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.
@article{119103,
author = {Gary M. Lieberman},
title = {Nonuniqueness for some linear oblique derivative problems for elliptic equations},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {477-481},
zbl = {1064.35508},
mrnumber = {1732488},
language = {en},
url = {http://dml.mathdoc.fr/item/119103}
}
Lieberman, Gary M. Nonuniqueness for some linear oblique derivative problems for elliptic equations. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 477-481. http://gdmltest.u-ga.fr/item/119103/
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