Balayage and 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 $$\int\kappa_\alpha(x,y)\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu,\text{ where $f:=-U^\omega:=-\int\kappa_\alpha(\cdot,y)\,d\omega(y)$},$$ $\omega$ being a given positive (Radon) measure on $\mathbb R^n$, and $\mu$ ranging over all positive measures of finite energy, concentrated on $A\subset\mathbb R^n$ and having unit total mass. We prove that if $A$ is a quasiclosed set of nonzero inner capacity $c_*(A)$, and if the inner balayage $\omega^A$ of $\omega$ onto $A$ is of finite energy, then the solution $\lambda_{A,f}$ to the problem exists if and only if either $c_*(A)<\infty$, or $\omega^A(\mathbb R^n)\geqslant1$. We also analyze the support $S(\lambda_{A,f})$ of $\lambda_{A,f}$, thereby discovering new surprising phenomena. To be precise, we say that $x\in\partial_{\mathbb R^n}A$ is inner $\alpha$-ultrairregular for $A$ if $c_*(A^*_x)<\infty$, $A^*_x$ being the inverse of $A$ with respect to $\{|y-x|=1\}$; let $A^u$ consist of all those $x$. We show that for any $x\in A^u\cup(\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A)$, there is $q_x\in[1,\infty)$ such that $\lambda_{A,f_q}$ do exist for all $q\in[q_x,\infty)$. Here $f_q:=-qU^{\varepsilon_x}$, $\varepsilon_x$ being the unit Dirac measure at $x$. Thus, for any $x\in A^u$ and $q\geqslant q_x$, no compensation effect occurs between two oppositely signed charges carried by the same conductor, which seems to contradict our physical intuition. Another interesting phenomenon is that, if $A$ is closed while $\partial_{\mathbb R^n}A$ unbounded, then for any $x\in A^u\cup(\mathbb R^n\setminus A)$, $S(\lambda_{A,f_{q_x}})$ is noncompact, whereas $S(\lambda_{A,f_{q_x+t}})$ is already compact for any $t\in(0,\infty)$ -- even arbitrarily small.Comment: 24 page