Inner Riesz Pseudo-Balayage and its Applications to Minimum Energy Problems with External Fields

Abstract

For the Riesz kernel $\kappa _\alpha (x,y):=|x-y|^{\alpha -n}$ of order $0<\alpha <n$ on $\mathbb R^n$, $n\geqslant 2$, we introduce the so-called inner pseudo-balayage $\widehat{\omega }^A$ 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

$$ \int \kappa _\alpha (x,y)\, d(\mu \otimes \mu )(x,y)-2\int \kappa _\alpha (x,y)\, d(\omega \otimes \mu )(x,y) $$

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 $\widehat{\omega }^A$ does exist, and it maintains the basic features of inner balayage for positive measures (defined when $\alpha \leqslant 2$), 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 $\widehat{\omega }^A$ 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 some of the latest results on this topic, by strengthening their formulations and/or by extending the areas of their applications. We prove e.g. that if A is a quasiclosed set of nonzero inner capacity $c_*(A)$, and if the signed measure $\omega$ is of finite total variation and of nonzero Euclidean distance between its support and the set A, then the problem in question is solvable if and only if either $c_*(A)<\infty$, or $\widehat{\omega }^A(\mathbb R^n)\geqslant 1$. In particular, if $c_*(A)=\infty$, then the solution fails to exist whenever $\omega ^+(\mathbb R^n)<1/C_{n,\alpha }$, where $C_{n,\alpha }$ equals 1 if $\alpha \leqslant 2$, or $2^{n-\alpha }$ otherwise; whereas $\omega ^-(\mathbb R^n)$, the total amount of the negative charge, has no influence on this phenomenon. The results obtained are illustrated by some examples.