The Dirichlet problem for modified complex Monge-Ampère equations (Q1270239)

Let \((X^{2m},g)\) be a strongly pseudoconvex Hermitian manifold with smooth boundary. For a function \(\phi\in C^2(X)\) consider the new Hermitian metrics defined by \[ \begin{align*}{ g_1(\phi) &= -\phi g + i\partial\overline\partial\phi, \cr g_2(\phi) &= g + i\partial\overline\partial\phi \pm \nabla\phi \otimes \nabla\phi. \cr }\end{align*} \] The authors prove the existence of globally smooth solutions of Dirichlet problems of the form \[ [\det g_i(\phi)] [\det g]^{-1} = F(x,\nabla\phi;\phi) \quad\text{in \(X\)}, \qquad \phi = u \quad\text{on \(\partial X\)}, \] where \(u\in C^\infty(\partial X)\) and \(F\in C^\infty(TX\times{\mathbb R})\) is a positive function satisfying appropriate conditions. The results are a generalization of those of \textit{Caffarelli} \textit{J. J. Kohn}, \textit{L. Nirenberg} and J. Spruck [Commun. Pure Appl. Math. 38, 209-252, (1985; Zbl 0598.35048)] concerning the Dirichlet problem on strongly pseudoconvex domains of \({\mathbb C}^n\) for the simpler Monge-Ampère equation \(\det \partial\overline\partial\phi = F(x,\nabla\phi;\phi)\). Analogous real Monge-Ampère equations on compact Riemannian manifolds without boundary have been considered by \textit{Ph. Delanoë} [J. Funct. Anal. 40, 358-386 (1981; Zbl 0466.58029); 41, 341-353 (1981; Zbl 0474.58023); 45, 403-430 (1982; Zbl 0497.58026)] and, on \({\mathbb S}^n\), by \textit{V. I. Oliker} [Commun. Partial Differ. Equ. 9, 807-838 (1984; Zbl 0559.58031)] in connection with hypersurfaces of \({\mathbb R}^{n+1}\) with prescribed Gauss curvature.