Calculus of Variations and Geometric Measure Theory

S. Dweik

The least gradient problem with Dirichlet and Neumann boundary conditions

created by dweik on 09 May 2023
modified on 18 May 2023



Inserted: 9 may 2023
Last Updated: 18 may 2023

Year: 2023


In this paper, we consider the BV least gradient problem with Dirichlet condition on a part $\Gamma \subset \partial\Omega$ and Neumann boundary condition on its complementary part $\partial\Omega\backslash\Gamma$. We will show that in the plane this problem is equivalent to an optimal transport problem with import-export taxes on $\partial\Omega\backslash\Gamma$. Thanks to this equivalence, we will be able to show existence and uniqueness of a solution to this mixed least gradient problem and, we will also prove some Sobolev regularity on this solution. We note that these results generalize those in $[7]$, where we studied the pure Dirichlet version of this problem.

Keywords: Mixed least gradient problem - Import-export optimal transport - 1-Laplacian