Invariance of Green equilibrium measure on the domain

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.