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European Mathematical Society Publishing House
2016-09-19 17:05:44
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
20
2004
1
Levi equation for almost complex structures
Giovanna
Citti
Università di Bologna, BOLOGNA, ITALY
Giuseppe
Tomassini
Scuola Normale Superiore, PISA, ITALY
Levi equation, almost complex structure, degenerate elliptic equation, anysotropic Sobolev spaces, foliation in holomorphic curves
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.
Partial differential equations
Several complex variables and analytic spaces
Global analysis, analysis on manifolds
General
151
182
10.4171/RMI/384
http://www.ems-ph.org/doi/10.4171/RMI/384