Fractional harmonic measure in minimum Riesz energy problems with external fields

For the Riesz kernel ${\kappa }_{\alpha }(x,y):=\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha -n}$ on ${\mathbb{R}}^n$, where $n⩾2$, $\alpha \in (0,2]$, and $\alpha <n$, we consider the problem of minimizing the Gauss functional $$\int {\kappa }_{\alpha }(x,y)d(\mu \otimes \mu )(x,y)+2\int f_{q,z}d\mu ,\text{where }{f}_{q,z}:=-q\int {\kappa }_{\alpha }(\cdot ,y)d{\epsilon }_z(y),$$ $q$ being a positive number, ${\epsilon }_z$ 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 $z\in A^u\cup ({\mathbb{R}}^n\setminus {\mathrm{C}\mathrm{l}}_{{\mathbb{R}}^n}A)$, where $A^u$ is the set of all inner $\alpha$-ultrairregular points for $A$, 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, $z\in {\mathrm{\partial }}_{{\mathbb{R}}^n}A$ is said to be inner $\alpha$-ultrairregular if the inner $\alpha$-harmonic measure ${\epsilon }_z^A$ of $A$ is of finite energy. We show that for any $z\in A^u\cup ({\mathbb{R}}^n\setminus {\mathrm{C}\mathrm{l}}_{{\mathbb{R}}^n}A)$, ${\lambda }_{A,f_{q,z}}$ exists if and only if either $A$ is of finite inner capacity, or $q⩾H_z$, where $H_z:=1\mathrm{/}{\epsilon }_z^A({\mathbb{R}}^n)\in [1,\mathrm{\infty })$. Thus, for any closed $A$, any $z\in A^u$, and any $q⩾H_z$ – even arbitrarily large, no compensation effect occurs between the two oppositely signed charges, $-q{\epsilon }_z$ and ${\lambda }_{A,f_{q,z}}$, carried by the same conductor $A$, which seems to contradict our physical intuition.