Short time existence and smoothness of the nonlocal mean curvature flow of graphs

Abstract

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an analytic semigroup approach, short time existence, uniqueness and optimal H\"older regularity in time and space of classical solutions of the nonlocal equation, depending on the regularity of the initial graph. The method also yields $C^\infty-$smoothness estimates of the evolving graphs for positive times.