Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

TL;DR

This paper studies a class of complex free boundary problems involving doubly degenerate fully non-linear PDEs with non-zero right-hand side, establishing regularity and classification results for solutions and free boundaries.

Contribution

It introduces a new analysis of non-homogeneous degenerate PDEs in free boundary problems, proving Lipschitz continuity, non-degeneracy, and $C^{1,\beta}$ regularity of free boundaries.

Findings

Solutions are Lipschitz continuous.

Solutions satisfy a non-degeneracy property.

Flat and Lipschitz free boundaries are $C^{1,\beta}$.

Abstract

We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such non-linear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli's classification scheme: Flat and Lipschitz free boundaries are locally $C^{1, \beta}$ for some $0 < \beta (universal) < 1$.