Fractional harmonic measure in minimum Riesz energy problems with external fields

Abstract

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$ on $\mathbb R^n$, where $n\geqslant2$, $\alpha\in(0,2]$, 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 $z\in A^u\cup(\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A)$, 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, $z\in\partial_{\mathbb R^n}A$ is said to be inner $\alpha$-ultrairregular if the inner $\alpha$-harmonic measure $\varepsilon_z^A$ of $A$ is of finite energy. We show that for any $z\in A^u\cup(\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A)$, $\lambda_{A,f_{q,z}}$ 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 seems to contradict our physical intuition.