Made available in DSpace on 2015-05-15T11:46:20Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 842217 bytes, checksum: 4549b711fa61f709fe2ff3b8c94c4bef (MD5) Previous issue date: 2014-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study the existence of positive solutions for a class of semilinear elliptic equations in a smooth bounded domain, with Dirichlet boundary condition and non-linear terms changing sign as well as with small perturbations. In order to obtain the positive solution, in the first case we use a version of the Mountain Pass Theorem in Ordered Banach spaces. In the second case, the main term is under assumptions that guarantee the application of the usual Mountain Pass Theorem and the perturbation term does not require any hypothesis. / Neste trabalho, estudamos existência de solução positiva para uma classe de equações elípticas semilineares em um domínio limitado suave, com condição de fronteira de Dirichlet, tanto com termos nao-lineares mudando de sinal, quanto com termos com pequenas perturbações. A fim de obtermos solução positiva, no primeiro caso, usamos uma versão do Teorema do Passo da Montanha para Espacos de Banach Ordenados. No segundo caso, o termo principal esta sob condições que garantem a aplicação do Teorema do Passo da Montanha usual e o termo de perturbação não requer nenhuma hipótese.

Orientador: Marcelo da Silva Montenegro / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T08:16:43Z (GMT). No. of bitstreams: 1 Queiroz_OlivaineSantanade_D.pdf: 886346 bytes, checksum: 5fe477c4619e746d923fc51e7d78f55c (MD5) Previous issue date: 2008 / Resumo: Estudamos a equação - D. u = x{ u>O} ( log u + )..1 (x, u)) em um domínio limitado e suave Ç1 C JR.n, com condições de fronteira u = O em é)Ç1. Demonstramos resultados de existência e regularidade da solução maximal. A positividade dessa solução depende do parâmetro ).. e de Ç1. Se a solução maximal se anula em partes de Ç1, obtemos uma estimativa local para a medida de Hausdorff da fronteira livre. Se a singularidade log u for trocada por -u-(3, com O < (3 < 1, então a teoria de Alt&Caffarelli e Alt&Phillips implica que a fronteira livre é regular. Também estudamos o problema de Neumann com não-linearidade logarítmica por meio de perturbações e técnicas variacionais / Abstract: We study the equation -D.u = X{u>O} (log u+Àf(x, u)) in a smooth bounded domain fl C JRn, with boundary conditions u = O on 8fl. We obtain existence and regularity of the maximal solution. The positivity of such a solution depends on the parameter À and on the domain fl. .If the maximal solution vanishes on a set of positive measure, then we obtain local estimates for the Hausdorff measure of the free boundary. If the singularity logu is replaced by -u-!3, with O < (3 < 1, the theory of Alt&Caffarelli and Alt&Phillips implies that the free boundary is regular. We also study the Neumann problem with logarithmic nonlinearity using perturbation techniques and variational methods / Doutorado / Doutor em Matemática

Orientadores: Francisco Odair Vieira de Paiva, Augusto César Ponce / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T10:14:29Z (GMT). No. of bitstreams: 1 Presoto_AdilsonEduardo_D.pdf: 2067267 bytes, checksum: 79c3ffe06a88b7cba190920dcf512036 (MD5) Previous issue date: 2011 / Resumo: No trabalho precursor de Brezis, Marcus e Ponce [15], estudou-se problemas semilineares elípticos com uma não linearidade não decrescente, contínua e dependendo apenas da variável dependente e com medidas como dados. Os autores estavam particularmente interessados no caso em que a equação não possuía solução. Numa das técnicas estudadas, eles aproximaram a medida por funções suaves através da convolução e, sob a condição adicional de convexidade da não linearidade, mostraram que as soluções correspondentes convergiam para a solução do mesmo problema com a maior medida menor do que ou igual a medida inicial tal que o problema tinha solução. O nosso objetivo é explorar profundamente este método. Ao invés de lidar com a convolução, consideramos sequências de medidas de Radon que convergem na topologia fraca-estrela e tais que o problema tem solução para cada termo. A pergunta que se põe é: as soluções convergem? Se sim, temos que o limite satisfaz a mesma equação com uma medida, em geral, distinta do limite-fraco, logo desejamos também determinar esta medida. Quando temos uma não linearidade, como descrita no parágrafo acima, as respostas têm um alto grau de variação, conforme os exemplos dados nos trabalhos de Ponce, e são inconclusivas. A proposta da tese é estudar a convergência dessas soluções para equações e sistemas semilineares elípticos com a não linearidade sendo do tipo exponencial. No caso em que temos a equação semilinear no plano, as soluções convergem para a solução do mesmo problema com uma medida que depende apenas do limite-fraco da sequência. Também, vemos que em dimensões superiores essas asserções não se verificam mais. Por fim, o sistema que aplicamos a técnica acima é o Sistema de Chern-Simons, surgido na física teórica e que representa o modelo de Chern-Simons Abeliano relativístico envolvendo duas partículas Higgs e dois campos calibrados / Abstract: In the pioneering work of Brezis, Marcus and Ponce [15], the authors studied elliptic semilinear problems with a continuous nondecreasing nonlinearity which vanishes at origin and depends only on dependent variable, and with measures as inicial data. They were particularly interested in the case which the equation does not have a solution. One of the techniques discussed was the approach of the measure by smooth functions via convolution. Under the additional condition of convexity, they showed that the corresponding solutions converge to the solution for the same problem with the largest measure less than inicial datum such that the problem admits a solution. Our aim is to explore deeply this method. Instead of dealing with the convolution, we consider sequences of Radon measures which converge in weak-star topology and such that the problem has solution for each term. The question posted is: the solutions converge? If yes, the limit solves the same problem with, in general distinct from the weak limit, another measure, thus, we also wish to determine this measure. The purpose of the thesis is to study the convergence of solutions for equations and systems with exponential nonlinearity. If we have the equation semilinear on the plane, the solutions converge to a solution for the same problem with a measure which depends only on weak limit of the sequence. We also see that in upper dimensions the results are no longer assured. In the end, the system concerned is the Chern-Simons System that comes from theoretical physics and it represents a relativistic Abelian Chern- Simons model with two Higgs particles and two gauge fields / Doutorado / Matematica / Doutor em Matemática

Orientadores: Lucas Catão de Freitas Ferreira, Marcelo da Silva Montenegro, José Antonio Carrillo de la Plata / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T15:13:15Z (GMT). No. of bitstreams: 1 Santos_MatheusCorreiados_D.pdf: 865476 bytes, checksum: 31a8b558231b701d81c20bf2712e4f50 (MD5) Previous issue date: 2015 / Resumo: Analisaremos dois problemas neste trabalho. Na primeira parte, estudaremos a existência de soluções para uma equação elíptica semilinear no espaço euclidiano todo e com dependência do gradiente e onde nenhuma restrição é imposta sobre o comportamento da não linearidade no infinito. Provaremos que existe uma solução que é localmente única e que herda muitas das propriedades de simetria da não linearidade. A positividade da solução e seu comportamento assintótico também são analisados. Os resultados obtidos também podem ser estendidos para outros casos como o de domínios exteriores ou o semiespaço e também para alguns operadores fracionários. Na segunda parte, analisaremos o comportamento assintótico das soluções da versão fracionária unidimensional da equações de meios porosos introduzida por Caffarelli e Vázquez e onde a pressão é obtida como a inversa do laplaciano fracionário da densidade. Devido à convexidade do núcleo do potencial de Riesz em dimensão um, mostraremos que a entropia associada à equação é displacement convex e satisfaz uma desigualdade funcional envolvendo a dissipação da entropia e a distância de transporte euclidiana. Um argumento por aproximação mostra que essa desigualdade funcional é suficiente para deduzir que a entropia das soluções converge exponencialmente para a entropia do estado estacionário. Também provaremos uma nova desigualdade de interpolação que permitirá obter a convergência exponencial das soluções em espaços Lp / Abstract: We analyse two problems in this work. In the first part we study the existence of solutions to a semilinear elliptic equation in the whole space and with dependence on the gradient and where no restriction is imposed on the behavior of the nonlinearity at infinity. We prove that there exists a solution which is locally unique and inherits many of the symmetry properties of the nonlinearity. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains like half-space and exterior domains and also to some fractional operators. For the second part, we analyse the asymptotic behavior of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vázquez and where the pressure is obtained as the inverse of the fractional Laplacian of the density. Due to the convexity of the kernel of the Riesz potential in one dimension, we show that the entropy associated with the equation is displacement convex and satisfies a functional inequality involving also entropy dissipation and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence, in the entropy level, of solutions to the unique steady state. A new interpolation inequality is also proved in order to obtain the exponential decay also in Lp spaces / Doutorado / Matematica / Doutor em Matemática

Cette thèse est consacrée à l'étude des équations et systèmes de réaction-diffusion dans des milieux hétérogènes. Elle est divisée en deux parties. La première est dédiée à l'étude des équations de réaction-diffusion dans des milieux périodiques. Nous nous intéressons en particulier aux équations posées dans des domaines qui ne sont pas l'espace entier $\mathbb{R}^{N}$, mais des domaines périodiques, avec des "obstacles". Dans un premier chapitre, nous étudions l'effet de la géométrie du domaine sur la vitesse d'invasion des solutions. Après avoir dérivé une formule de type Freidlin-Gartner, nous construisons des domaines où la vitesse d'invasion est strictement inférieure à la vitesse critique des fronts. Nous donnons également des critères géométriques qui garantissent l'existence de directions où l'invasion se produit à la vitesse critique. Dans le chapitre suivant, nous donnons des conditions nécessaires et suffisantes pour garantir que l'invasion ait lieu, après quoi nous construisons des domaines où des phénomènes intermédiaires (blocage, invasion orientée) se produisent. La deuxième partie de cette thèse est consacrée à l'étude de modèles décrivant l'influence de lignes à diffusion rapide (une route, par exemple) sur la propagation d'espèces invasives. Il a en effet été observé que certaines espèces, dont le moustique-tigre, envahissent plus rapidement que prévu certaines zones proches du réseau routier. Nous étudions deux modèles : le premier décrit l'influence d'une route courbe sur la propagation. Nous nous intéressons en particulier au cas de deux routes non-parallèles. Le second modèle décrit l'influence d'une route sur une niche écologique, en présence d'un changement climatique. Le résultat principal est que l'effet de la route est ambivalent : si la niche est stationnaire, alors l'effet de la route est délétère. Cependant, si la niche se déplace, suite à un changement climatique, nous montrons que la route peut permettre à une population de survivre. Pour étudier ce second modèle, nous développons une notion de valeur propre principale généralisée pour des systèmes de type KPP, et nous dérivons une inégalité de Harnack, qui est nouvelle pour ce type de systèmes. / This thesis is dedicated to the study of reaction-diffusion equations and systems in heterogeneous media. It is divided into two parts. The first one is devoted to the study of reaction-diffusion equations in periodic media. We pay a particular attention to equations set on domains that are not the whole space $\mathbb{R}^{N}$, but periodic domains, with "obstacles". In a first chapter, we study how the geometry of the domain can influence the speed of invasion of solutions. After establishing a Freidlin-Gartner type formula, we construct domains where the speed of invasion is strictly less than the critical speed of fronts. We also give geometric criteria to ensure the existence of directions where the invasion occurs with the critical speed. In the second chapter, we give necessary and sufficient conditions to ensure that invasion occurs, and we construct domains where intermediate phenomena (blocking, oriented invasion) occur. The second part of this thesis is dedicated to the study of models describing the influence of lines with fast diffusion (a road, for instance) on the propagation of invasive species. Indeed, it was observed that some species, such as the tiger mosquito, invade faster than expected some areas along the road-network. We study two models : the first one describes the influence of a curved road on the propagation. We study in particular the case of two non-parallel roads. The second model describes the influence of a road on an ecological niche, in presence of climate change. The main result is that the effect of the road is ambivalent: if the niche is stationary, then effect of the road is deleterious. However, if the niche moves, because of a shifting climate, the road can actually help the population to persist. To study this model, we introduce a notion of generalized principal eigenvalue for KPP-type systems, and we derive a Harnack inequality, that is new for this type of systems.

Ce travail s'inscrit dans l'étude mathématique d'écoulements de fluides environnementaux. Nous en abordons deux aspects, à travers deux contextes distincts d'application.En lien avec la simulation des écoulements en milieux poreux, on s'intéresse dans une première partie à la discrétisation d'opérateurs de diffusion anisotropes hétérogènes par des méthodes de volumes finis sur des maillages généraux. Dans le but d'obtenir des solutions approchées qui respectent les bornes physiques des modèles, notre attention se porte sur la conservation du principe du maximum pour les opérateurs elliptiques. Nous présentons des mécanismes généraux permettant de corriger tout schéma volumes finis afin de garantir un principe du maximum discret tout en préservant certaines de ses propriétés principales. On étudie en particulier les propriétés de coercivité et de convergence des schémas corrigés.La deuxième partie est consacrée à la construction de modèles approchés pour la propagation des vagues en eaux peu profondes et sur des topographies irrégulières. A cet effet, nous proposons tout d'abord une adaptation de la démarche d'étude classique à des écoulements bidimensionnels sur des topographies polygonales. Dans un cadre plus général, nous développons ensuite une démarche formelle qui débouche sur des alternatives non locales à quelques modèles classiques (équations de Saint-Venant, équations de Serre, système de Boussinesq). Ces nouveaux modèles contiennent des termes régularisants pour les contributions du fond. / This work investigates two research questions associated with environmental flows and their mathematical modeling.The first part is devoted to the development of finite volume methods for anisotropic and heterogeneous diffusion operators arising in models of porous media flows. To ensure that the approximate solutions lie within physical bounds, we aim at maintaining a discrete analogous of the maximum principle for elliptic operators. Starting from any given cell-centered finite volume scheme, we present a general approach to devise non-linear corrections providing a discrete maximum principle while retaining some main properties of the scheme. In particular, we study the coercivity and convergence properties of the modified schemes.The second part of this work focuses on the derivation of approximate models for shallow water wave propagation over rough topographies. In the particular case of one-dimensional polygonal bottom profiles, we first propose an adaptation of the usual derivation method using complex analysis tools. We then develop a formal approach to account for more general topographies. We propose nonlocal alternatives to some classical models (namely Saint-Venant equations, Serre equations and Boussinesq system). All these alternative models only involve smoothing contributions of the bottom.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-05T12:59:20Z No. of bitstreams: 1 AILTON RODRIGUES DA SILVA - DISSERTAÇÃO PPGMAT 2012..pdf: 874312 bytes, checksum: 1dc2f2515ff17b649766c1fa11f76b11 (MD5) / Made available in DSpace on 2018-08-05T12:59:20Z (GMT). No. of bitstreams: 1 AILTON RODRIGUES DA SILVA - DISSERTAÇÃO PPGMAT 2012..pdf: 874312 bytes, checksum: 1dc2f2515ff17b649766c1fa11f76b11 (MD5) Previous issue date: 2012-02 / CNPq / Nesta dissertação, estudamos a existência de solução blow-up para uma classe de problemas e sistemas elípticos. A principal ferramenta usada foi o Método de Sub e Supersolução, além de Regularidade Elíptica e alguns resultados de Equações Diferenciais Ordinárias. / In this dissertation, we study the existence of blow-up solution for some classes of elliptic problem, which include scalar problem and elliptic systems. The main tool used is the sub and super-solution methods combined with elliptic regularity and some results of Ordinary Differential Equations.

Cette thèse est constituée de trois parties. Dans la première partie, on s’intéresse au problème de trace au bord d’une solution positive de l’équation (E1) - Δu + g(∇u) = 0 dans un domaine borné Ω. Si g(r) ≥ rq avec q > 1, on prouve que toute solution positive de (E1)admet une trace au bord considérée comme une mesure de Borel régulière. Si g(r) = rq avec1 < q < qc = N+1/N , on montre l’existence d’une solution positive dont la trace au bord est une mesure de Borel régulière. Si g(r) = rq avec qc ≤ q < 2, on établit une condition nécessaire de résolution en terme de capacité de Bessel C2-q/q ,q’ . On étudie aussi des ensembles éliminables au bord pour des solutions modérées et sigma-modérées. La deuxième partie est consacrée à étudier la limite, lorsque k → ∞, de solutions d’équation ∂tu - Δu + f(u) = 0 dans ℝN × (0,∞) avec donnée initiale kδ0. On prouve qu’il existe essentiellement trois types de comportement possible et démontre un résultat général d’existence de trace initiale et quelques résultats d’unicité et de non-unicité de solutions dont la donnée initiale n’est pas bornée. Dans la troisième partie, on considère l’équation ∂tu - Δu + f(u) = 0 dans ℝN × (0,∞) où p > 1. Si p > 2N/N+1, on fournit une condition suffisante portant sur f pour l’existence et l’unicité des solutions fondamentales et on étudie la limite lorsque k → ∞. On donne aussi de nouveaux résultats de non-unicité de solutions avec donnée initiale non bornée. Si p ≥ 2, on prouve que toute solution positive admet une trace initiale dans la classe des mesures de Borel régulières positives. Finalement on applique les résultats ci-dessus au cas f(u) = uα lnβ(u + 1) avec α,β > 0. / This thesis is divided into three parts. In the first part, we study the boundary trace of positive solutions of the equation (E1) - Δu + g(∇u) = 0 in a bounded domain . When g(r) ≥ rq with q > 1, we prove that any positive function of (E1) admits a boundary trace which is an outer regular Borel measure. When g(r) ≥ rq with 1 < q < qc = N+1/N, we prove the existence of a positive solution with a general outer regular Borel measure as boundary trace.When g(r) ≥ rq with qc ≤ q < 2, we establish a necessary condition for solvability in term of the Bessel capacity C2-q/q ,q’ . We also study boundary removable sets for moderate and sigma-moderate solutions. The second part is devoted to investigate the limit, when k → ∞, of the solutions of ∂tu - Δu + f(u) = 0 in ℝN × (0,∞) with initial data kδ0. We prove that there exist essentially three types of possible behaviour and provide a new and more general construction of the initial trace and some uniqueness and non-uniqueness results for solutions with unbounded initial data. In the third part, we consider the equation ∂tu - Δu + f(u) = 0 in ℝN × (0,∞) where p > 1. If p > 2N/N+1we provide a sufficient condition on f for existence and uniqueness of the fundamental solutions and we study their limit when k → ∞. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If p ≥ 2, we prove that any positive solution admits an initial trace in the class of positive Borel measures. Finally we apply the above results to the case f(u) = uα lnβ(u + 1) with α,β > 0.

Les travaux de cette thèse sont regroupés en trois grandes parties traitant respectivement-des ondes stationnaires des systèmes de Schr"odinger-Maxwell-Proca et de Klein-Gordon-Maxwell-Proca sur une variété riemannienne fermée (compacte sans bord dans toute la thèse),-de systèmes elliptiques de Kirchhoff sur une variété riemannienne fermée,-de phénomènes d'explosion propres aux petites dimensions. / This thesis, divided into three main parts, deals with-standing waves for Schrödinger-Maxwell-Proca and Klein-Gordon-Maxwell-Proca systems on a closed Riemannian manifold (compact without boundary during all the thesis),-elliptic Kirchhoff systems on a closed manifold,-low-dimensional blow-up phenomena.

La Partie I présente l’obtention du développement asymptotique topologique pour une classe d’équations elliptiques quasilinéaires. Un point central réside dans la possibilité de définir la variation de l’état direct à l’échelle 1 dans R^N. Après avoir défini un cadre fonctionnel approprié faisant intervenir les normes L^p et L^2, et avoir justifié la classe d’équations considérée, la méthode se poursuit par l’étude du comportement asymptotique de la solution du problème d’interface non linéaire dans R^N et par une mise en dualité appropriée des états direct et adjoint aux différentes étapes d’approximation.La Partie II traite d’estimations et de développements asymptotiques de p-capacités de condensateurs, dont l’obstacle est d’intérieur vide et de codimension > ou = 2. Après les résultats préliminaires, les condensateurs équidistants permettent de donner deux illustrations de l’anisotropie engendrée par un segment dans l’équation de p-Laplace, puis d’établir une minoration de la p-capacité N-dimensionnelle d’un segment, qui fait intervenir les p-capacités d’un point, respectivement en dimensions N et (N-1). Les condensateurs elliptiques permettent d’établir que le gradient topologique de la 2-capacité n’est pas un outil approprié pour distinguer les courbes des obstacles d’intérieur non vide en 2D / Part I deals with obtaining topological asymptotic expansions for a class of quasilinear elliptic equations. A key point lies in the ability to define the variation of the direct state at scale 1 in R^N. After setting up an appropriate functional framework involving both the L^p and the L^2 norms, and then justifying the chosen class of equations, the approach goes on with the study of the asymptotic behavior of the solution of the nonlinear interface problem in R^N and by setting up an adequate duality scheme between the direct and adjoint states at each step of approximation. Part II deals with estimates and asymptotic expansions of condenser p-capacities and focuses on obstacles with empty interiors and with codimensions > ou = 2. After preliminary results, equidistant condensers are introduced to point out the anisotropy caused by a segment in the p-Laplace equation, and to provide a lower bound to the N-dimensional condenser p-capacity of a segment, by means of the N-dimensional and of the (N-1)-dimensional condenser p-capacities of apoint. Introducing elliptical condensers, it turns out that the topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with nonempty interior in 2D