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The Dirichlet problem for nonlinear second-order elliptic equations. I: Monge-Ampère equation. (English) Zbl 0598.35047
In der vorliegenden Arbeit behandeln die Verff. in einem beschränkten, strikt konvexen Gebiet \(\Omega \subset {\mathbb{R}}^ n\) mit \(C^{\infty}\)- Rand \(\partial \Omega\) die Monge-Ampèresche Gleichung \[ \det (u_{ij})=\psi \quad mit\quad u| \partial \Omega =\phi | \Omega. \] Hierbei sind \(u_{ij}:=\partial_ i\partial_ ju\), \(\psi\in C^{\infty}({\bar \Omega})\), \(\psi >0\) und \(\phi \in C^{\infty}({\bar \Omega})\). Es wird die eindeutige Existenz der Lösung u des Dirichletproblems in der Klasse der strikt konvexen Funktionen und \(u\in C^{\infty}({\bar \Omega})\) gezeigt.
Zum Beweis wird die Kontinuitätsmethode benutzt. Dazu wird eine a priori-Abschätzung von der Form \(| u|_{2+\alpha}\leq K(\Omega,\psi,\phi)\) benötigt. Der Nachweis dieser Abschätzung ist wesentlicher Inhalt dieser Arbeit. Dazu werden das Maximumsprinzip und einseitige Abschätzungen der dritten Ableitungen bis zum Rand verwandt. Abschließend werden auch allgemeinere Monge-Ampère-Gleichungen untersucht.
Reviewer: R.Leis

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs
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