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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

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
Full Text: DOI
[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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