Variational Nonlinear and Nonlocal Curvature Flows

TL;DR

This paper proves convergence of a variational scheme to level-set solutions for a nonlinear, nonlocal curvature flow with time-dependent forcing, extending the framework of variational curvatures and establishing uniqueness under certain conditions.

Contribution

It introduces a convergence proof for a variational scheme applied to a nonlinear nonlocal curvature flow with minimal assumptions, and establishes uniqueness of solutions.

Findings

Convergence of the minimizing movements scheme to level-set solutions.

Extension of the variational curvature framework to nonlinear, nonlocal flows.

Conditions for uniqueness of solutions.

Abstract

We prove that the minimizing movements scheme \'a la Almgren-Taylor-Wang converges towards level-set solutions to a nonlinear version of nonlocal curvature flows with time-depending forcing term, in the rather general framework of variational curvatures introduced in \cite{ChaMorPon15}. The nonlinearity involved is assumed to satisfy minimal assumptions, namely continuity, monotonicity, and vanishing at zero. Under additional assumptions only on the curvatures involved, we establish uniqueness for level-set solutions.