# Article

 Title: Nonuniqueness for some linear oblique derivative problems for elliptic equations (English) Author: Lieberman, Gary M. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 40 Issue: 3 Year: 1999 Pages: 477-481 . Category: math . Summary: It is well-known that the standard'' oblique derivative problem, $\Delta u = 0$ in $\Omega$, $\partial u/\partial \nu-u=0$ on $\partial\Omega$ ($\nu$ is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen. (English) Keyword: elliptic equations Keyword: uniqueness Keyword: a priori estimates Keyword: linear problems Keyword: boundary value problems MSC: 35A05 MSC: 35B65 MSC: 35J25 idZBL: Zbl 1064.35508 idMR: MR1732488 . Date available: 2009-01-08T18:54:17Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119103 . Reference: [1] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order.Springer-Verlag Berlin-Heidelberg-New York (1983). Zbl 0562.35001, MR 0737190 Reference: [2] Lieberman G.M.: Local estimates for subsolutions and supersolutions of oblique derivative problems for general second-order elliptic equations.Trans. Amer. Math. Soc. 304 (1987), 343-353. Zbl 0635.35037, MR 0906819 Reference: [3] Lieberman G.M.: Oblique derivative problems in Lipschitz domains I. Continuous boundary values.Boll. Un. Mat. Ital. 1-B (1987), 1185-1210. MR 0923448 Reference: [4] Lieberman G.M.: Oblique derivative problems in Lipschitz domains II. Discontinuous boundary values.J. Reine Angew. Math. 389 (1988), 1-21. MR 0953664 .

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