Weak solutions for the Levi equation and envelope of holomorphy.

*(English)*Zbl 0744.35015Given a bounded domain \(\Omega\subset\mathbb{R}^ 3\) and a continuous function \(k\) on \(\Omega\times\mathbb{R}\) the authors consider the so-called Levi equation
\[
\begin{split} L(u;k)=(1+u^ 2_ 3)(u_{11}+u_{22})+(u^ 2_ 1+u^ 2_ 2)u_{33}+2(u_ 2-u_ 1u_ 2)u_{13}-\\ -2(u_ 1 +u_ 2u_ 3)u_{23}+k(\cdot ;u)(1+| Du|^ 2)^{3/2}=0,\end{split}
\]
where \(u_ j=\partial u/\partial x_ j\). This is a quasilinear strongly degenerate elliptic equation. The Dirichlet problem for this equation with \(u=g\) on \(\partial\Omega\) is considered. The solutions under consideration are weak (in the sence of viscosity) solutions. The main result is an existence theorem in the space \(\hbox{Lip}(\overline\Omega)\) of the Lipschitz functions in the case, when \(\partial\Omega\) and \(g\) are of the class \(C^{2,\alpha}\) \((0<\alpha<1)\) and \(\Omega\) is “strictly pseudoconvex”. In the case \(k=0\) it is stated that the result remains true if the function \(g\) is assumed to be continuous. As an application the existence of the envelope of holomorphy for certain compact 2-manifolds is obtained.

Reviewer: A.Pankov (Vinnitsa)

##### MSC:

35J70 | Degenerate elliptic equations |

35A15 | Variational methods applied to PDEs |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

32D10 | Envelopes of holomorphy |

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\textit{Z. Slodkowski} and \textit{G. Tomassini}, J. Funct. Anal. 101, No. 2, 392--407 (1991; Zbl 0744.35015)

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