The smoothness problem of eigenvalues of the Laplace operator on the plane
Julián Haddad, Marcos Montenegro · arXiv (Cornell University) · 2015
A classical open problem involving the Laplace operator on symmetric domains in R n , such as balls and cubes, is whether all its Dirichlet eigenvalues vary smoothly upon one-parameter C 1 perturbations of the domain. We provide a fairly complete answer to this question in dimension n = 2 on disks and squares and also for the second eigenvalue on balls in R n with n ≥ 3. The proofs are based on a suitable framework of problem and on a new degenerate implicit function theorem on