Title: Boundary problem for Levi flat graphs

Authors: Giuseppe Tomassini, Pierre Dolbeault and Dmitri Zaitsev

Issue: Volume 60 (2011), Issue 1, 161-170


In [P. Dolbeault, G. Tomassini and D. Zaitsev, \textit{On Levi-flat hypersurfaces with prescribed boundary}, Pure Appl. Math. Q. \textbf{6} (2010), no. 3, 725--753] the authors provided general conditions on a real codimension $2$ submanifold $S\subset\mathbb{C}^{n}$, $n \ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$ bounded by $S$. In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $\Omega\subset\mathbb{C}^{n-1}\times\mathbb{R}$ and show that in this case $M$ is necessarily a graph of a smooth function over $\Omega$. In particular, $M$ is non-singular.