On the lack of directional quasiconcavity of the fundamental mode in the clamped membrane problem

Abstract

We consider the principal Dirichlet eigenfunction u of the Laplacian in a bounded region in \({\mathbb{R}^2}\) which is convex in one direction, say in x1. It has been asked by Kawohl (Remarks on some old and current eigenvalue problems, Cambridge University Press, pp 165–183, 1994) whether in this case u is quasiconcave in x1, i.e., all superlevel sets of u are convex in x1. In this note we provide a negative answer to this question by giving an explicit counterexample.