Accepted Paper
Inserted: 9 may 2023
Last Updated: 26 sep 2024
Journal: Advances in Calculus of Variations
Year: 2023
Abstract:
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
Download: