| 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 |
| . |
