## Levi equation for almost complex structures.(English)Zbl 1061.35025

The paper deals with the problem of finding a $$J$$-Levi flat hypersurface in $$\mathbb{R}^{4}$$ with prescribed boundary. The considered hypersurface is a graph $M=\left\{ x_{4}=u(x_{1},x_{2},x_{3}):(x_{1},x_{2},x_{3})\in\Omega \subset\mathbb{R}^{3}\right\}$ and the almost complex structure $$J$$ is defined by the following matrix $\begin{pmatrix} 0 & -1 & 0 & 0\\ 1 & 0 & 0 & 0\\ f & 0 & 0 & -1\\ 0 & -f & 1 & 0 \end{pmatrix} ,$ where $$f$$ is a regular function. In this situation, the boundary problem for $$J$$-Levi flat hypersurfaces amounts to solve the following Dirichlet problem for a nonlinear, second order, elliptic degenerate operator $$\mathcal{L}^{J} ,$$ called $$J$$-Levi operator for the almost complex structure $$J,$$ $\begin{cases} \mathcal{L}^{J}u=D_{1}^{2}u+D_{2}^{2}u-D_{1}f=0&\text{ in }\Omega\\ u=g&\text{ on }b\Omega, \end{cases} \tag{Pb}$ where $$g$$ is a given regular function on the boundary $$b\Omega$$ of $$\Omega$$ and $D_{1} =\partial_{1}+a(u)\partial_{3}\text{ , }D_{2}=\partial _{2}+b(u)\partial_{3}$
$a(u) =\frac{\partial_{2}u-\partial_{3}u(\partial_{1}u-f)}{1+(\partial _{3}u)^{2}} ,\;b(u)=\frac{\partial_{2}u\partial_{3}u+\partial_{1} u-f}{1+(\partial_{3}u)^{2}}.$ The main result of the paper is the following theorem.
Assume that $$\Omega$$ is bounded and $$b\Omega$$ is strictly $$J$$-pseudoconvex. Let $$f\in C^{m+1}(\overline{\Omega}),$$ $$g\in C^{2}(b\Omega)$$ and either $$f=0$$ or $$\sup\partial_{1}f<0$$ in $$\overline{\Omega}.$$ Then the problem (Pb) has a unique (viscosity) solution $$u\in \text{Lip}(\overline{\Omega} )$$ whose Lie derivatives of order $$k\leq m,$$ in the directions of the vector fields $$D_{i},$$ are of class $$C_{loc}^{\alpha}$$ for all $$\alpha<1.$$ The solution $$u$$ may not be regular in the usual sense but its graph is foliated by complex curves.
Reviewer: C. Bouzar (Oran)

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35H10 Hypoelliptic equations 32Q60 Almost complex manifolds
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### References:

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