The C^r dependence problem of eigenvalues of the Laplace operator on domains in the plane
Julian Haddad, Marcos Montenegro
TL;DR
The paper investigates how Dirichlet Laplacian eigenvalues depend on smooth domain perturbations in a $C^r$ sense, focusing on cases where eigenvalues have multiplicity. It introduces a degenerate implicit function theorem on Banach spaces as the key technical tool and formulates an abstract perturbation result (plus an extension to non-self-adjoint operators) that yields $C^r$-smooth evolution of eigenvalues along one-parameter deformations. The main result shows that if a multiple eigenvalue corresponds to a simple real zero $\mu$ of $\chi(s)=\det(A - sB)$, then there exist $C^r$-regular curves $t \mapsto \lambda(t)$ with $\lambda'(0) = -\mu$, and associated eigenfunctions $u(t)$, under $C^{r+1}$ domain perturbations. Applying this to disks, squares, and balls gives almost-everywhere $C^r$ (and sometimes $C^{\infty}$) dependence of the spectrum under perturbations, with explicit Fourier-analytic criteria on boundary deformations that guarantee this regularity.
Abstract
The C^r dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding smooth one-parameter families of C^1 perturbations of domains in R^n. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares and also for its second eigenvalue on balls in R^n for any n >= 3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces of independent interest.