The smoothness problem of eigenvalues of the Laplace operator on the plane

Julian Haddad, Marcos Montenegro

Published 2015-05-10Version 1

A classical open problem involving the Laplace operator on symmetric domains in Rn, n >= 2, such as balls and cubes, is whether all its Dirichlet eigenvalues vary smoothly upon one-parameter C1 perturbations of the domain. We provide a fairly complete answer to this question in dimension n = 2 on disks and squares. We also give a positive answer to the problem for the second eigenvalue on balls in Rn. The proofs are based on a suitable framework of problem and on a new degenerate implicit function theorem on Banach spaces.