The least gradient problem with Dirichlet and Neumann boundary conditions
Year: 2024
Type: article
Abstract: In this paper, we consider the BV least gradient problem with Dirichlet condition on a part Γ ⊂ ∂ Ω {\Gamma\subset\partial\Omega} and Neumann boundary condition on its complementary part ∂ Ω \ Γ {\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 ∂ Ω \ Γ {\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 [S. Dweik, W 1 , p W^{1,p} regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary, Nonlinear Anal. 223 2022, Article ID 113012], where we studied the pure Dirichlet version of this problem.
Author: Samer Dweik
Institution: Qatar University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,489
OpenAlex ID: W4404501011