Global $W^{2,\delta}$ estimates for a type of singular fully nonlinear elliptic equations

TL;DR

This paper establishes global $W^{2, ext{delta}}$ regularity estimates for a class of singular fully nonlinear elliptic equations with bounded right-hand side, using a geometric touching method.

Contribution

It introduces a novel approach involving paraboloid sliding to derive regularity estimates for singular elliptic equations with bounded data.

Findings

Proves global $W^{2, ext{delta}}$ estimates for the equations.

Develops a geometric method based on paraboloid sliding.

Provides bounds on the measure of contact point sets.

Abstract

We obtain global $W^{2,\delta}$ estimates for a type of singular fully nonlinear elliptic equations where the right hand side term belongs to $L^\infty$. The main idea of the proof is to slide paraboloids from below and above to touch the solution of the equation, and then to estimate the low bound of the measure of the set of contact points by the measure of the set of vertex points.