An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a by-product of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) "flat flow" in the case of general, possibly crystalline, anisotropies is settled.

中文翻译：

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各向异性和晶体平均曲率流的存在性和唯一性

证明了具有强制和任意（凸）迁移率的晶体平均曲率流的存在性和唯一性结果，直到增肥。这是通过向相应的水平集公式引入新的解决方案概念来实现的。这样的解决方案通过适当正则化的问题满足关于近似的比较原理和稳定性特性。结果在任何维度和任意的，可能是无界的，初始闭集都是有效的。该方法考虑了可能存在与时间相关的有界强迫项，具有空间 Lipschitz 连续性。作为分析的副产品，Almgren-Taylor-Wang 最小化运动方案在一般的、可能是结晶的、各向异性的情况下收敛到独特的（直至育肥）“平坦流”的问题得到解决。