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On some nonlinear elliptic equations involving derivatives of the nonlinearity. (English) Zbl 0586.35044
Die Arbeit enthält Existenz- und Eindeutigkeitsresultate für schwache beschränkte Lösungen von \[ Au+(\partial /\partial \nu_ i)\beta_ i(u)=f_ 0+\partial f_ i/\partial x_ i\quad in\quad \Omega \subset {\mathbb{R}}^ n,\quad u=\phi \quad auf\quad \partial \Omega, \] wobei die \(\beta_ i\) nicht notwendig differenzierbar zu sein brauchen. A ist ein linearer elliptischer Operator in Divergenzform und \(\nu_ i\) sind konstante Vektoren. Von Interesse sind insbesondere die Eindeutigkeitsresultate für stetige (und nicht Lipschitz stetige) \(\beta_ i\). Die Autoren geben hinreichende Kriterien zur Eindeutigkeit an und zeigen anhand von Gegenbeispielen die Notwendigkeit solcher Kriterien auf.
Reviewer: B.Kawohl

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
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References:
[1] Kinderlehrer, An Introduction to Variational Inequalities and their Applications (1980) · Zbl 0457.35001
[2] Gilbarg, Elliptic Partial Differential Equations of Second Order (1977)
[3] Brezis, Free Boundary Problems: Theory and Applications 1 (1983)
[4] Ladyzhenskaya, Linear and Quasilinear Elliptic Equations (1968)
[5] Brezis, C. R. Acad. Sci. Paris, Sér. A 287 pp 711– (1978)
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