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On the hot spots conjecture in higher dimensions
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abstract
We prove a strong form of the hot spots conjecture for a class of domains in $\mathbb{R}^d$ 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 $\mathbb{R}^d$ 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.
Forward citations
Cited by 2 Pith papers
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Sharp bounds on the failure of the hot spots conjecture
The exact hot spots ratio is eta_d(0), extremizers do not exist, and it tends to sqrt(e) as d approaches infinity.
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Hot spots in domains of constant curvature
The hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature, with additional critical point and monotonicity results for other constant curvature triangles and polygons.
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