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Neumann problems for p -harmonic functions, and induced nonlocal operators in metric measure spaces
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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en
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59
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American Journal of Mathematics, Volume 147, issue 6, pp. 1653-1711
Abstract
Following ideas of Caffarelli and Silvestre in 2007, and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p)θ with 0 < θ < 1 on any compact, doubling metric measure space (Z,d,ν), and prove existence, regularity and stability for the non-homogenous nonlocal equation (−∆p)θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Existence of solutions for the Neumann boundary value problem in the metric setting was first demonstrated in 2018 by Malý and Shanmugalingam for the case that the Neumann data f is bounded, and here we relax the boundedness requirement of f with a more general integrability requirement. Our work also includes as special cases much of the previous results by other authors in the context of compact subsets in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z,d,ν) supports a Poincaré inequality.
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Publisher Copyright: © 2025 by Johns Hopkins University Press.
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Capogna, L, Kline, J, Korte, R, Shanmugalingam, N & Snipes, M 2025, 'Neumann problems for p -harmonic functions, and induced nonlocal operators in metric measure spaces', American Journal of Mathematics, vol. 147, no. 6, pp. 1653-1711. https://doi.org/10.1353/ajm.2025.a975705