Global Lipschitz estimates for fully non-linear singular perturbation problems with non-homogeneous degeneracy

Abstract

This manuscript investigates the global regularity of singularly perturbed unbalanced models with variable exponents. In this context, we aim to find a non-negative function $u^{ϵ}$ that satisfies the following equation for each fixed $ϵ>0$$\{\begin{array}{ccccc}H(x,\nabla u^{ϵ}(x\left)\right)\left[\Delta u^{ϵ}\right(x)+b(x)\cdot \nabla u^{ϵ}(x\left)\right]& =& {\zeta }_{ϵ}\left(u^{ϵ}\right)+f_{ϵ}\left(x\right)& \text{in}& \Omega ,\\ u^{ϵ}\left(x\right)& =& g\left(x\right)& \text{on}& \partial \Omega ,\end{array}$ where Ω is a bounded regular domain in $R^n$, and $H$ represents a function exhibiting a variable degeneracy signature. Additionally, ${\zeta }_{ϵ}$ approaches the Dirac measure ${\delta }_0$ as ϵ tends to zero, and $f_{ϵ}$ remains bounded away from zero and infinity. We aim to establish global gradient bounds that are unaffected by the parameter ϵ. Specifically, this family of solutions converges uniformly to a Lipschitz limiting profile associated with a one-phase Bernoulli-type free boundary problem.