An upper bound on the hot spots constant
Stefan Steinerberger
University of Washington, Seattle, USA
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Abstract
Let be a bounded, connected domain with smooth boundary, and let 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 .
Cite this article
Stefan Steinerberger, An upper bound on the hot spots constant. Rev. Mat. Iberoam. 39 (2023), no. 4, pp. 1373–1386
DOI 10.4171/RMI/1350