Orientadores: Djairo Guedes de Figueiredo, Francisco Odair de Paiva / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T16:14:26Z (GMT). No. of bitstreams: 1 Presoto_AdilsonEduardo_M.pdf: 798633 bytes, checksum: be0d791852ff6c964303c43174b753c4 (MD5) Previous issue date: 2008 / Resumo: Abordaremos diferentes métodos da Teoria do Potencial desenvolvidos no fim do século XIX e no começo do século XX para solucionar o Problema de Dirichlet. Iniciamos o primeiro capítulo com o Método da Varredura de Poincaré que transcendeu os anteriores e focalizou o problema sob uma nova óptica. Neste método, uma função harmônica, num domínio geral, era obtida, uma vez que condição de contorno fosse dada. Então condições na fronteira eram analizadas afim de que a função harmônica fosse, de fato, a solução do Problema de Dirichlet. Até então, as principais resoluções se baseavam no Princípio de Dirichlet que admitia soluções minimizantes para integrais de energia, se fundamentando em argumentos físicos. Contudo, tais argumentos continham alguns deslizes matemáticos como a admissão do mínimo para essas integrais. Posteriormente, surgiram os métodos de Perron e de Wiener dentro do espírito o Método do Poincaré. Ainda no primeiro capítulo, apresentamos um antecessor do método de Poincaré: o "Método de Schwarz. O segundo capítulo é dedicado ao Método das Equações Intregrais de Fredholm, no qual a Análise FUncional e as Equações Diferenciais Parciais caminharam lado a lado. Por fim, no último capítulo temos um resultado devido a Wiener que caracteriza os pontos regulares em termos de convergência de uma série envolvendo a capacidade de alguns conjuntos / Abstract: We will present different methods of Potential Theory developed at the end of the nineteenth century and the beginning of the twentieth century to solve the Dirichlet Problem. We start in the first chapter, with the Poincaré's Sweepping out Method, which transcended the former ones and focused the problem in a new insight. In this method, a harmonic function in a general domain is obtained, once a boundary condition is given. Then, conditions in the boundary are discussed so that this harmonic function is indeed the solution of the Dirichlet Problem. Until then, the key results were based on Dirichlet PrincipIe which admitted minimizing solutions to energy integraIs, by using some physical arguments. However, such arguments contained a few Mathematical gaps like the admission of a minimun to these integrals. Later, it appeared the Perron and Wiener Methods in the spirit of the Poincaré Method. Even in the first chapter, we discuss a predecessor of Poincaré's Method: the Schwarz's Method. The second chapter is devoted to the Integral Equations Method, where the Functional Analysis and Differential Equations walked side by side. Finally, the last chapter is a result due to Wiener that characterizes the regular points in terms of covergence of a series involving the capacity of some sets / Mestrado / Matematica - Analise- Equações Diferenciais e Parciais / Mestre em Matemática

Orientador: Djairo Guedes de Figueiredo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-08-04T21:17:17Z (GMT). No. of bitstreams: 1 Diniz_HugoAlexCarneiro_D.pdf: 770650 bytes, checksum: 55f077fc4cf6042e72a4b852d549e423 (MD5) Previous issue date: 2005 / Resumo: Neste trabalho estudamos a unicidade e a não-degenerescência de soluções positi-vas radiais para problemas não-autônomos envolvendo o p-Iaplaciano em anéis e bolas, com condição de Neumann na parte interna do anel, e condição de Dirichlet na parte externa. Quando o domínio é uma bola, temos apenas a condição de Dirichlet. Consideraremos três perfis diferentes para o problema: sublinear, superlinear e positivo, superlinear com parte negativa. Utilizando a técnica de Coffman, a qual consiste em estudar os zeros da solu-ção do problema linearizado, através de argumentos de comparação de Sturm, provamos primeiramente a não-degenerescência. Pelo método de "shooting", obtemos a unicidade. Como aplicação, demonstramos um resultado de unicidade para o laplaciano em domínios não-simétricos (até mesmo não-convexos) "próximos" a uma bola / Abstract: In this work, we study uniqueness and non-degeneracy of positive radial solutions for non-autonomous problems involving p-Iaplacian in annuli and balls, with Neumann condition in the inner part of annulus, and Dirichlet condition in the outer part. We consider three different problems: sublinear, superlinear and positive, superlinear with a negative part. Using the Coffman's technique, which consists in studying the zeros of the solution of the linearized problem, through Sturm comparison arguments we prove non-degeneracy. By the "shooting" method, we prove uniqueness. As an application, we demonstrate a uniqueness result for laplacian in non-symmetric (even non-convex) domains ''near'' a baIl / Doutorado / Doutor em Matemática

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:12:48Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:13:03Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:13:19Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Made available in DSpace on 2017-02-20T14:13:19Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) Previous issue date: 2015-12-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Let M be a compact orientable Riemannian manifold with boundary. We first study in this thesis the transversality of a family of metrics via a family of Bilaplacian opera-tors parametrised by the metrics themselves. We next consider a family of Bilaplacian operators with variation of their dominion in flat Riemannian manifolds and establish a condition of transversality in this case. In the case of arbitrary Riemannian manifold, we obtain that the spectrum of 2g. is simple and the set of parameters is residual. We complete this thesis by studying the generic propeties of the eigenvalues of the Laplacian of a family of hyper-surfaces of revolution in the vector space Rn+1 / Nesta tese primeiramente tratamos de transversalidade de famílias de métricas, em que foi tomada uma família de operadores Bilaplacianos parametrizada pelas métricas de uma variedade Riemanniana orientável M compacta com bordo. Em seguida foi considerada uma família de operadores Bilaplacianos em que o parâmetro é o domínio de definição do operador, no caso da variação do domínio em variedades Riemannianas flat foi mostrada a condição de transversalidade para a família de operadores Bilaplacianos parametrizados por tais domínios. Porém no caso de variedades Riemannianas quaisquer, obtemos a simplicidade genérica dos autovalores associados ao 2g. Por último, estudamos a situação genérica dos autovalores do Laplaciano numa família de hipersuperficies de rotação no espaço vetorial Rn+1.

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-05-03T18:56:20Z No. of bitstreams: 1 Tese - Juliana Ferreira Ribeiro de Miranda.pdf: 5725957 bytes, checksum: 608e325c9c09666cd44e53ddfba72e23 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-05-03T18:56:33Z (GMT) No. of bitstreams: 1 Tese - Juliana Ferreira Ribeiro de Miranda.pdf: 5725957 bytes, checksum: 608e325c9c09666cd44e53ddfba72e23 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-05-03T18:57:02Z (GMT) No. of bitstreams: 1 Tese - Juliana Ferreira Ribeiro de Miranda.pdf: 5725957 bytes, checksum: 608e325c9c09666cd44e53ddfba72e23 (MD5) / Made available in DSpace on 2016-05-03T18:57:02Z (GMT). No. of bitstreams: 1 Tese - Juliana Ferreira Ribeiro de Miranda.pdf: 5725957 bytes, checksum: 608e325c9c09666cd44e53ddfba72e23 (MD5) Previous issue date: 2015-03-31 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This view is composed of two parts. Initially we present an open way the principle of the weak maximum for a specific class of elliptic operators with applications hypersurface height estimates immersed mean curvature k constant warped products, whose base is a line interval. Later, we display estimates involving eigenvalues of an operator in n-divergently in limited domains with boundary conditions of Dirichlet, a Riemannian manifold Full and found a sharp estimate, compared to asymptotic formula Weyl, to the eigenvalues of the n-Laplacian operator / Esta tese é composta de duas partes. Inicialmente apresentamos uma forma aberta do princípio do máximo fraco para uma classe específica de operadores elípticos com aplicações em estimativas de altura de hipersuperfícies imersas com k-curvatura média constante em produtos warped, cuja base é um intervalo da reta. Posteriormente, exibimos estimativas envolvendo autovalores de um operador na forma n-divergente em domínios limitados, com condições de bordo de Dirichlet, de uma variedade riemanniana completa e encontramos uma estimativa sharp, em relação a fórmula assintótica de Weyl, para os autovalores do operador n-laplaciano

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / This work shows results existence and uniqueness of graphs with prescribed mean curvature. We demonstrate that a natural fixation Dirichlet problem for graphs of average curvature is required to consider those graphs like leaves on a Riemannian submersion Killing transversal cylinder, the cylinder given by flow lines of a Killing vector field. Using this approach, we are able to solve the problem in a way more comprehensive, giving a unified proof and existence results. / O objetivo deste trabalho à exibir resultados de existÃncia e unicidade de grÃficos com curvatura mÃdia prescrita. Demonstraremos que uma fixacÃo natural do problema de Dirichlet para grÃficos de curvatura mÃdia prescrita à considerar esses grÃficos como folhas em uma submersÃo Riemanniana transversal ao cilindro de Killing, isto Ã, ao cilindro dado pelas linhasde fluxo de um campo de vetores de Killing. Usando essa aproximaÃÃo, somos capazes de resolver o problema em um modo mais compreensivo, dando uma prova unificada e resultados de existÃncia para uma ampla gama do ambiente de variedades Riemannianas.

Orientador: Djairo Guedes de Figueiredo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-08-04T17:25:30Z (GMT). No. of bitstreams: 1 Pereira_FabioRodrigues_D.pdf: 1468332 bytes, checksum: 3e4d6ad380a672eddddbebbfbe9c85f4 (MD5) Previous issue date: 2005 / Doutorado / Doutor em Matemática

The thesis studies homogenization of Dirichlet boundary value problems for divergence type elliptic operators, and the associated boundary layer issues. This type of problems for operators with periodically oscillating coeffcients, and fixed boundary data are by now a classical topic largely due to the celebrated work by Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet boundary data exhibit periodic oscillations simultaneously was a longstanding open problem, and a progress in this direction has been achieved only very recently, in 2012, by Gerard-Varet and Masmoudi who proved a homogenization result for the simultaneously oscillating case with an algebraic rate of convergence in L2. Aimed at understanding the homogenization process of oscillating boundary data, in the first part of the thesis we introduce and develop Fourier-analytic ideas into the study of homogenization of Dirichlet boundary value problems for elliptic operators in divergence form. In smooth and bounded domains, for fixed operator and periodically oscillating boundary data we prove pointwise, as well as Lp convergence results the homogenization problem. We then investigate the optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we establish optimal Lp bounds for homogenization in some class of operators. For domains with non smooth boundary, we study similar boundary value homogenization problems for scalar equations set in convex polygonal domains. In the vein of smooth boundaries, here as well for problems with fixed operator and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and study the optimality of our Lp bounds. Although the statements are somewhat similar with the smooth setting, challenges for this case are completely different due to a radical change in the geometry of the domain. The second part of the work is concerned with the analysis of boundary layers arising in periodic homogenization. A key difficulty toward the homogenization of Dirichlet problem for elliptic systems in divergence form with periodically oscillating coefficients and boundary condition lies in identification of the limiting Dirichlet data corresponding to the effective problem. This question has been addressed in the aforementioned work by Gerard-Varet and Masmoudi on the way of proving their main homogenization result. Despite the progress in this direction, some very basic questions remain unanswered, for instance the regularity of this effective data on the boundary. This issue is directly linked with the up to the boundary regularity of homogenized solutions, but perhaps more importantly has a potential to cast light on the homogenization process. We initiate the study of this regularity problem, and prove certain Lipschitz continuity result. The work also comprises a study on asymptotic behaviour of solutions to boundary layer systems set in halfspaces. By a new construction we show that depending on the normal direction of the hyperplane, convergence of the solutions toward their tails far away from the boundaries can be arbitrarily slow. This last result, combined with the previous studies gives an almost complete picture of the situation.

by Chu Lap-foo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 68-69). / Introduction --- p.iii / Chapter 1 --- Basic Properties of Singular Solutions --- p.1 / Chapter 1.1 --- An Asymptotic Radial Result --- p.2 / Chapter 1.2 --- Local Uniqueness of Radial Solutions --- p.8 / Chapter 2 --- Dirichlet Problem : Existence Theory I --- p.11 / Chapter 2.1 --- Formulation --- p.12 / Chapter 2.2 --- Explicit Solutions on Balls --- p.14 / Chapter 2.3 --- The Moser Inequality --- p.19 / Chapter 2.4 --- Existence of Solutions in General Domains --- p.24 / Chapter 2.5 --- Spectrum of the Problem --- p.26 / Chapter 3 --- Dirichlet Problem : Existence Theory II --- p.29 / Chapter 3.1 --- Mountain Pass Lemma --- p.29 / Chapter 3.2 --- Existence of Second Solution --- p.31 / Chapter 4 --- Dirichlet Problem : Non-Existence Theory --- p.36 / Chapter 4.1 --- Upper Bound of λ* in Star-Shaped Domains --- p.36 / Chapter 4.2 --- Numerical Values --- p.41 / Chapter 5 --- The Neumann Problem --- p.42 / Chapter 5.1 --- Existence Theory I --- p.43 / Chapter 5.2 --- Existence Theory II --- p.47 / Chapter 6 --- The Schwarz Symmetrization --- p.49 / Chapter 6.1 --- Definitions and Basic Properties --- p.49 / Chapter 6.2 --- Inequalities Related to Symmetrization --- p.58 / Chapter 6.3 --- An Application to P.D.E --- p.63 / Bibliography --- p.68

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2014-08-29T19:24:13Z No. of bitstreams: 2 Dissertação Corrigida e Finalisada.pdf: 2280692 bytes, checksum: fa3c7d92b5ed8a39139ceeb3abb80551 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-08-29T19:24:13Z (GMT). No. of bitstreams: 2 Dissertação Corrigida e Finalisada.pdf: 2280692 bytes, checksum: fa3c7d92b5ed8a39139ceeb3abb80551 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-12-13 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this dissertation we analyze questions of existence of a weak solution for a class of superlineares elliptic Dirichlet problems. Here we do not consider the Ambrosseti Rabinovitz condition , which restricts some nonlinearities. We obtain main results of this dissertation via Variational Methods, such as Mountain Pass Theorem and Linking Theorem. Furthermore, weusePalais-Smalecondition(P.S.) or Cerami condition(Ce) / Nesta dissertação analisamos questões de existência de uma solução fraca para uma classe de problemas de Dirichlet elípticos superlineares. Aqui não consideramos a condição deAmbrosetti-Rabinowitz,a qual restringealgumasfunçõesnão lineares. Obtemos os principais resultados desta dissertação via Métodos variacionais, tais como o Teorema do Passo da Montanha e um Teorema de Linking. Além disso, utilizamos a TeoriaEspectral e ascondições dePalais-Smale(P.S.) eCerami(Ce).

In this thesis we are interested in some problems regarding harmonic functions. The topics are divided into three chapters. Chapter 2 concerns singularities developed by solutions of the Cauchy problem for a holomorphic elliptic equation, especially Laplace's equation. The principal motivation is to locate the singularities of the Schwarz potential. The results have direct applications to Laplacian growth (or the Hele-Shaw problem). Chapter 3 concerns the Dirichlet problem when the boundary is an algebraic set and the data is a polynomial or a real-analytic function. We pursue some questions related to the Khavinson-Shapiro conjecture. A main topic of interest is analytic continuability of the solution outside its natural domain. Chapter 4 concerns certain complex-valued harmonic functions and their zeros. The special cases we consider apply directly in astrophysics to the study of multiple-image gravitational lenses.