Regularity estimates for fully nonlinear dead-core problems with a Hamiltonian Term

TL;DR

This paper investigates the regularity and growth behavior of solutions to fully nonlinear elliptic equations with Hamiltonian terms, especially near free boundaries and plateau zones, providing new optimality results and applications.

Contribution

It introduces new regularity estimates and growth rate optimality for fully nonlinear dead-core problems with Hamiltonian terms, advancing understanding of free boundary behavior.

Findings

Improved regularity along the free boundary.

Proved optimal growth rates under certain conditions.

Established positivity and Hausdorff measure estimates for the boundary.

Abstract

In this paper, we present a problem involving fully nonlinear elliptic operators with Hamiltonian, which can present a singularity or degenerate as the gradient approaches the origin. The model studied here, allows the appearance of plateau zones, i.e. unknown regions of the domain in which the non-negative solutions vanishes. We show an improvement in regularity along the free boundary of the problem, and with some hypotheses on the exponents of the equation we proved the optimality of the growth rate with the help of the non-degeneracy also obtained here. In addition, some more applications of the growth results on the free boundary are obtained, such as: the positivity of solutions and also information on the Hausdorff measure of the boundary of the coincidence set.