Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems
Pier Domenico Lamberti, Massimo Lanza de Cristoforis · Journal of the Mathematical Society of Japan · 2006
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.