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.

中文翻译：

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凸域中的平面最小梯度问题：不连续情况

我们研究平面\（\ Omega \）中的凸多边形集的二维最小梯度问题。我们显示了在边界意义上获得边界数据f时解的存在。这里的主要困难是f的可能不连续性。此外，由于缺乏\（\ Omega \）的严格凸性，因此经典结果不适用。我们在边界数据f上陈述可容许条件，该条件足以确定存在结果。其中之一是\（f \ in BV（\ partial \ Omega）\）。解决方案是通过限制过程构建的，该过程使用已知问题的解决方案。