# Existence and uniqueness for anisotropic and crystalline mean curvature flows

## TL;DR

This paper establishes existence and uniqueness of crystalline mean curvature flows with forcing and arbitrary mobilities, introducing a new solution concept that ensures comparison and stability, applicable in any dimension.

## Contribution

It introduces a novel solution notion for crystalline mean curvature flows, proving existence, uniqueness, and stability, and confirms convergence of a minimizing movements scheme to a flat flow.

## Key findings

- Proved existence and uniqueness of solutions up to fattening.
- Developed a new level set solution satisfying comparison and stability.
- Established convergence of the minimizing movements scheme to a flat flow.

## Abstract

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.