The least gradient problem with Dirichlet and Neumann boundary conditions - Samer Dweik (22047329) (2024)
In this paper, we consider the BV least gradient problem with Dirichlet condition on a part Γ⊂∂Ω and Neumann boundary condition on its complementary part ∂Ω\Γ. We will show that in the plane this problem is equivalent to an optimal transport problem with import/export taxes on ∂Ω\Γ. 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, W1,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. Other Information Published in: Advances in Calculus of Variations License: http://creativecommons.org/licenses/by/4.0 See article on publisher's website: https://dx.doi.org/10.1515/acv-2023-0067