O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / We investigate the existence of hypersurfaces with prescribed curvature in a wide context. First we study the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on a Riemannian manifold, which are closely related with the existence of hypersurfaces with prescribed curvature and boundary. In this setting we prove some existence results which extend to a Riemannian manifold previous results by Caffarelli, Nirenberg,Spruck and Bo Guan for the Euclidean space. We also study the existence of hypersurfaces with prescribed anisotropic mean curvature. We prove existence results for the Dirichlet problem related to the anisotropic mean curvature equation. This ensures the existence of Killing graphs with prescribed anisotropic mean curvature and boundary in a Riemannian manifold endowed with a nonsingular Killing vector field. Finally, we prove the existence of hyperspheres with prescribed anisotropic mean curvature in the Euclidean space, extending a previous result of Treibergs and Wei. / Neste trabalhamos investigamos a existÃncia de hipersuperfÃcies com curvatura prescrista num contexto amplo. Inicialmente estudamos o problema de Dirichlet para uma equaÃÃo totalmente nÃo-linear do tipo curvatura, definida em uma variedade Riemanniana. Este problema està intimamente relacionado a existÃncia de hipersuperfÃcies com curvatura e bordo prescritos. Neste contexto obtemos alguns resultados que estendem para uma variedade Riemanniana resultados obtidos anteriormente por Caffarelli, Nirenberg, Spruck e Bo Guan para o espaÃo Euclideano. Investigamos tambÃm a existÃncia de hipersuperfÃcies com curvatura mÃdia anisotrÃpica prescrita. Estabelecemos a solubilidade do problema de Dirichlet relacionado a equaÃÃo da curvatura mÃdia anisotrÃpica prescrita. Este resultado assegura a existncia de grÃficos de Killing com curvatura mÃdia anisotrÃpica e bordo prescritos numa variedade Riemanniana dotada com um campo de Killing sem singularidades. Finalmente, provamos a existÃncia de hiperesferas com curvatura mÃdia anisotrÃpica prescrita no espaÃo Euclideano, estendendo o resultado obtido Treibergs e Wei para a curvatura mÃdia usual.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Apresentamos uma extensÃo do Teorema de Barta devido a G. P. Bessa and J. F. Montenegro e fazemos algumas aplicaÃÃes geomÃtricas do resultado obtido. A primeira aplicaÃÃo geomÃtrica da extensÃo do Teorema de Barta à uma extensÃo do Teorema de Cheng sobre estimativas inferiores de autovalores do Laplaciano em bolas geodÃsicas normais. A segunda aplicaÃÃo geomÃtrica à uma generalizaÃÃo do Teorema de Cheng-Li-Yau de estimativas de autovalores para uma subvariedade mÃnima do espaÃo forma. / We present an extension to Barta's Theorem due to G. P. Bessa and J. F. Montenegro and we show some geometric applications of the obtained result. As first application, we extend Chang's lower eigenvalue estimates of the Laplacian in normal geodesic balls. As second application, we generalize Cheng-Li-Yau's eigenvalue estimates to a minimal submanifold of the space forms.

No description available.

In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also present a regularity result on linear elliptic equation where a coefficient has critical growth.

In this paper we consider a radially symmetric nonlinear Dirichlet problem in a ball, where the nonlinearity is "superlinear" and "superlinear with jumping."

Das Hauptgebiet der Arbeit stellt die Approximation der Lösungen partieller Differentialgleichungen mit Dirichlet-Randbedingungen durch Splinefunktionen dar. Partielle Differentialgleichungen finden ihre Anwendung beispielsweise in Bereichen der Elektrostatik, der Elastizitätstheorie, der Strömungslehre sowie bei der Untersuchung der Ausbreitung von Wärme und Schall. Manche Approximationsaufgaben besitzen keine eindeutige Lösung. Durch Anwendung der Penalized Least Squares Methode wurde gezeigt, dass die Eindeutigkeit der gesuchten Lösung von gewissen Minimierungsaufgaben sichergestellt werden kann. Unter Umständen lässt sich sogar eine höhere Stabilität des numerischen Verfahrens gewinnen. Für die numerischen Betrachtungen wurde ein umfangreiches, effizientes C-Programm erstellt, welches die Grundlage zur Bestätigung der theoretischen Voraussagen mit den praktischen Anwendungen bildete. / This work focuses on approximating solutions of partial differential equations with Dirichlet boundary conditions by means of spline functions. The application of partial differential equations concerns the fields of electrostatics, elasticity, fluid flow as well as the analysis of the propagation of heat and sound. Some approximation problems do not have a unique solution. By applying the penalized least squares method it has been shown that uniqueness of the solution of a certain class of minimizing problems can be guaranteed. In some cases it is even possible to reach higher stability of the numerical method. For the numerical analysis we have developed an extensive and efficient C code. It serves as the basis to confirm theoretical predictions with practical applications.

In this thesis we focus on Dirichlet's problem for the complex Monge-Ampère equation. That is, for a given non-negative Radon measure µ we are interested in the conditions under which there exists a plurisubharmonic function u such that (ddcu)n=µ, where (ddc)n is the complex Monge-Ampère operator. If this function u exists, then can it be chosen with given boundary values? Is this solution uniquely determined within a given class of functions?

Würzburg, Univ., Diss., 2008