Fully Nonlinear Singularly perturbed models with non-homogeneous degeneracy

TL;DR

This paper investigates complex nonlinear elliptic models with double degeneracy, establishing existence, regularity, and geometric properties of solutions, and analyzing their asymptotic behavior in relation to free boundary problems in combustion theory.

Contribution

It introduces new results on existence, Lipschitz regularity, measure-theoretic properties, and asymptotic analysis of solutions for fully nonlinear degenerate singularly perturbed models.

Findings

Existence of solutions with prescribed boundary conditions.

Solutions are locally Lipschitz continuous and grow linearly.

Finiteness of the Hausdorff measure of level sets for certain nonlinearities.

Abstract

In our work we study non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In such a scenario, we establish the existence of solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Moreover, for a restricted class of non-linearities, we prove the finiteness of the (N-1)-dimensional Hausdorff measure of level sets. We also address a complete analysis concerning the asymptotic limit as the singular parameter, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory.