Weighted minimum $\hat{I}\pm$-Green energy problems

Abstract

For the $\hat{I}\pm$-Green kernel $g^{\hat{I}}{\pm }_D$ on a domain $D\subset {\mathbb{R}}^n$, $n⩾2$, associated with the $\hat{I}\pm$-Riesz kernel $|x-y{|}^{\hat{I}\pm -n}$, where $\hat{I}\pm \in (0,n)$ and $\hat{I}\pm ⩽2$, and a relatively closed set $F\subset D$, we investigate the problem on minimizing the Gauss functional \[\int g^α_D(x,y)\,d(μ\otimesμ)(x,y)-2\int g^α_D(x,y)\,d(\vartheta\otimesμ)(x,y),\] $\vartheta$ being a given positive (Radon) measure concentrated on $D\setminus F$, and $\hat{I}¼$ ranging over all probability measures of finite energy, supported in $D$ by $F$. For suitable $\vartheta$, we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when $F$ is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the $\hat{I}\pm$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).

21 pages