An upper bound on the hot spots constant

Abstract

Let $D\subset {\mathbb{R}}^d$ be a bounded, connected domain with smooth boundary, and let $-\Delta u={\mu }_{1}u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove that

and we emphasize that this constant is uniform among all connected domains with smooth boundary in all dimensions. In particular, the hot spots conjecture cannot fail by an arbitrarily large factor. The inequality also holds for other (Neumann-) eigenfunctions (possibly with a different constant) provided their eigenvalue is smaller than the first Dirichlet eigenvalue. An example of Kleefeld shows that the optimal constant is at least $1+10^{-3}$.