PreprintNonconcavity of the Robin ground stateBen Andrews, Julie Clutterbuck, Daniel HauerAbstractOn a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is logconcave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper, we show that this is false, by analysing the perturbation problem from the Neumann case. In particular, we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground state is not logconcave (and indeed even has some superlevel sets which are nonconvex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance. Keywords: Eigenfunction, eigenvalue problem, Robin boundary condition, concavity, quasiconcavity.AMS Subject Classification: Primary 35B65; secondary 35J15, 35J25, 35P15, 47A75.
This paper is available as a pdf (600kB) file. It is also on the arXiv: arxiv.org/abs/1711.02779.
