On the Dirichlet problem for Monge-Ampère type equations

Abstract

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampère type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in their study of global regularity in optimal transportation [28] as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.

A note on global regularity in optimal transportion

“…The last mentioned alternative sufficient condition follows from a duality argument in Section 3 of [20] which does not seem to extend to general dependence of A on (x, p), (as envisaged in an earlier version of [20]). An alternative sub-solution condition for these second derivative bounds is also given in the recent paper [6]. In the next section we indicate the necessary modifications to obtain the full generality.…”

On Pogorelov estimates in optimal transportation and geometric optics

“…(1.1) is uniformly elliptic and use it to prove an appropriate version, Lemma 2.2, of the key Lemma 2.1 in [4]. From here we obtain Theorems 1.1 and 1.2 by following the same arguments as in [4]. In Sect.…”

“…In the optimal transportation case, Theorem 1.2 improves Theorem 1.1 in [9] and Theorem 2.1 in [4] by removing the barrier and subsolution hypotheses assumed there. As in [9], we may also replace g 0 by any strict supersolution.…”

“…When we use both A3w and A4w, we would also assume P x to be convex. We note that in [4], U = R n × R × R n so that these conditions are automatically satisfied and in the optimal transportation case J x = R. Similar conditions are also needed for the convexity theory in [16].…”

“…Note that the inequality u ≤ g 0 is trivially satisfied if there exists a g-affine function on¯ which is greater than sup u and moreover can be dispensed with in the optimal transportation case, where Theorem 1.1 improves Theorems 3.1 and 3.2 in [20] and Theorem 1.1 in [4]. Also we may assume in place of A4w the reversed monotonicity condition: A4*w: The matrix A is monotone decreasing with respect to u in U, that is…”