Boundary Value Problems on S(n) for Surfaces of Constant Gauss Curvature

摘要

The organization of the paper is as follows. In Section 2, the authors reexamine the local regularity theory for classical Monge-Ampere equations in R(sup n) near the boundary of a domain with arbitrary geometry. In particular, they derive C(sup 2) estimates for solutions assuming the structure conditions (locally) of Theorem 1.5. These estimates are used in Sections 4 and 5 in the study of boundary value problems on S(sup n) and also suffice to prove Theorem 1.5. In Section 3, they formulate the problem of Monge-Ampere type for radial graphs over S(sup n) that they will study in Sections 4 and 5. Section 4 contains a proof of the strong compactness of solutions (and their strict local convexity) in C(sup(2 + alpha)). Because the kernel of the linearized operator may be nontrivial, these estimates do not immediately yield the desired existence result. Therefore, in Section 5 they construct the solution directly by a monotone iteration scheme starting from the given strict subsolution. They modify the estimates of Section 4 and show the convergence of the sequence of approximations in C(sup(2 + alpha)) to the (unique) admissible solution closest to the subsolution.