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Keywords:
elliptic equations; uniqueness; a priori estimates; linear problems; boundary value problems
elliptic equations; uniqueness; a priori estimates; linear problems; boundary value problems
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.
References:
[1] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag Berlin-Heidelberg-New York (1983). MR 0737190 | Zbl 0562.35001
[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. MR 0906819 | Zbl 0635.35037
[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
[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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