Weighted minimum $\alpha$-Green energy problems

For the $\alpha$-Green kernel $g_D^{\alpha }$ on a domain $D\subset {\mathbb{R}}^n$, $n⩾2$, associated with the $\alpha$-Riesz kernel $\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha -n}$, where $\alpha \in (0,n)$ and $\alpha ⩽2$, and a relatively closed set $F\subset D$, we investigate the problem on minimizing the Gauss functional $$\int g_D^{\alpha }(x,y)d(\mu \otimes \mu )(x,y)-2\int g_D^{\alpha }(x,y)d(\vartheta \otimes \mu )(x,y),$$ $\vartheta$ being a given positive (Radon) measure concentrated on $D\setminus F$, and $\mu$ 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 $\alpha$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).