English

Planar least gradient problem: existence, regularity and anisotropic case

Analysis of PDEs 2017-09-29 v2

Abstract

We show existence of solutions to the least gradient problem on the plane for boundary data in BV(Ω)BV(\partial\Omega). We also provide an example of a function fL1(Ω)\(C(Ω)BV(Ω))f \in L^1(\partial\Omega) \backslash (C(\partial\Omega) \cup BV(\partial\Omega)), for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.

Keywords

Cite

@article{arxiv.1608.02617,
  title  = {Planar least gradient problem: existence, regularity and anisotropic case},
  author = {Wojciech Górny},
  journal= {arXiv preprint arXiv:1608.02617},
  year   = {2017}
}
R2 v1 2026-06-22T15:15:22.870Z