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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:
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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