THE DIRICHLET PROBLEM FOR MODIFIED COMPLEX MONGE-AMPERE EQUATIONS

Let (X-2m, g) be a strongly pseudoconvex hermitian manifold with C-inf inity boundary. If phi belongs to C-2(X), let us consider the changes of hermitian metric defined by g(1)(phi) = -phi g + i partial derivati ve partial derivative phi and g(2)(+/-)(phi) = g + i partial derivativ e partial derivative phi +/- del phi x del phi. We solve problems of t he form [detg(i)(phi)][det(g)](-1) = F(x, del phi; phi) in X and phi = u on partial derivative X, where F is an element of C-infinity (TX x R) is an everywhere strictly positive function satisfying some assumpt ions and u is an element of C-infinity(partial derivative X). (C) 1998 Academic Press.