Geometric approach to nonvariational singular elliptic equations

TL;DR

This paper introduces a geometric framework for analyzing fully nonlinear elliptic equations with singular absorption, establishing existence, regularity, and geometric properties of solutions and their free boundaries.

Contribution

It develops a systematic geometric approach to study singular elliptic equations, proving existence, regularity, non-degeneracy, and detailed geometric-measure properties of free boundaries.

Findings

Established existence and sharp regularity of solutions.

Proved minimal solutions are non-degenerate.

Derived Hausdorff estimates and perimeter finiteness of the free boundary.

Abstract

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.