourth chapters, we propose a general approach to determine algebraic rates of convergence of solutions of approximation schemes to the viscosity solution of fully nonlinear elliptic or parabolic equations under the assumption of uniform ellipticity of the operator. We first give the rate associated to the elliptic schemes derived by dynamic programming principles and proposed by Kohn and Serfaty. We then prove a rate of convergence for finite-difference schemes implicit in time associated to fully nonlinear parabolic equations.