Soluciones simétricas de algunos problemas elípticos

In this paper we study solutions to the Neumann problem (I) ∆u= F(u) in Ω, ∂u/∂n = G(u) on Ω, and the Dirichlet problema (II) ∆u=F(u) in Ω, u=c n ∂Ω where Ω is a bounded domain in Rn with a smooth bo...

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Autores:: Quintero H., José Raúl

Tipo de recurso:: Article of journal

Fecha de publicación:: 1993

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: spa

Description
Summary:	In this paper we study solutions to the Neumann problem (I) ∆u= F(u) in Ω, ∂u/∂n = G(u) on Ω, and the Dirichlet problema (II) ∆u=F(u) in Ω, u=c n ∂Ω where Ω is a bounded domain in Rn with a smooth boundary ∂ Ω ∂/ ∂n is the derivative with respect to the outward normal n and c ϵ R. If Ω is the unit ball and if either F(t) = f(t) and G(t) = g(t) or F(t) = /(t) . t and G(t) = 9(t) . t where f is a strictly increasing continuous function and g is a strictly decreasing continuous function, we prove that solutions to problems (I) and (II) are radially symmetric about the origen. If Ω is the unit ball and F is a continuous function that does not change sign, we prove that solutions of (II) are radially symmetric about the origen. If Ω ⊂ Rn is a symmetric bounded domain with respect to a hyperplane T and f ϵ C(Ω x R,R), g ϵC (∂Ω x R, R) are functions that satisfy the same monoton..icity properties in the second variable as before, then we prove that solutions are symmetric with respect to the hyperplane T. If F satisfies the same condition as in the first case and G ≡ 0, we prove that the only solutions of (I) are constant functions. Furthermore, we find a formula for solutions of (I) in the unitary ball that allow us to deduce some non-existence results. We find conditions on F and G in order for (I) to have no solutions in any bounded domain.