The least gradient problem with Dirichlet and Neumann boundary conditions.

Dweik, Samer

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, 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.

NEUMANN boundary conditions; DIRICHLET problem; EXPORT duties; NEUMANN problem

Advances in Calculus of Variations, 2025, Vol 18, Issue 1, p151

1864-8258

Academic Journal

10.1515/acv-2023-0067