Minimum Riesz Energy Problems for a Condenser with Touching Plates

Abstract

We consider minimum energy problems in the presence of an external field for a condenser with touching plates A 1 and A 2 in Rn, n⩾3, relative to the α-Riesz kernel |x−y| α−n , 0\u3cα⩽2. An intimate relationship between such problems and minimal α-Green energy problems for positive measures on A 1 is shown. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The approach applied is mainly based on the establishment of a perfectness-type property for the α-Green kernel with 0\u3cα⩽2 which enables us, in particular, to analyze the existence of the α-Green equilibrium measure of a set. The results obtained are illustrated by several examples