tailieunhanh - Đề tài " Boundary regularity for the Monge-Amp`ere and affine maximal surface equations "

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Amp`re equation when the inhomogee neous term is only assumed to be H¨lder continuous. As a consequence of our o approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation. | Annals of Mathematics Boundary regularity for the Monge-Amp ere and affine maximal surface equations By Neil S. Trudinger and Xu-Jia Wang Annals of Mathematics 167 2008 993-1028 Boundary regularity for the Monge-Ampere and affine maximal surface equations By Neil S. Trudinger and Xu-Jia Wang Abstract In this paper we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation. 1. Introduction In a landmark paper 4 Caffarelli established interior W2 p and C2 a estimates for solutions of the Monge-Ampere equation detD2u f in a domain Q in Euclidean n-space Rra under minimal hypotheses on the function f. His approach in 3 and 4 pioneered the use of affine invariance in obtaining estimates which hitherto depended on uniform ellipticity 2 and 19 or stronger hypotheses on the function f 9 13 18 . If the function f is only assumed positive and Holder continuous in Q that is f E Ca Q for some a E 0 1 then one has interior estimates for convex solutions of in C2 a Q in terms of their strict convexity. When f is sufficiently smooth such estimates go back to Calabi and Pogorelov 9 and 18 . The estimates are not genuine interior estimates as assumptions on Dirichlet boundary data are needed to control the strict convexity of solutions 4 and 18 . Our first main theorem in this paper provides the corresponding global estimate for solutions of the Dirichlet problem u p on dQ. Supported by the Australian Research .