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.

This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111]. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the time-harmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. {In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical time-dependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fully-discrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models time-Harmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.

On s'intéresse dans ce mémoire aux équations de la magnétohydrodynamique (MHD) dans des milieux hétérogènes, i.e. dans des milieux pouvant présenter des variations (éventuellement brutales) de propriétés physiques. En particulier, on met ici l'accent sur la résolution des équations de Maxwell dans des milieux avec des propriétés magnétiques inhomogènes. On présentera une méthode non standard pour résoudre ce problème à l'aide d'éléments finis de Lagrange. On évoquera ensuite l'implémentation dans le code SFEMaNS, développé depuis 2002 par J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre et A. Ribeiro, ainsi que les premiers résultats obtenus dans les simulations de dynamo. Nous nous intéresserons par exemple au cas de la dynamo dite de Von Kármán, afin de comprendre l'expérience VKS2. En outre, nous aborderons des cas de dynamo en précession, ou encore le problème de la dynamo au sein d'un écoulement de Taylor-Couette. / We focus on the magnetohydrodynamics (MHD) equations in hetereogeneous media, i.e. media with (possibly brutal) variations on the physical properties. In particular, we are interested in solving the Maxwell equations with discontinuous magnetic properties. We introduce a method that is, to the best of our knowledge, new to solve this problem using only Lagrange Finite Elements. We then discuss its implementation in SFEMaNS, a numerical code developped since 2002 by J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre and A. Ribeiro. We show the results of the first dynamo simulations we have been able to make with this solver. For instance, we present a kinematic dynamo in a VKS setup, as well as some results about dynamo action induced either by a Taylor-Couette flow, or by a precessionnally driven flow.

Neste trabalho estudamos propriedades de integrais concentradas, ou seja, integrais cujo integrando atua apenas em uma vizinhança do domínio em questão. Tais termos são utilizados para conhecer o comportamento do integrando em regiões cuja medida de Lebesgue se aproxima de zero quando um parâmetro tende a zero. Ilustraremos estes resultados abstratos através de duas aplicações, ambas em domínios Lipschitz de R2, onde adicionamos um termo de concentração em problemas semilineares elípticos: domínio com fronteira oscilante que tende a um domínio limite fixo; e domínio do tipo fino com fronteira oscilante. Em ambos os casos, provamos a semicontinuidade superior e inferior da família de soluções dos problemas. / In this work we study concentrating integrals properties, in other words, we analyze integrals which function that is been integrated acts only in a neighborhood of the boundary of the domain. Such terms are use to know the behaviour of the integrand in regions which Lebesgue measure tends to zero when a parameter goes to zero. We will illustrate these abstract results through two applications, both in Lipschitz domains of R2, where we add a concentration term in semi linear elliptic problems: oscillating boundary domain which tends to a fixed limit domain; and a thin domain with a oscillatory boundary. In both cases we prove the upper and lower semicontinuity of the family of solutions from these problems.

Neste trabalho estudamos propriedades de integrais concentradas, ou seja, integrais cujo integrando atua apenas em uma vizinhança do domínio em questão. Tais termos são utilizados para conhecer o comportamento do integrando em regiões cuja medida de Lebesgue se aproxima de zero quando um parâmetro tende a zero. Ilustraremos estes resultados abstratos através de duas aplicações, ambas em domínios Lipschitz de R2, onde adicionamos um termo de concentração em problemas semilineares elípticos: domínio com fronteira oscilante que tende a um domínio limite fixo; e domínio do tipo fino com fronteira oscilante. Em ambos os casos, provamos a semicontinuidade superior e inferior da família de soluções dos problemas. / In this work we study concentrating integrals properties, in other words, we analyze integrals which function that is been integrated acts only in a neighborhood of the boundary of the domain. Such terms are use to know the behaviour of the integrand in regions which Lebesgue measure tends to zero when a parameter goes to zero. We will illustrate these abstract results through two applications, both in Lipschitz domains of R2, where we add a concentration term in semi linear elliptic problems: oscillating boundary domain which tends to a fixed limit domain; and a thin domain with a oscillatory boundary. In both cases we prove the upper and lower semicontinuity of the family of solutions from these problems.

Dans cette thèse, nous donnons des résultats d’existence, de non-existence, d’unicité et de multiplicité de solutions pour des équations aux dérivées partielles avec croissance critique dans le gradient. Les principales méthodes utilisées dans nos preuves sont des arguments variationnels, la théorie des sous et sur-solutions, des estimations à priori et la théorie de la bifurcation. La thèse se compose de six chapitres. Dans le chapitre 0 nous introduisons le sujet de thèse et nous présentons les résultats principaux. Le chapitre 1 porte sur l’´étude d’une équation du type p-Laplacien avec croissance critique dans le gradient et dépendant d’un paramètre. En fonction de l’intervalle où se trouve le paramètre, nous obtenons l’existence et l’unicité d’une solution ou nous montrons l’existence et la multiplicité de solutions. Dans les chapitres 2 et 3, nous poursuivons notre étude dans le cas où l’opérateur utilisé est le Laplacien mais, contrairement au chapitre 1, nous étudions le cas où les coefficients changent de signe. Nous obtenons à nouveau des résultats d’existence et de multiplicité de solutions. Dans le chapitre 4, nous étudions des problèmes nonlocaux du type Laplacien fractionnaire avec différents termes de gradient non-local. Nous montrons des résultats d’existence et de non-existence de solutions pour différentes équations de ce type. Finalement, dans le chapitre 5 nous présentons quelques problèmes ouverts liés au contenu de la thèse et des perspectives de recherche. / In this thesis, we provide existence, non-existence, uniqueness and multiplicity results for partial differential equations with critical growth in the gradient. The principal techniques employed in our proofs are variational techniques, lower and upper solution theory, a priori estimates and bifurcation theory. The thesis consists of six chapters. In chapter 0, we introduce the topic of the thesis and we present the main results. Chapter 1 deals with a p-Laplacian type equation with critical growth in the gradient. This equation will depend on a real parameter. Depending on the interval where this parameter lives, we obtain the existence and uniqueness of one solution or we prove the existence and multiplicity of solutions. In chapters 2 and 3, we continue our study in the case where the operator is the Laplacian. However, unlike chapter 1, we study the case where the coefficient functions may change sign. We obtain again existence and multiplicity results. In chapter 4, we study non-local problems of fractional Laplacian type with different non-local gradient terms. We prove existence and non-existence results for different equations of this type. Finally, in chapter 5, we present some open problems related to the content of the thesis and some research perspectives.

I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-24T16:01:03Z No. of bitstreams: 1 GEIZANE LIMA DA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 596736 bytes, checksum: d02e34d40e7147e46c734ba297c181bf (MD5) / Made available in DSpace on 2018-07-24T16:01:03Z (GMT). No. of bitstreams: 1 GEIZANE LIMA DA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 596736 bytes, checksum: d02e34d40e7147e46c734ba297c181bf (MD5) Previous issue date: 2010-03 / Capes / Neste trabalho estudamos a existência de soluções positivas do tipo blow-up para uma classe de equações elípticas semilineares. Usamos argumentos desenvolvidos por Cîrstea & Radulescu [6], Lair & Wood [20] e as técnicas empregadas são o Método de Sub e Supersolução, Teoremas de Ponto Fixo e em alguns resultados exploramos a simetria radial e algumas estimativas para equações elípticas. / In this work we studied the existence of blow-up positive solutions for the class of semilinear elliptic equations. We used arguments developed by Cîrstea & Radulescu [6], and by Lair & Shaker [20] and the techniques used are the method of Sub and Supersolution, Fixed point theorems and some results explored radial symmetry and some estimates for elliptic equations.

Made available in DSpace on 2015-05-15T11:46:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 467058 bytes, checksum: ebe1089b4399fe71150fc70fa81ea4ed (MD5) Previous issue date: 2012-04-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study a fixed point theorem for increasing operators in ordered normed spaces and we apply it in order to obtain results of existence of weak solution for semilinear elliptic equations of type 8<: ---u = f(x; u) + h; in u = 0; on @ ; where - RN is a smooth domain, f : -R --! R satisfies some convenient conditions and h 2 H--1(. / Neste trabalho, estudamos um teorema de ponto fixo para operadores crescentes em espaços vetoriais ordenados e o aplicamos para obter resultados de existência de solução fraca para problemas elípticos semilineares do tipo 8<: ---u = f(x; u) + h; em u = 0; sobre @ em que - RN é um domínio suave, f : - R ! R satisfaz algumas condições convenientes e h 2 H- -1(:).

Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 517535 bytes, checksum: 44009b0bc09a5af772f82b9303aa5e7b (MD5) Previous issue date: 2010-11-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Here we study a class of semilinear elliptic equations with nonlinearity of an inverse square type. This equations arise, in applications, on the modeling of certain electrostatic devices from microtechnology, MEMS - Micro Electro Mechanical Systems. More precisely, these equations characterizes the function that represents the deformation of a deformable capacitor under the influence of an applied voltage. The Mathematical tools used on the study of such problems involve a bit of Nonlinear Analysis and Partial Differential Equations' methods as sub and supersolutions, sign preserving Theorems (Maximum Principle, Boggio's Principle), energy estimates via Sobolev spaces, etc. In a parallel way we wish to emphasize the importance of this investigation, in Mathematics, on helping the understanding on the class of singular problems in Partial Differential Equations. / Estudamos aqui uma classe de equações elípticas semilineares com singularidade do tipo inverso do quadrado. Estas equações aparecem, na modelagem de certos dispositivos eletrostáticos da microtecnologia, MEMS - Micro Electro Mechanical Systems (sistemas microeletromecânicos). Mais precisamente tais equações caracterizam a função que descreve a deformação de um capacitor deformável sob a influência de uma voltagem aplicada. A Matemática necessária ao estudo de tais problemas envolve um bom aparato de métodos da Análise não Linear e das Equações Diferenciais Parciais tais como Método de Sub- e Supersolução, Teoremas de Preservação de Sinal (Princípio do Máximo, Princípio de Boggio), estimativas de Energia via Espaços de Sobolev, entre outros. Em paralelo destacamos a importância desta investigação em Matemática, para entendermos como se comportam as soluções de problemas supercríticos em Equações Diferenciais Parciais.