We study the planar least gradient problem with respect to an anisotropic norm for continuous boundary data. We prove existence of minimizers for strictly convex domains. Furthermore, we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of the norm and the domain. The results are independent from the regularity of the norm.

中文翻译：

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关于非严格凸范数的最小梯度问题

我们针对各向异性边界连续数据的平面最小梯度问题进行了研究。我们证明了存在于严格凸域中的极小子。此外，我们仅根据范数和域的凸模来检验最小化器的唯一性和规则性问题。结果与规范的规律性无关。