Invariance of Green equilibrium measure on the domain

TL;DR

This paper establishes conditions under which Green equilibrium measures and energies are invariant across different domains, linking measure equality to domain similarity and level sets of Green potentials.

Contribution

It provides a characterization of when Green equilibrium measures are invariant under domain changes, relating measure equality to domain nearly identicalness and level sets of potentials.

Findings

Green equilibrium measures are equal if and only if the domains are nearly equal.

Equality of measures corresponds to the domain being a level set of the Green potential.

The results apply to a broad class of compact sets.

Abstract

We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set K relative to the domains D and G are the same if and only if D is nearly equal to G, for a wide class of compact sets K. Also, we prove that equality of Green equilibrium measures arises if and only if the one domain is related with a level set of the Green equilibrium potential of K relative to the other domain.