Abstract We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is 1 in a neighborhood of a point on the boundary of the domain, then the solution is −1 in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.

中文翻译：

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度量设置中 1-Laplace 方程的 Neumann 问题的模拟：存在性、边界正则性和稳定性

摘要 我们研究了度量空间中有界域上的最小梯度函数的非齐次 Neumann 边值问题，这些函数配备了加倍测度并支持 Poincaré 不等式。我们表明在域的某些规律性假设下存在解决方案，但通常是非唯一的。我们还表明，解可以被视为两个特征函数的差异，并且当边界为正平均曲率时，它们直到边界是规则的。通过正则到边界，我们的意思是，如果域边界上某个点的邻域中的边界数据为 1，则该域与该点的可能较小邻域的交集处的解为 -1。最后，我们考虑解决方案相对于边界数据的稳定性。