Local pointwise second derivative estimates for strong solutions to the $\sigma_k$-Yamabe equation on Euclidean domains

TL;DR

This paper establishes local second derivative bounds for positive solutions to the $\sigma_k$-Yamabe equation on Euclidean domains, extending to augmented Hessian equations, thus advancing regularity theory in geometric PDEs.

Contribution

It provides new local second derivative estimates for solutions to the $\sigma_k$-Yamabe equation, including both positive and negative cases, and generalizes results to augmented Hessian equations.

Findings

Established local second derivative estimates for positive solutions.

Extended estimates to both positive and negative cases.

Generalized results to augmented Hessian equations.

Abstract

We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $\sigma_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.