First mixed Laplace eigenfunctions with no hot spots

Lawford Hatcher

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arXiv · 2401.01514

First mixed Laplace eigenfunctions with no hot spots

Lawford Hatcher

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Abstract

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.

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Lawford Hatcher. 2024-01-03. First mixed Laplace eigenfunctions with no hot spots. https://arxiv.org/abs/2401.01514

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