Invariance of Green Equilibrium Measure on the Domain

Stamatis Pouliasis

We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set $K$ relative to the domains $D$ and $\Omega$ are the same if and only if $D$ is nearly equal to $\Omega$, 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.