Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries

TL;DR

This paper investigates fully nonlinear singularly perturbed parabolic equations, analyzing their solutions' regularity, free boundary behavior, and deriving conditions for rotationally invariant operators, revealing the free boundary's porous nature.

Contribution

It provides a detailed analysis of the regularity and free boundary properties of solutions to singularly perturbed nonlinear parabolic equations, including the measure and structure of the free boundary.

Findings

Solutions are uniformly Lipschitz continuous in space.

Solutions are Hölder continuous in time.

The free boundary is a porous set of Lebesgue measure zero.

Abstract

We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and H\"{o}lder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.