Nonlinear equations related to the Levi form
type: Item
instance of: scholarly article
title: Nonlinear equations related to the Levi form
zbMATH Open document ID: 0836.35056
DOI: 10.1007/BF02844692
author: Giuseppe Tomassini
publication date: 1992-09-27T00:00:00Z
review text: In the geometric theory of functions of several complex variables, nonlinear equations related to the Levi form often occur. The most famous among them is perhaps the Monge-Ampère equation \(\text{det} (u_{\alpha \overline \beta}) = f\). Another equation which seems to play an important role in the complex analysis of CR-structures is \[ \text{det} \left( \begin{matrix} 0 & & u_1 &\ldots & u_n \\ u_{\overline 1} & & u_{\overline 11} &\ldots & u_{\overline 1n} \\ \vdots & & \vdots & & \vdots \\ u_{\overline n} &\ldots & u_{\overline n1} &\ldots & u_{\overline nn} \end{matrix} \right) = f. \tag{L} \] We call it the Levi equation. There is a great difference between these two equations. Indeed the former is elliptic in the family of strictly plurisubharmonic functions while the latter is elliptic degenerate in all reasonable classes of functions we may choose. It becomes clear that the traditional methods based on ``a priori estimates'' do not apply to (L) so that, in order to study the general problem, a preliminary analysis of the specific features of the solutions of (L) is necessary. This paper, which can be seen as a continuation of [the author, Ann. Mat. Pura Appl., IV. Ser. 152, 331-344 (1988; Zbl 0681.35017)], has just this motivation. In the first part, various aspects of the maximum principle are analysed. The second part is devoted to removable singularities.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 60573
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: Levi equation
zbMATH Keywords: removable singularities
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 60573