A note on second derivative estimates for Monge-Ampe`re type equations

TL;DR

This paper revisits second derivative estimates for Monge-Ampère type equations, removing previous monotonicity assumptions and leveraging recent convexity regularity results to improve classical solvability and regularity outcomes.

Contribution

It extends second derivative estimates to more general generated Jacobian equations by removing monotonicity constraints, building on recent convexity regularity advances.

Findings

Removed monotonicity assumptions in second derivative estimates.

Achieved improved classical solvability results.

Enhanced understanding of global regularity for boundary value problems.

Abstract

In this note we revisit previous Pogorelov type interior and global second derivative estimates of the author, F. Jiang and J. Liu for solutions of Monge-Amp`ere type partial differential equations. Taking account of recent strict convexity regularity results of Guillen, Kitagawa and Rankin, and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.