The Dirichlet problem for the Monge-Ampère equations in Hermitian metric
For \(X\) being a \(C^\infty\) strongly pseudoconvex hermitian compact manifold with boundary, the authors prove the existence of solutions to the Dirichlet problem \(\text{ det}\nabla^\lambda_\mu \varphi =F(x,\nabla \varphi, \varphi)\) on \(X\) and \(\varphi=u\) on \(\partial X\). They assume \(F\in C^\infty (TX\times {\mathbb R})\) to be an everywhere positive function and \(u\in C^\infty (\partial X)\). The methods, which they use to obtain a priori estimates for solutions extend ideas from the classical work by \textit{L. Caffarelli, J. J. Kohn, L. Nirenberg} and \textit{J. Spruck} [Commun. Pure Appl. Math. 38, 209-252 (1985; Zbl 0598.35048)].
pseudoconvex Hermitian compact manifolds
scientific article; zbMATH DE number 1388957