Boundary problem for Levi flat graphs

In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, 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 $Ω\subset C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a smooth function over $Ω$. In particular, $M$ is non-singular.

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