The Hadamard variational formula and the Minkowski problem for p-capacity

Abstract

A Hadamard variational formula for p-capacity of convex bodies in Rn is established when 1<p<n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge–Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1<p<n and existence and regularity when 1<p<2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p=2).