Interior $W^{2,\delta}$ type estimates for degenerate fully nonlinear elliptic equations with $L^n$ data

TL;DR

This paper proves interior $W^{2, ext{delta}}$ estimates for certain degenerate fully nonlinear elliptic equations with $L^n$ data, using a novel approach involving sliding $C^{1, ext{alpha}}$ cones to analyze the solution's regularity.

Contribution

It introduces a new method of sliding $C^{1, ext{alpha}}$ cones to obtain Hessian estimates for degenerate elliptic equations with minimal data assumptions.

Findings

Solutions have tangent $C^{1, ext{alpha}}$ cones almost everywhere.

Establishes a counterpart to divergence-structure quasilinear elliptic estimates.

Develops a measure-theoretic approach to regularity for degenerate equations.

Abstract

We establish interior $W^{2,\delta}$ type estimates for a class of degenerate fully nonlinear elliptic equations with $L^n$ data. The main idea of our approach is to slide $C^{1,\alpha}$ cones, instead of paraboloids, vertically to touch the solution, and estimate the contact set in terms of the measure of the vertex set. This shows that the solution has tangent $C^{1,\alpha}$ cones almost everywhere, which leads to the desired Hessian estimates. Accordingly, we are able to develop a kind of counterpart to the estimates for divergent structure quasilinear elliptic problems.