Existence and uniqueness for planar anisotropic and crystalline curvature flow

TL;DR

This paper establishes short-time existence and uniqueness of solutions for planar anisotropic and crystalline curvature flows, even with complex forcing terms like white noise, using variational schemes and approximation methods.

Contribution

It introduces a novel approach combining implicit variational schemes and regularization to handle existence and uniqueness in anisotropic curvature flows with irregular forcing.

Findings

Proved short-time existence of solutions for anisotropic curvature flow.

Established uniqueness when anisotropy is smooth and elliptic.

Handled discontinuous and unbounded forcing terms such as white noise.

Abstract

We prove short-time existence of \phi-regular solutions to the planar anisotropic curvature flow, including the crystalline case, with an additional forcing term possibly unbounded and discontinuous in time, such as for instance a white noise. We also prove uniqueness of such solutions when the anisotropy is smooth and elliptic. The main tools are the use of an implicit variational scheme in order to define the evolution, and the approximation with flows corresponding to regular anisotropies.