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.