Levi equation for almost complex structures

Abstract

In this paper we are dealing with the boundary problem for Levi flat graphs in the space $\mathbb{R}^4$, endowed with an almost complex structure $J$. This problem can be formalized as a Dirichlet problem for a quasilinear degenerate elliptic equation, called Levi equation. The Levi equation has the form $$D_1^2 + D^2_2 - D_1f = 0,$$ where $D_1$ and $D_2$ are nonlinear vector fields. Under geometrical assumptions on the boundary a lipschitz continuous viscosity solution is found. The regularity of the viscosity solution is studied in suitable anisotropical Sobolev spaces, and it is proved that the solution has derivatives of any order in the direction of the vectors $D_1$ and $D_2$ i.e. it is of class $C^\infty$ in these directions, but not necessary regular in the third direction of the space. Finally, after proving a weak version of the Frobenius theorem, we show that the graph of the solution is foliated in holomorphic curves.