Levi-equation in higher dimensions
The authors announce two results about the Dirichlet problem for the Levi equation in \(\mathbb C^ 3\). Let \(\Omega\) be a bounded domain in \(\mathbb R^ 5\) with connected boundary, and let \(u\) be a continuous real-valued function on \(\Omega\). The graph of \(u\) is then a surface in \(\mathbb C^ 3\), and this surface has a Levi form. The problem is to find a function \(u\), with given continuous boundary values on \(\partial\Omega\), such that the determinant of the Levi form equals \(k(\cdot,u)(1+| Du|^ 2)^ 2\) in \(\Omega\), where \(k\) is a given continuous function on \(\Omega\times\mathbb R\). The authors' first proposition states that if \(u\) is a weak solution of this problem with \(k=0\), then the graph of \(u\) is neither strictly pseudoconvex nor strictly pseudoconcave at any point. The second proposition states that the problem can be reduced to a certain ``Bellman problem'' for a family of quasilinear degenerate elliptic operators.
scientific article; zbMATH DE number 61694