On the hot spots conjecture in higher dimensions
Abstract
We prove a strong form of the hot spots conjecture for a class of domains in R d \mathbb {R}^{d} which are a natural generalization of the lip domains of Atar and Burdzy [J. Amer. Math. Soc. 17 (2004), pp. 243–265] in dimension two, as well as for a class of symmetric domains in R d \mathbb {R}^{d} generalizing the domains studied by Jerison and Nadirashvili [J. Amer. Math. Soc. 13 (2000), pp. 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.
Fields of Study & Subjects
Advanced Differential Equations and Dynamical Systems
Mathematical Approximation and Integration
Advanced Banach Space Theory
Publication details
- Journal / source
- Transactions of the American Mathematical Society Series B
- Volume
- 13
- Issue
- 4
- Pages
- 108–131
- Publication date
- 22 April 2026
- Language
- EN