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The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. (English) Zbl 0708.35019
Der Autor betrachtet voll nichtlineare elliptische Differentialgleichungen \(F(D^ 2u,Du,u)=0\) zweiter Ordnung. Er weist nach, daß Viskositätslösungen in \(W^{1,\infty}(\Omega)\cap C({\bar \Omega})\) eindeutig sind sofern a) \(F\) entartet elliptisch und monoton fallend in \(u\) ist, oder b) \(F\) gleichmäßig elliptisch und monoton nicht wachsend in \(u\) ist. Beim Beweis werden Regularisierungen von \(u\) benutzt, welche Viskositäts- Ober- und Unterlösungen in Viskositäts- Ober- und Unterlösungen überführen. Frühere Arbeiten von P. L. Lions hatten Konvexität oder Konkavität von \(F\) und lineares Wachstum in \((D^ 2u,Du,u)\) vorausgesetzt, allerdings auch Abhängigkeit von \(x\) zugelassen.
Reviewer: B.Kawohl

MSC:
35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
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