Nonuniqueness for some linear oblique derivative problems for elliptic equations
Commentationes Mathematicae Universitatis Carolinae (1999)
- Volume: 40, Issue: 3, page 477-481
- ISSN: 0010-2628
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topLieberman, Gary M.. "Nonuniqueness for some linear oblique derivative problems for elliptic equations." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 477-481. <http://eudml.org/doc/248391>.
@article{Lieberman1999,
abstract = {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.},
author = {Lieberman, Gary M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {elliptic equations; uniqueness; a priori estimates; linear problems; boundary value problems; nonuniqueness; oblique derivative problem; elliptic equations},
language = {eng},
number = {3},
pages = {477-481},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Nonuniqueness for some linear oblique derivative problems for elliptic equations},
url = {http://eudml.org/doc/248391},
volume = {40},
year = {1999},
}
TY - JOUR
AU - Lieberman, Gary M.
TI - Nonuniqueness for some linear oblique derivative problems for elliptic equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 3
SP - 477
EP - 481
AB - 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.
LA - eng
KW - elliptic equations; uniqueness; a priori estimates; linear problems; boundary value problems; nonuniqueness; oblique derivative problem; elliptic equations
UR - http://eudml.org/doc/248391
ER -
References
top- Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin-Heidelberg-New York (1983). (1983) Zbl0562.35001MR0737190
- 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. (1987) Zbl0635.35037MR0906819
- Lieberman G.M., Oblique derivative problems in Lipschitz domains I. Continuous boundary values, Boll. Un. Mat. Ital. 1-B (1987), 1185-1210. (1987) MR0923448
- Lieberman G.M., Oblique derivative problems in Lipschitz domains II. Discontinuous boundary values, J. Reine Angew. Math. 389 (1988), 1-21. (1988) MR0953664
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