Characterizations of $p$-Parabolicity on Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918428902686720 |
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| author | Adriani, Andrea Fischer, Florian Setti, Alberto G. |
| author_facet | Adriani, Andrea Fischer, Florian Setti, Alberto G. |
| contents | We study $p$-energy functionals on infinite locally summable graphs for $p\in (1,\infty)$ and show that many well-known characterizations for a parabolic space are also true in this discrete, non-local and non-linear setting. Among the characterizations are an Ahlfors-type, a Kelvin-Nevanlinna-Royden-type, a Khas'minski\uı-type and a Poincaré-type characterization. We also illustrate some applications and describe examples of graphs which are locally summable but not locally finite. Finally, we study the obstacle problem for the $p$-Laplacian using an approximation procedure by finite graphs in the summable, not necessarily locally finite, case. This is then utilized to give an alternative proof of the Khas'minski\uı-type characterization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizations of $p$-Parabolicity on Graphs Adriani, Andrea Fischer, Florian Setti, Alberto G. Functional Analysis Differential Geometry We study $p$-energy functionals on infinite locally summable graphs for $p\in (1,\infty)$ and show that many well-known characterizations for a parabolic space are also true in this discrete, non-local and non-linear setting. Among the characterizations are an Ahlfors-type, a Kelvin-Nevanlinna-Royden-type, a Khas'minski\uı-type and a Poincaré-type characterization. We also illustrate some applications and describe examples of graphs which are locally summable but not locally finite. Finally, we study the obstacle problem for the $p$-Laplacian using an approximation procedure by finite graphs in the summable, not necessarily locally finite, case. This is then utilized to give an alternative proof of the Khas'minski\uı-type characterization. |
| title | Characterizations of $p$-Parabolicity on Graphs |
| topic | Functional Analysis Differential Geometry |
| url | https://arxiv.org/abs/2507.13696 |