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\(C^ \infty\) regularity of solutions of a quasilinear equation related to the Levi operator. (English) Zbl 0872.35018
The author studies the regularity of solutions of the equation \[ {\mathcal L}u = q {{1+|\nabla u|^2}\over{1+u_t^2}} \quad\hbox{in}\quad \Omega\subset {\mathbb{R}}^3 \] where \((x,y,t)\in \Omega\), \(q\in C^\infty(\Omega)\) and \({\mathcal L}\) is the Levi operator, which is a second order quasilinear degenerate elliptic operator arising in complex analysis. The main result is that if \(q\) is never zero in \(\Omega\), then any solution \(u\in C^{2,\alpha}(\Omega)\) with \(\alpha>1/2\) belongs to \(C^\infty(\Omega)\). A regularity result in the case \(q=0\) was proved by A. Debiard and B. Gaveau [Bull. Sci. Math., II. Ser. 102, 369-386 (1978; Zbl 0413.35067)].
Reviewer: J.Urbas (Bonn)

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
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References:
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