Two inequalities for convex equipotential surfaces
Differential Geometry
2021-10-11 v3 Mathematical Physics
math.MP
Abstract
We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green's functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics.
Cite
@article{arxiv.1912.12778,
title = {Two inequalities for convex equipotential surfaces},
author = {Yajun Zhou},
journal= {arXiv preprint arXiv:1912.12778},
year = {2021}
}
Comments
13 pages; (v2) References updated; paragraph after Theorem 3.4 rewritten. (v3) Revised according to referees' reports