Two inequalities for convex equipotential surfaces

TL;DR

This paper derives two geometric inequalities involving curvature and potential derivatives for convex equipotential surfaces, extending classical results and solving free boundary problems in electrostatics.

Contribution

It introduces new inequalities for convex equipotential surfaces that generalize known conservation laws and address free boundary problems in electrostatics.

Findings

Established inequalities involving mean and Gaussian curvature.

Generalized a conservation law for equipotential curves.

Provided solutions to free boundary problems in electrostatics.

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.