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arXiv · 2610.19908

A starlike 3-dimensional counterexample to the hot spots conjecture

Rami Atar·Krzysztof Burdzy
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Abstract

We construct a bounded Lipschitz domain in $\mathbb{R}^3$ that is starlike about the origin and whose first positive Neumann eigenvalue is simple. A corresponding eigenfunction, after a choice of sign, has its unique global maximum at the origin. The construction uses domains with increasingly fine angular structure and convergence to a weighted eigenvalue problem whose first positive eigenfunction is radial.

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Rami Atar, Krzysztof Burdzy. 2026-10-14. A starlike 3-dimensional counterexample to the hot spots conjecture. https://arxiv.org/abs/2610.19908

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