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Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient

Abstract

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.

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Correspondence to Nageswari Shanmugalingam.

Additional information

The authors thank Estibalitz Durand-Cartagena, Marie Snipes and Manuel Ritoré for fruitful discussions about the subject of the paper. The research of N.S. was partially supported by the NSF Grant #DMS-1500440 (U.S.). The research of P.L. was supported by the Finnish Cultural Foundation. The research of L.M. was supported by the Knut and Alice Wallenberg Foundation (Sweden). Part of this research was conducted during the visit of N.S. and P.L. to Linköping University. The authors wish to thank this institution for its kind hospitality.

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Lahti, P., Malý, L., Shanmugalingam, N. et al. Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient. J Geom Anal 29, 3176–3220 (2019). https://doi.org/10.1007/s12220-018-00108-9

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Keywords

  • Finite perimeter
  • Least gradient
  • Mean curvature
  • Dirichlet problem

Mathematics Subject Classification

  • 31E05
  • 30L99
  • 51F99
  • 26A45