For the Riesz kernel \(\kappa _\alpha (x,y):=|x-y|^{\alpha -n}\) on \(\mathbb {R}^n\) , where \(n\geqslant 2\) , \(\alpha \in (0,2]\) , and \(\alpha <n\) , we consider the problem of minimizing the Gauss functional \( \int \kappa _\alpha (x,y)\,d(\mu \otimes \mu )(x,y)+2\int f_{q,z}\,d\mu ,\quad \text {where}\,\, f_{q,z}:=-q\int \kappa _\alpha (\cdot ,y)\,d\varepsilon _z(y), \) q being a positive number, \(\varepsilon _z\) the unit Dirac measure at \(z\in \mathbb {R}^n\) , and \(\mu \) ranging all probability measures of finite energy, concentrated on quasiclosed \(A\subset \mathbb {R}^n\) . For any \(z\in A^u\cup (\mathbb {R}^n\setminus \textrm{Cl}_{\mathbb {R}^n}A)\) , where \(A^u\) is the set of all inner \(\alpha \) -ultrairregular points for A (the concept of \(\alpha \) -ultrairregularity being newly introduced), we provide necessary and sufficient conditions for the existence of the minimizer \(\lambda _{A,f_{q,z}}\) , establish its alternative characterizations, and describe its support, thereby discovering new interesting phenomena. In detail, \(z\in \partial _{\mathbb {R}^n}A\) is said to be inner \(\alpha \) -ultrairregular for A if the inner \(\alpha \) -harmonic measure \(\varepsilon _z^A\) is of finite energy. We show that for any \(z\in A^u\cup (\mathbb {R}^n\setminus \textrm{Cl}_{\mathbb {R}^n}A)\) , \(\lambda _{A,f_{q,z}}\) exists if and only if either A is of finite inner capacity, or \(q\geqslant H_z\) , where \(H_z:=1/\varepsilon _z^A(\mathbb {R}^n)\in [1,\infty )\) . Thus, for any closed A, any \(z\in A^u\) , and any \(q\geqslant H_z\) — even arbitrarily large, no compensation effect occurs between the two oppositely signed charges, \(-q\varepsilon _z\) and \(\lambda _{A,f_{q,z}}\) , carried by the same conductor A, which at the first glance seems to contradict our physical intuition. Another interesting phenomenon is that, if closed A has unbounded boundary, then for any \(z\in A^u\cup (\mathbb {R}^n\setminus A)\) , the support of \(\lambda _{A,f_{H_z,z}}\) is noncompact, whereas that of \(\lambda _{A,f_{H_z+\beta ,z}}\) is already compact for any \(\beta \in (0,\infty )\) — even arbitrarily small. The results obtained substantially improve some of the latest ones on the problem in question, e.g. those by Dragnev et al. (Constr. Approx. 57, 1–43, 2023), and are illustrated by means of some examples.
