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Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient
Dept. of Mathematical Sciences, University of Cincinnati.
Dept. of Mathematical Sciences, University of Cincinnati.ORCID iD: 0000-0003-2083-9180
Dept. of Mathematical Sciences, University of Cincinnati.ORCID iD: 0000-0002-2891-5064
Dept. of Mathematical Sciences, University of Cincinnati.ORCID iD: 0000-0002-3028-1558
2018 (English)In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 29, no 4, p. 3176-3220Article in journal (Refereed) Published
Abstract [en]

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions), and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg et al. (J Reine Angew Math 430:35–60, 1992) to the non-smooth setting. Via counterexamples, we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.

Place, publisher, year, edition, pages
2018. Vol. 29, no 4, p. 3176-3220
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-182908DOI: 10.1007/s12220-018-00108-9OAI: oai:DiVA.org:liu-182908DiVA, id: diva2:1637368
Funder
Knut and Alice Wallenberg FoundationAvailable from: 2022-02-14 Created: 2022-02-14 Last updated: 2022-02-14

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Malý, Lukáš

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