Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and H\'{e}non-type equations

TL;DR

This paper advances the understanding of degenerate or singular fully nonlinear systems with absorption, providing improved regularity results, free boundary analysis, and sharp estimates for Hénon-type equations, even for models with degenerate Laplacians.

Contribution

It introduces new regularity and measure estimates for solutions and free boundaries in degenerate or singular nonlinear systems, including Hénon-type equations, with novel sharp regularity results.

Findings

Enhanced regularity of viscosity solutions along free boundaries

Measure estimates for free boundary behavior

Sharp regularity estimates for degenerate Hénon-type equations

Abstract

In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary, non-degeneracy, a measure estimate of the free boundary, Liouville-type results, and the behavior of blow-up solution. We also derive sharp regularity estimates for viscosity solutions to H\'{e}non-type equations with a degenerate weight and strong absorption, governed by a degenerate fully nonlinear operator. Our results are new even for the model equations involving degenerate Laplacian operators.