A note on second derivative estimates for Monge-Ampère-type equations

Abstract

Abstract In this article, we revisit previous Pogorelov-type interior and global second derivative estimates of N. S. Trudinger, F. Jiang, and J. Liu for solutions of Monge-Ampère-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.