Neste trabalho estudamos existência, multiplicidade e não existência de soluções não triviais para o seguinte problema elíptico: { - \'DELTA\' = f(x, u), em \'OMEGA\' u = 0, sobre \'\\PARTIAL\' \'OMEGA\', onde \'OMEGA\' é um conjunto limitado de \'R POT. 2\' com fronteira suave e a função f possui crescimento exponencial. Para a existência de soluções são aplicados métodos variacionais combinados com as desigualdades de Trudinger-Moser. O resultado de não-existência é restrito ao caso de soluções radiais positivas e \'OMEGA\' = \'B IND.1\'(0). A prova usa técnicas de equações diferenciais ordinárias / In this work we study the existence, multiplicity and non-existence of non-trivial solutions to the following elliptic problem: { - \'DELTA\' u = f(x; u); in \'OMEGA\', ; u = 0; on \'\\PARTIAL\' \'OMEGA\' where \"OMEGA\' is a bounded and smooth domain in \'R POT. 2\' and f possesses exponential growth. The existence results are proved by using variational methods and the Trudinger- Moser inequalities. The non-existence result is restricted to the case of positive radial solutions and \'OMEGA\' = \'B IND. 1\'(0). The proof uses techniques of the theory of ordinary differential equations.

We study Monge-Ampère-type equations on compact complex manifolds. We prove a C² estimate for solutions to a general class of non-concave parabolic equations, extending work from the Kähler setting. Next we prove C⁰, C², and curvature estimates for solutions to a particular continuity path of elliptic equations on specific examples of non-Kähler manifolds, adapting work on the Chern-Ricci flow. In each case the estimates give a certain type of convergence of the solutions. The estimates are obtained by maximum principle arguments, and in the first part of this work we set up a general framework that facilitates the various C² estimates which follow.

Orientador: Sônia Maria Gomes, Philippe Remy Bernard Devloo, Igor Mozolevski / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T03:23:02Z (GMT). No. of bitstreams: 1 Goncalves_JoaoLuis_D.pdf: 15054031 bytes, checksum: 23ef9ef75ca5a7ae7455135fc552a678 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho, estudamos indicadores a posteriori para o erro na aproximação de funcionais das soluções das equações biharmônica e de Poisson obtidas pelo método de Galerkin descontínuo. A metodologia usada na obtenção dos indicadores é baseada no problema dual associado ao funcional, que é conhecida por gerar os indicadores mais eficazes. Os dois principais indicadores de erro com base no problema dual já obtidos, apresentados para problemas de segunda ordem, são estendidos neste trabalho para problemas de quarta ordem. Também propomos um terceiro indicador para problemas de segunda e quarta ordem. Estudamos as características dos diferentes indicadores na localização dos elementos com as maiores contribuições do erro, na caracterização da regularidade das soluções, bem como suas consequências na eficiência dos indicadores. Estabelecemos uma estratégia hp-adaptativa específica para os indicadores de erro em funcionais. Os experimentos numéricos realizados mostram que a estratégia hp-adaptativa funciona adequadamente e que o uso de espaços de aproximação hp-adaptados resulta ser eficiente para a redução do erro em funcionais com menor úmero de graus de liberdade. Além disso, nos exemplos estudados, a qualidade dos resultados varia entre os indicadores, dependendo do tipo de singularidade e da equação tratada, mostrando a importância de dispormos de uma maior diversidade de indicadores / Abstract: In this work we study goal-oriented a posteriori error indicators for approximations by the discontinuous Galerkin method for the biharmonic and Poisson equations. The methodology used for the indicators is based on the dual problem associated with the functional, which is known to generate the most effective indicators. The two main error indicators based on the dual problem, obtained for second order problems, are extended to fourth order problems. We also propose a third indicator for second and fourth order problems. The characteristics of the different indicators are studied for the localization of the elements with the greatest contributions of the error, and for the characterization of the regularity of the solutions, as well as their consequences on indicators efficiency. We propose an hp-adaptive strategy specific for goal-oriented error indicators. The performed numerical experiments show that the hp-adaptive strategy works properly, and that the use of hp-adapted approximation spaces turns out to be efficient to reduce the error with a lower number of degrees of freedom. Moreover, in the examples studied, a comparison of the quality of results for the different indicators shows that it may depend on the type of singularity and of the equation treated, showing the importance of having a wider range of indicators / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

[Truncated abstract] The study reported in this thesis extends the scaled boundary finite-element method to firstorder and second-order wave diffraction problems. The scaled boundary finite-element method is a newly developed semi-analytical technique to solve systems of partial differential equations. It works by employing a special local coordinate system, called scaled boundary coordinate system, to define the computational field, and then weakening the partial differential equation in the circumferential direction with the standard finite elements whilst keeping the equation strong in the radial direction, finally analytically solving the resulting system of equations, termed the scaled boundary finite-element equation. This unique feature of the scaled boundary finite-element method enables it to combine many of advantages of the finite-element method and the boundaryelement method with the features of its own. ... In this thesis, both first-order and second-order solutions of wave diffraction problems are presented in the context of scaled boundary finite-element analysis. In the first-order wave diffraction analysis, the boundary-value problems governed by the Laplace equation or by the Helmholtz equation are considered. The solution methods for bounded domains and unbounded domains are described in detail. The solution process is implemented and validated by practical numerical examples. The numerical examples examined include well benchmarked problems such as wave reflection and transmission by a single horizontal structure and by two structures with a small gap, wave radiation induced by oscillating bodies in heave, sway and roll motions, wave diffraction by vertical structures with circular, elliptical, rectangular cross sections and harbour oscillation problems. The numerical results are compared with the available analytical solutions, numerical solutions with other conventional numerical methods and experimental results to demonstrate the accuracy and efficiency of the scaled boundary finite-element method. The computed results show that the scaled boundary finite-element method is able to accurately model the singularity of velocity field near sharp corners and to satisfy the radiation condition with ease. It is worth nothing that the scaled boundary finite-element method is completely free of irregular frequency problem that the Green's function methods often suffer from. For the second-order wave diffraction problem, this thesis develops solution schemes for both monochromatic wave and bichromatic wave cases, based on the analytical expression of first-order solution in the radial direction. It is found that the scaled boundary finiteelement method can produce accurate results of second-order wave loads, due to its high accuracy in calculating the first-order velocity field.

Cette thèse est consacrée à l'étude de plusieurs problèmes d'équations aux dérivées partielles non-linéaires.<p><p>La première partie (chapitres 1-2-3) traite d'un problème trouvant son origine en biologie mathématique, à savoir l'étude de la survie à long terme d'une population dont l'évolution est gouvernée par une équation parabolique non-linéaire. Dans le modèle considéré, le mécanisme de diffusion est contrôlé par le p-Laplacien, la non-linéarité est de type logistique et fait intervenir un poids m pouvant changer de signe, et les conditions aux limites sont de flux nul. Le poids m correspond à une répartition des ressources devant permettre la survie de la population. Dans le chapitre 1, nous déterminons entre autres un critère de survie à long terme faisant intervenir la valeur propre principale du p-Laplacien avec poids m. Cette valeur propre apparait, plus précisément, comme la valeur limite d'un paramètre en-dessous de laquelle toute solution positive de l'équation converge vers zéro lorsque t tend vers l'infini. Ceci nous conduit naturellement au problème de minimiser la valeur propre en question lorsque m varie dans une classe adéquate de poids. Dans le chapitre 2, nous prouvons l'existence de minimiseurs et montrons que ces derniers satisfont une propriété de type “bang-bang”. Plusieurs propriétés de montonie sont aussi étudiées dans des situations géométriques particulières, et une caractérisation complète est donnée en dimension 1. Le chapitre 3 est consacré à l'élaboration de simulations numériques, où l'algorithme utilisé combine un méthode de plus grande pente avec une représentation de certains ensembles comme ensembles de niveaux.<p><p>La deuxième sujet de cette thèse (chapitre 4) est un problème elliptique faisant intervenir l'opérateur de courbure moyenne. Nous nous intéressons à l'existence et à la multiplicité de solutions nodales de ce problème. Nous montrons que, si un certain paramètre de l'équation est suffisamment grand, il existe une solution nodale qui change de signe exactement deux fois. Nous établissons également l'existence d'un nombre arbitrairement grand de solutions nodales. Enfin, dans le cas particulier où le domaine est une boule, un résultat de brisure de symétrie est obtenu, résultat qui induit l'existence d'au moins deux solutions à deux domaines nodaux. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Note de l'administrateur du service :le résumé de cette thèse est disponible dans le fichier déposé par l'auteur. Il ne peut techniquement pas être placé sous cette rubrique, dans la mesure où il contient des formules mathématiques avec des caractères grecs. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished