For the \(\alpha\) -Green kernel \(g^\alpha_D\) on a domain \(D\subset\mathbb R^n\) , \(n \geq 2\) , associated with the \(\alpha\) -Riesz kernel \(|x-y|^{\alpha-n}\) , where \(\alpha\in(0,n)\) and \(\alpha \leq 2\) , and a relatively closed set \(F\subset D\) , we investigate the problem on minimizing the Gauss functional \(\int g^\alpha_D(x,y)\,d(\mu\otimes\mu)(x,y)-2\int g^\alpha_D(x,y)\,d(\vartheta\otimes\mu)(x,y),\) \(\vartheta\) 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).
