Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

Abstract

We aim to provide a uniform way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems in real analytic domains. The exemplary examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The nodal sets of eigenfunctions are derived from doubling inequalities and a complex growth lemma. The novel idea is to obtain the doubling inequalities in an extended domain by a real analytic continuation and Carleman estimates.

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

“…Their result was generalized to eigenfunctions of elliptic operators with real analytic coefficients by Kukavica [9]. Similar estimates were recently obtained by Lin and Zhu [13] for eigenfunctions of the bi-Laplace operator with various boundary conditions under the assumption that the boundary is real analytic. Also, the polynomial (in the eigenvalue) upper bounds for the area of the zero set of the Dirichlet, Neumann, and Robin eigenfunctions in smooth bounded domains in R n were proved by Zhu in [21].…”