Problemas variacionais geometricos
AUTOR(ES)
Gil Ramos Cavalcanti
DATA DE PUBLICAÇÃO
2000
RESUMO
In this dissertation we treat two variational geometric problems: The Isoperimetric Problem and the Faber-Krahn Inequality. By means of the notions of functions of bounded variation and sets of finite perimeter (a la de Giorgi). we present the resolution of the first problem in the Euclidean space. We also make the computations for the variation formulae, which describe, in a Riemannian manifold, which domains are the possible candidates for solution of the problem and. in the end. we prove the Gromov-Levy Theorem which gives an isoperimetric profile for a manifold whose Ricci curvatures are bounded from below by a positive constant. In the case of the sphere this theorem gives us the solution of the isoperimetric problem. The Faber-Krahn inequality is extended to rotationally symmetric manifolds with extra hypothesis concerning the solutions of the isoperimetric problem. Among the manifolds satisfying the necessary hypothesis for the resolution are all simply-connected space forms and two dimensional paraboloids and some ovaloids. We also have results comparing the Faber-Krahn inequality for manifolds with some kind of limitation on the curvature with the Faber-Krahn inequality for simply-connected space forms
ASSUNTO(S)
geometria diferencial desigualdades isoperimetricas
ACESSO AO ARTIGO
http://libdigi.unicamp.br/document/?code=000433538Documentos Relacionados
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