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

Albert Chau, Ben Weinkove

TL;DR

The paper extends smooth short-time existence for degenerate parabolic equations on fixed domains to nonlinear free-boundary problems, specifically the $p$-Laplacian evolution and the $\alpha$-Gauss curvature flow with a flat side. It achieves this by a standard moving-domain to fixed-domain change of variables, followed by linearization to a linear degenerate parabolic operator whose boundary degeneracy is controlled by Fichera-type conditions; applying the authors' prior CW Nash–Moser framework yields short-time regularity up to the free boundary. Under natural nondegeneracy and geometric constraints (e.g., $\alpha>1/n$ and $\tfrac{2}{\sigma}\in\mathbb{Z}^+$ with $\sigma=n-\frac{1}{\alpha}$), the free boundaries are shown to be smooth and the solutions smooth near the boundary. This provides a unified, potentially extensible method for other nonlinear degenerate parabolic problems with moving boundaries, complementing existing Schauder and viscosity approaches.

Abstract

The $p$-Laplacian evolution equation and the $α$-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.

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

TL;DR

The paper extends smooth short-time existence for degenerate parabolic equations on fixed domains to nonlinear free-boundary problems, specifically the -Laplacian evolution and the -Gauss curvature flow with a flat side. It achieves this by a standard moving-domain to fixed-domain change of variables, followed by linearization to a linear degenerate parabolic operator whose boundary degeneracy is controlled by Fichera-type conditions; applying the authors' prior CW Nash–Moser framework yields short-time regularity up to the free boundary. Under natural nondegeneracy and geometric constraints (e.g., and with ), the free boundaries are shown to be smooth and the solutions smooth near the boundary. This provides a unified, potentially extensible method for other nonlinear degenerate parabolic problems with moving boundaries, complementing existing Schauder and viscosity approaches.

Abstract

The -Laplacian evolution equation and the -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.

Paper Structure

This paper contains 8 sections, 4 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Short time existence and boundary conditions for nonlinear equations
  3. Change of variables and linearization
  4. The $p$-Laplacian evolution equation
  5. The $\alpha$-Gauss curvature flow with a flat side
  6. Computing the linearization
  7. The $p$-Laplacian evolution equation
  8. The $\alpha$-Gauss curvature flow with a flat side

Key Result

Theorem 1.1

There exists $T>0$ such that for $t\in [0,T]$, the sets $\Omega_t$ have smooth boundaries $\partial \Omega_t$. The function $g=g(x,t)$ on ${\cup_{t \in [0,T]} \overline{\Omega}_t \times \{ t\} }$ is smooth in a neighborhood of the boundary ${\cup_{t \in [0,T]} \partial \Omega_t \times \{ t\} }$, sat

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • proof