Singularly perturbed equations of degenerate type

Abstract

This work is devoted to the study of nonvariational, singularly perturbed elliptic equations of degenerate type. The governing operator is anisotropic and ellipticity degenerates along the set of critical points. The singular behavior is of order $\text{O}\left(\frac{1}{ϵ}\right)$ along ϵ-level layers $\{u_{ϵ}\sim ϵ\}$, and a non-homogeneous source acts in the noncoincidence region $\{u_{ϵ}>ϵ\}$. We obtain the precise geometric behavior of solutions near ϵ-level surfaces, by means of optimal regularity and sharp geometric nondegeneracy. We further investigate Hausdorff measure properties of ϵ-level surfaces. The analysis of the asymptotic limits as the ϵ parameter goes to zero is also carried out. The results obtained are new even if restricted to the uniformly elliptic, isotropic setting.