Global $W^{2,\delta}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right hand side terms

TL;DR

This paper proves global $W^{2, ext{delta}}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right-hand side terms, using novel geometric and measure-theoretic techniques.

Contribution

It introduces a new method involving paraboloid sliding and barrier functions to obtain global estimates for singular equations with $L^n$ data, extending previous results.

Findings

Established global $W^{2, ext{delta}}$ estimates for singular equations.

Developed a new covering argument and measure estimate technique.

Provided a direct proof of known estimates for nonsingular equations.

Abstract

We establish in this paper \emph{a priori} global $W^{2,\delta}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right hand side terms. The method is to slide paraboloids and barrier functions vertically to touch the solution of the equation, and then to estimate the measure of the contact set in terms of the measure of the vertex point set. To derive global estimates from $L^n$ data, the Hardy-Littlewood maximal functions, appropriate localizations and a new type of covering argument are adopted. These methods also provide us a more direct proof of the $W^{2,\delta}$ estimates for (nonsingular) fully nonlinear elliptic equations established by L. A. Caffarelli and X. Cabr\'{e}.