Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Two inequalities for convex equipotential surfaces

Pages: 437 – 451



Yajun Zhou (Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, New Jersey, U.S.A.; and Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing, China)


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.


harmonic function, level sets, curvature

2010 Mathematics Subject Classification

31B05, 53A05

This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

Received 1 January 2020

Accepted 12 September 2020

Published 12 April 2021