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

We consider the principal Dirichlet eigenfunction u of the Laplacian in a bounded region in \mathbbR²{\mathbb{R}^2} which is convex in one direction, say in x ₁. 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 x ₁, i.e., all superlevel sets of u are convex in x ₁. In this note we provide a negative answer to this question by giving an explicit counterexample.