Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields

Abstract

For the Riesz kernel ${\kappa }_{\alpha }(x,y):=|x-y{|}^{\alpha -n}$, $0<\alpha <n$, on ${\mathbb{R}}^n$, $n⩾2$, we introduce the inner pseudo-balayage ${\hat{\omega }}^A$ of a (Radon) measure $\omega$ on ${\mathbb{R}}^n$ to a set $A\subset {\mathbb{R}}^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class ${\mathcal{E}}^{+}(A)$ of all positive measures $\mu$ of finite energy, concentrated on $A$. For quite general signed $\omega$ (not necessarily of finite energy) and $A$ (not necessarily closed), such ${\hat{\omega }}^A$ does exist, and it maintains the basic features of inner balayage for positive measures (defined when $\alpha ⩽2$), except for those implied by the domination principle. (To illustrate the latter, we point out that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure may increase its total mass.) The inner pseudo-balayage ${\hat{\omega }}^A$ is further shown to be a powerful tool in the problem of minimizing the Gauss functional over all $\mu \in {\mathcal{E}}^{+}(A)$ with $\mu ({\mathbb{R}}^n)=1$, which enables us to improve substantially many recent results on this topic, by strengthening their formulations and/or by extending the areas of their applications. For instance, if $A$ is a quasiclosed set of nonzero inner capacity $c_{\ast }(A)$, and if $\omega$ is a signed measure, compactly supported in ${\mathbb{R}}^n\setminus {\mathrm{Cl}}_{{\mathbb{R}}^n}A$, then the problem in question is solvable if and only if either $c_{\ast }(A)<\infty$, or ${\hat{\omega }}^A({\mathbb{R}}^n)⩾1$.

27 pages