Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates

TL;DR

This paper establishes local second derivative estimates for solutions to augmented Hessian equations on Riemannian manifolds, using integral methods, without structural assumptions or with convexity conditions, enhancing understanding of regularity in geometric PDEs.

Contribution

It provides new a priori pointwise second derivative estimates for augmented Hessian equations on Riemannian manifolds, including cases with minimal assumptions and convexity conditions, using integral estimates.

Findings

Derived local second derivative estimates using integral methods.

Established results without structural assumptions on augmenting terms.

Showed dependence on $W^{2,p}$ norms can be eliminated under convexity and ellipticity.

Abstract

We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any $W^{2,p}$ norm can be dropped. Our results are derived using integral estimates.