An upper bound on the hot spots constant

Abstract

Let D ⊂ R be a bounded, connected domain with smooth boundary and let −∆u = μ1u be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove ‖u‖L∞(D) ≤ 60 · ‖u‖L∞(∂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.