An upper bound on the Hot Spots constant

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 $$ |u|_{L^{infty}(D)} leq 60 cdot |u|_{L^{infty}(partial D)}.$$ This shows that the Hot Spots Conjecture cannot fail by an arbitrary factor. An example of Kleefeld shows that the optimal constant is at least $1 + 10^{-3}$.