Two inequalities for convex equipotential surfaces

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.

13 pages; (v2) References updated; paragraph after Theorem 3.4 rewritten. (v3) Revised according to referees' reports

References in corpus (2)

• Symmetry of singular solutions of degenerate quasilinear elliptic equations
• A Simple Formula for Scalar Curvature of Level Sets in Euclidean Spaces

Cited by in corpus (2)

• Some geometric relations for equipotential curves
• Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics