Abstract We consider two notions of functions of bounded variation in complete metric measure spaces,one due to Martio and the other due to Miranda Jr. We show that these two notionscoincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.
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NEUMANN PROBLEMS FOR p-HARMONIC FUNCTIONS, AND INDUCED NONLOCAL OPERATORS IN METRIC MEASURE SPACES
Following ideas of Caffarelli and Silvestre in [20], 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 non-local 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. Our work also includes as special cases much of the previous results by other authors 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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- Award ID(s):
- 2141297
- PAR ID:
- 10519882
- Publisher / Repository:
- Arxiv
- Date Published:
- Format(s):
- Medium: X
- Institution:
- Arxiv
- Sponsoring Org:
- National Science Foundation
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