An upper bound on the hot spots constant

Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary, and let $-\Delta u = \mu\_1 u$ the first nontrivial eigenfunction of Laplace operator Neumann boundary conditions. We prove that $$ \max\_{x \in D} \~u(x) \leq 60 \cdot \partial we emphasize this constant is uniform among all domains in dimensions. In particular, hot spots conjecture cannot fail by an arbitrarily large factor. The inequality also holds for other (Neumann-) eigenfunctions (possibly different constant) provided their eigenvalue smaller than Dirichlet eigenvalue. An example Kleefeld shows optimal at least $1 + 10^{-3}$.