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arXiv · 2007.06361

The planar Least Gradient problem in convex domains: the discontinuous case

Piotr Rybka·Ahmad Sabra
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Abstract

We study the two dimensional least gradient problem in convex polygonal sets in the plane, $\Omega$. We show the existence of solutions when the boundary data $f$ are attained in the trace sense. The main difficulty here is a possible discontinuity of $f$. Moreover, due to the lack of strict convexity of $\Omega$, the classical results are not applicable. We state the admissibility conditions on the boundary datum $f$, that are sufficient for establishing an existence result. One of them is that $f\in BV(\partial\Omega)$. The solutions are constructed by a limiting process, which uses solutions to known problems

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Piotr Rybka, Ahmad Sabra. 2020-07-13. The planar Least Gradient problem in convex domains: the discontinuous case. https://arxiv.org/abs/2007.06361

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