Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries

TL;DR

This paper develops geometric regularity estimates for fully nonlinear elliptic equations with free boundaries, improving understanding of boundary behavior and solution properties under strong absorption conditions.

Contribution

It introduces new geometric regularity estimates along free boundaries and establishes a sharp Liouville theorem for solutions with controlled growth.

Findings

Enhanced regularity estimates at free boundaries

Non-degeneracy and measure-theoretic properties established

A sharp Liouville theorem for entire solutions

Abstract

In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value depending only on structural parameters. Non degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.