Geometric aspects of p-capacitary potentials

TL;DR

This paper develops monotonicity formulas for p-Laplace solutions outside convex domains, leading to geometric inequalities and characterizations of symmetric solutions, using conformal splitting techniques.

Contribution

It introduces new monotonicity formulas for p-Laplace equations and derives geometric consequences, including Minkowski inequality and symmetry characterizations.

Findings

Established monotonicity formulas for p-Laplace solutions

Derived classical Minkowski inequality from the formulas

Provided new characterizations of symmetric solutions and domains

Abstract

We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani.