Smooth regularity for solutions of the Levi Monge-Ampère equation

Lascialfari, Francesca ; Montanari, Annamaria

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 12 (2001), p. 115-123 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a real hypersurface in $\mathbb{C}^{n+1}$, the Levi Monge-Ampère equation arises in the study of envelopes of holomorphy and has important applications in the theory of holomorphic functions of several complex variables.

Résumé

In questa Nota presentiamo un risultato di regolarità $\mathcal{C}^{\infty}$ delle soluzioni strettamente Levi convesse dell’equazione di Levi Monge-Ampère. Si tratta di un’equazione alle derivate parziali totalmente non lineare e ellittico-degenere. Per studiare questa equazione non si possono utilizzare le tecniche classiche delle equazioni ellittiche, pertanto costruiamo una teoria della regolarità ad hoc in spazi di funzioni hölderiane modellati sulla geometria del problema. La nostra tecnica si ispira a quella introdotta in [3] e [8] per operatori quasilineari ellittico-degeneri che intervengono nello studio dell’equazione con assegnata curvatura media di Levi. Quando il secondo membro ha il significato di curvatura totale di un’ipersuperficie reale in $\mathbb{C}^{n+1}$, l’equazione di Levi Monge-Ampère contiene informazioni sui domini di olomorfia ed ha importanti applicazioni nella teoria delle funzioni olomorfe di più variabili complesse.

Détail

Publié le : 2001-06-01

MR 1898454 | Zbl 1097.35061

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