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Transactions of the American Mathematical Society

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Movement of centers with respect to various potentials

Author: Shigehiro Sakata
Journal: Trans. Amer. Math. Soc. 367 (2015), 8347-8381
MSC (2010): Primary 31C12, 35K05, 35J05; Secondary 35B38, 51M16, 51M25, 52A40
Published electronically: August 18, 2015
MathSciNet review: 3403058
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Abstract: We investigate a potential with a radially symmetric and strictly decreasing kernel depending on a parameter. We regard the potential as a function defined on the upper half-space $ \mathbb{R}^m \times (0,+\infty )$ and study some geometric properties of its spatial maximizer. To be precise, we give some sufficient conditions for the uniqueness of a maximizer of the potential and study the asymptotic behavior of the set of maximizers.

Using these results, we imply geometric properties of some specific potentials. In particular, we consider applications for the solution of the Cauchy problem for the heat equation, the Poisson integral (including a solid angle) and $ r^{\alpha -m}$-potentials.

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Additional Information

Shigehiro Sakata
Affiliation: Global Education Center, Waseda University, 1-104 Totsuka-machi, Shinjuku-ku, Tokyo 169-8050, Japan

Keywords: Hot spot, Poisson integral, illuminating center, solid angle, $r^{\alpha-m }$-potential, radial center, Newton potential, Riesz potential, Alexandrov's reflection principle, moving plane method, minimal unfolded region, heart.
Received by editor(s): May 21, 2012
Received by editor(s) in revised form: March 14, 2013
Published electronically: August 18, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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