On the hot spots conjecture in higher dimensions

2024/10/01 by Kennedy, James B., Rohleder, Jonathan · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2410.00816

Abstract

We prove a strong form of the hot spots conjecture for a class of domains in ℝ^d which are a natural generalization of the lip domains of Atar and Burdzy [J. Amer. Math. Soc. 17 (2004), 243-265] in dimension two, as well as for a class of symmetric domains in ℝ^d generalizing the domains studied by Jerison and Nadirashvili [J. Amer. Math. Soc. 13 (2000), 741-772]. Our method of proof is based on studying a vector-valued Laplace operator whose spectrum contains the spectrum of the Neumann Laplacian. This proof is essentially variational and does not require tools from stochastic analysis, nor does it use deformation arguments. In particular, it contains a new proof of the main result of Jerison and Nadirashvili.

Cited by

• Sharp bounds on the failure of the hot spots conjecture
• Hot spots on cones and warped product manifolds
• Hot spots in domains of constant curvature

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