Elliptic Equations in Divergence Form, Geometric Critical Points of Solutions, and Stekloff Eigenfunctions

Abstract

The Stekloff eigenvalue problem (1.1) has a countable number of eigenvalues $\{ p_n \} n = 1,2, \ldots $, each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of $p_n $, and the number of critical points and of nodal domains of the eigenfunctions corresponding to $p_n $. In view of a possible application to inverse conductivity problems, the result for the general case of elliptic equations with discontinuous coefficients in divergence form is proven by replacing the classical concept of critical point with the more suitable notion of geometric critical point.