Smoothness up to the free boundary for the $p$-Laplacian evolution equation and the $\alpha$-Gauss curvature flow

TL;DR

This paper proves short-time smooth existence of solutions up to free boundaries for the degenerate parabolic equations of the p-Laplacian evolution and alpha-Gauss curvature flow with flat sides.

Contribution

It establishes the first rigorous proof of smooth solutions up to free boundaries for these degenerate flows using linear degenerate equation techniques.

Findings

Short-time smooth solutions exist up to free boundaries.

Solutions are smooth at the free boundary.

Method applies to degenerate parabolic equations with evolving free boundaries.

Abstract

The $p$-Laplacian evolution equation and the $\alpha$-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.