Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser (Q2304620)

Let \((A_i)_i\) be finitely many closed sets in \(\mathbb R^n\), \(n\geq 3\), and let \(s_i=\pm 1\) the sign of \(A_i\) such that that the oppositely charged plates are mutually disjoint. The authors consider the minimum energy problem relative to the Riesz kernel \(|x-y|^{-n+\alpha}\), \(0<\alpha\leq 2\), over positive vector Radon measures \((\mu_i)_i\) such that \(\mathrm{supp}(\mu_i)\subset A_i\) and \(\mu(A_i)=a_i<\infty\). The interaction between the measures \(\mu_i\) is described by a matrix \((s_is_j)_{ij}\). It is shown that, although the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity and even in the presence of an external field, this problem has a solution if the measures \(\mu_i\) satisfy a certain constraint of the form \(\mu_i\leq \xi_i\) with suitable measures \(\xi_i\) supported by \(A_i\). It is shown that the sufficient conditions on the solvability are sharp, moreover descriptions of the weighted vector Riesz potentials of the solutions are obtained and their characteristic properties are found; finally the support of the solution is analyzed. The approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the \(\alpha\)-Riesz energy on a set of vector measures associated with \((A_i)_i\), as well as on the establishment of an intimate relationship between the constrained minimum \(\alpha\)-Riesz energy problem and a constrained minimum \(\alpha\)-Green energy problem. Examples are given.