The least gradient problem with Dirichlet and Neumann boundary conditions

Abstract In this paper, we consider the BV least gradient problem with Dirichlet condition on a part <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:math> {\Gamma\subset\partial\Omega} and Neumann boundary condition on its complementary part <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>\</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>\</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {\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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> 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.