An upper bound on the Hot Spots constant

Understand

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 $$ \max_{x \in D} ~u(x) \leq 60 \cdot \max_{x \in \partial D} ~u(x)$$ and 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 arbitrary factor.
• The inequality also holds for other (Neumann-)eigenfunctions (possibly with a different constant) provided the eigenvalue is smaller than the first Dirichlet eigenvalue.