We consider the Dirichlet and the Neumann eigenvalue problem for the Laplace operator on a variable nonsmooth domain, and we prove that the elementary symmetric functions of the eigenvalues splitting from a given eigenvalue upon domain deformation have a critical point at a domain with the shape of a ball. Correspondingly, we formulate overdetermined boundary value problems of the type of the Schiffer conjecture.
Publié le : 2006-01-14
Classification:
Dirichlet and Neumann eigenvalues and eigenfunctions,
Laplace operator,
overdetermined problems,
domain perturbation,
special nonlinear operators,
35P15,
35N05,
47H30
@article{1145287100,
author = {LAMBERTI, Pier Domenico and LANZA DE CRISTOFORIS, Massimo},
title = {Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems},
journal = {J. Math. Soc. Japan},
volume = {58},
number = {3},
year = {2006},
pages = { 231-245},
language = {en},
url = {http://dml.mathdoc.fr/item/1145287100}
}
LAMBERTI, Pier Domenico; LANZA DE CRISTOFORIS, Massimo. Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Japan, Tome 58 (2006) no. 3, pp. 231-245. http://gdmltest.u-ga.fr/item/1145287100/