2017/08/07 by Panu Lahti, Lukáš Malý, Lahti, Panu +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1708.02346
openalex publication_date 2017/08/07 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We study an inhomogeneous Neumann boundary value problem for functions of\nleast gradient on bounded domains in metric spaces that are equipped with a\ndoubling measure and support a Poincar 'e inequality. We show that solutions\nexist under certain regularity assumptions on the domain, but are generally\nnonunique. We also show that solutions can be taken to be differences of two\ncharacteristic functions, and that they are regular up to the boundary when the\nboundary is of positive mean curvature. By regular up to the boundary we mean\nthat if the boundary data is 1 in a neighborhood of a point on the boundary\nof the domain, then the solution is -1 in the intersection of the domain with\na possibly smaller neighborhood of that point. Finally, we consider the\nstability of solutions with respect to boundary data.\n