On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian (Q1769370)

Let \( D\) be a smoothly bounded domain in \( {\mathbb C}^n \). The author investigates the existence and uniqueness of solutions \( u\) for Dirichlet problems \( f(\lambda(H(u)))=\psi \) in \( \overline{D} \), \( u=\phi \) on \( \partial D \), where \( \lambda(H(u)) \) denotes the eigenvalues of the complex Hessian of \( u \) and \( f \) is a symmetric function satisfying suitable concavity, positivity and growth assumptions. In the real case, such problems were treated by \textit{L. Caffarelli, L. Nirenberg} and \textit{J. Spruck} [Acta Math. 155, 261--301 (1985; Zbl 0654.35031)]. In the complex case, the author obtains, among other results, the following statement in the particular situation of elementary symmetric functions \( \sigma^{(k)} \) for \( k=2,\dots,n \): assume that \( \partial D \) is connected and that the eigenvalues \( \lambda({\mathcal L}) \) of the Levi form \( \mathcal L \) satisfy \( \sigma^{(k-1)}(\lambda({\mathcal L}))>0 \) on \( \partial D \). Then, for smooth data \( \psi \) and \( \phi \) with \( \psi>0 \), the problem \( \sigma^{(k)}(\lambda(H(u)))=\psi \) in \( \overline{D} \), \( u=\phi \) on \( \partial D \) has a unique solution \( u\in C^\infty(\overline{D}) \).