Convex sets can have interior hot spots

2024/12/09 by Jaume de Dios Pont, Pont, Jaume de Dios · 3 citations
Computer Science · #Optimization and Variational Analysis #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2412.06344

Abstract

The hot spots conjecture asserts that for any convex bounded domain Ω in \mathbb R^d, the first non-trivial Neumann eigenfunction of the Laplace operator in Ω attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of d. The construction is based on an extension of the conjecture from convex sets to log-concave measures.

Cited by

• Sharp bounds on the failure of the hot spots conjecture
• Hot spots in domains of constant curvature
• Potential Theory and the Boundary of Combinatorial Graphs

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