Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and Hénon-type equations
Jiangwen Wang, Feida Jiang
TL;DR
The paper studies degenerate or singular dead-core systems governed by fully nonlinear operators with strong absorption, establishing sharp regularity along the free boundary, non-degeneracy, density/measure estimates, and Liouville-type results, along with a blow-up analysis. A novel flatness-improvement method and a weak comparison principle for non-variational, degenerate systems underpin the approach, yielding $|(u,v)|$-growth rates of order $|x-x_{0}|^{2/\bigl((1+p)(1+q)-\lambda_{1}\lambda_{2}\bigr)}$ at free boundary points and improved exponents for $u$ and $v$. The work extends to Hardy–Henon type equations with degenerate weights, delivering sharp local growth and gradient estimates and showing that the results remain valid even in the classical degenerate Laplacian setting. These findings advance understanding of regularity and geometric properties for degenerate fully nonlinear dead-core problems and related free boundary phenomena, with implications for both theory and non-variational free boundary problems.
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 Hénon-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.