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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 $F(D\sp 2u,Du,u)=0$ zweiter Ordnung. Er weist nach, daß Viskositätslösungen in $W\sp{1,\infty}(\Omega)\cap C({\bar \Omega})$ 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 $(D\sp 2u,Du,u)$ vorausgesetzt, allerdings auch Abhängigkeit von $x$ zugelassen.
Reviewer: B.Kawohl

35B50Maximum principles (PDE)
35J60Nonlinear elliptic equations
Full Text: DOI
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[9] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 3. Regularity of the optimal cost function. Collège de France Seminar, Vol. V, Pitman, London, 1983. · Zbl 0716.49024
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