Recognition: unknown
Characterizations of p-Parabolicity on Graphs
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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\u{\i}-type and a Poincar\'{e}-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\u{\i}-type characterization.
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Forward citations
Cited by 2 Pith papers
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A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs
Nonnegative solutions to -Δ_p u ≥ u^σ on non-p-parabolic weighted graphs are zero whenever the divergent sum condition on weighted ball volumes holds.
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A Wiener criterion at infinity for $p$-massiveness on weighted graphs
Under volume doubling and weak (1,p)-Poincaré inequalities, p-massiveness of infinite connected sets on weighted graphs is equivalent to a dyadic relative p-capacity condition in nested balls.
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