On Pogorelov estimates in optimal transportation and geometric optics

Abstract

In this paper, we prove global and interior second derivative estimates of Pogorelov type for certain Monge–Ampère type equations, arising in optimal transportation and geometric optics, under sharp hypotheses. Specifically, for the case of generated prescribed Jacobian equations, as developed recently by the second author, we remove barrier or subsolution hypotheses assumed in previous work by Trudinger and Wang (Arch Ration Mech Anal 192:403–418, 2009), Liu and Trudinger (Discret Contin Dyn Syst Ser A 28:1121–1363, 2010), Jiang et al. (Calc Var Partial Differ Equ 49:1223–1236, 2014), resulting in new applications to optimal transportation and near field geometric optics.