Existence and regularity in the fully nonlinear one-phase free boundary problem

TL;DR

This paper establishes existence and regularity results for viscosity solutions to fully nonlinear one-phase free boundary problems, including higher regularity of free boundaries using advanced mathematical techniques.

Contribution

It proves existence of solutions using Perron's method and demonstrates $C^{2,eta}$ regularity of flat free boundaries with a quadratic improvement of flatness, advancing understanding of free boundary regularity.

Findings

Existence of viscosity solutions via Perron's method.

$C^{2,eta}$ regularity of flat free boundaries.

Higher regularity achieved through hodograph transform.

Abstract

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove $C^{2,\alpha}$ regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.