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The hot spots conjecture on a class of domains in \(\mathbb R^n\) with \( n \geqslant 3\). (English) Zbl 1259.35150

Summary: In this paper, we define a class of domains in \(\mathbb R^n\). Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.

MSC:

35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K05 Heat equation
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