Classical solvability of the Dirichlet problem for the Monge-Ampère equation

Regularity of generalized solutions of Monge-Amp�re equations

“…Using the continuity of u, we see that there is a constant ee(0, a/2) so small that U~ lies between two parallel hyperplanes separated by a distance less than 60, where 6 o is given by (2.16) with #o and #1 replaced by ft. Now let {12,.} be an increasing sequence of smooth uniformly convex subdomains of U, such that U~= ~) O,.. By the results of [8] or [15] there is a rn=I convex solution u,.e C z (~,.) of the Dirichlet problem detD2u,~=f(X, Um,DUm) in O,,, um=e on 0(2m, and by Lemma 2.2 and our choice of e, {Urn} converges to u uniformly on compact subsets of U~.…”

Classical boundary value problems for Monge-Ampère type equations

“…More general results are in fact formulated in [5] , [12] but the condition 6<n+1 cannot be improved [26]. The situation with regard to oblique boundary value problems of the form…”

The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature

“…When Ω is a strictly convex domain, this problem has received considerable study both in the non-degenerate case (ψ > 0) and in the degenerate case (ψ = 0 somewhere). A well known theorem (see Caffarelli, Nirenberg and Spruck [6], Ivochkina [17] and Krylov [19]) states that in the non-degenerate case ψ > 0, (1.1) has a strictly convex solution in C ∞ (Ω), provided Ω is strictly convex and there exists a strictly convex subsolution in C 2 (Ω). (Please see, for example, [6], [12] and [22] for further references, including the earlier work of, among others, Pogorelov, Cheng and Yau, and P. L.…”