A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition

TL;DR

This paper investigates a free-boundary problem for the evolution of the p-Laplacian equation with specific boundary conditions, establishing existence, uniqueness, regularity, and smoothness of the solution and its free boundary over time.

Contribution

It provides new results on the existence, uniqueness, and regularity of solutions to a p-Laplacian evolution equation with over-determined boundary conditions, including free boundary smoothness.

Findings

Unique solution exists for initial data with certain properties.

Solution remains regular and the free boundary is smooth over time.

Solution vanishes after finite time.

Abstract

We study the existence, uniqueness and regularity of solutions of the equation $f_t = \Delta_p f = \text{div} (|Df|^{p-2} Df)$ under over-determined boundary conditions $f = 0$ and $|Df| = 1$. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until it vanishes identically. Furthermore, the free-boundary of the support of $f$ is smooth for all positive time.