Weighted minimum $α$-Green energy problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909601161543680 |
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| author | Zorii, Natalia |
| author_facet | Zorii, Natalia |
| contents | For the $α$-Green kernel $g^α_D$ on a domain $D\subset\mathbb R^n$, $n\geqslant2$, associated with the $α$-Riesz kernel $|x-y|^{α-n}$, where $α\in(0,n)$ and $α\leqslant2$, 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 $μ$ 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 $α$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted minimum $α$-Green energy problems Zorii, Natalia Classical Analysis and ODEs 31C15 For the $α$-Green kernel $g^α_D$ on a domain $D\subset\mathbb R^n$, $n\geqslant2$, associated with the $α$-Riesz kernel $|x-y|^{α-n}$, where $α\in(0,n)$ and $α\leqslant2$, 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 $μ$ 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 $α$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). |
| title | Weighted minimum $α$-Green energy problems |
| topic | Classical Analysis and ODEs 31C15 |
| url | https://arxiv.org/abs/2505.02260 |