The Dirichlet problem for a complex Monge-Ampere equation

Abstract

Communicated by P. R. Halmos, September 25, 1975On C 1 , write d = 3 + 3, d c = i(3 -3) so that dd c u = 2/33w, and let

Weak solutions to the complex Hessian equation

“…-For m = n Theorem 3.7 was proved by Bedford and Taylor [1] with the help of an interior C 1,1 estimate ([1, Theorem 6.7]), which, together with later simplifications due to Demailly [11], gives an overall simpler and more elementary proof than the one presented here (not employing strong solutions at all and thus not using the EvansKrylov theory and estimate (3.7)). It relied however on the following, rather rare, property: the group of smooth diffeomorphisms of the unit ball in C n preserving plurisubharmonic functions is transitive.…”

“…The proofs of the following three results for m = n can be essentially found in [1]. PROPOSITION 3.3.…”

“…[1], [2], [5], [11], [18]). The aim of this paper is to concentrate on those problems related to the Hessian operator where the methods of the complex Monge-Ampère operator cannot be automatically repeated.…”

Cauchy–Riemann meet Monge–Ampère

“…The complex Monge-Ampère operator (dd c ) n plays the central role in pluripotential theory, it has been developed in this context by Bedford and Taylor [1,2]. For example, Demailly [43] characterized the pluricomplex Green function as a solution to the Monge-Ampère equation with point-mass on the right-hand side.…”

A characterization of bounded plurisubharmonic functions

“…Denote by PSH(Ω) the plurisubharmonic (psh) functions on Ω. The complex Monge-Ampère operator (dd c ) n is well defined over the class of locally bounded psh functions, according to the fundamental work of BedfordTaylor in [BT1], [BT2]. Recently Cegrell has introduced in [Ce1], [Ce2] new classes of psh functions on which the complex Monge-Ampère operator can be defined and enjoys important properties, e.g.…”