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.

Samer Dweik

Advances in Calculus of Variations

ADVANCES IN CALCULUS OF VARIATIONS 发表了高质量的原创研究，重点关注变分法和相关应用的部分，将偏微分方程的工具和方法与几何技术相结合。

The journal "Advances in Calculus of Variations" publishes high-quality original research focusing on the calculus of variations and its related applications, integrating tools and methods from partial differential equations with geometric techniques.

《Advances in Calculus of Variations》期刊发表高质量的原创研究，主要关注变分法及其相关应用，结合偏微分方程的工具与几何技术。

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数学	3
数学, 数学	3
数学, 应用数学	3

The least gradient problem with Dirichlet and Neumann boundary conditions

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Category	Quartile
MATHEMATICS, APPLIED	2