Geodesic
Nonuniqueness for some linear oblique derivative problems for elliptic equations
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 3, pp. 477-481 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 35A05, 35B65, 35J25
Keywords: elliptic equations; uniqueness; a priori estimates; linear problems; boundary value problems 
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     author = {Lieberman, Gary M.},
     title = {Nonuniqueness for some linear oblique derivative problems for elliptic equations},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {477--481},
     year = {1999},
     volume = {40},
     number = {3},
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     zbl = {1064.35508},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1999_40_3_a6/}
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Lieberman, Gary M. Nonuniqueness for some linear oblique derivative problems for elliptic equations. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 3, pp. 477-481. http://geodesic.mathdoc.fr/item/CMUC_1999_40_3_a6/