Gary M. Lieberman
Nonuniqueness for some linear oblique derivative problems for elliptic equations

Comment.Math.Univ.Carolinae 40,3 (1999) 477-481.

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.

Keywords: elliptic equations, uniqueness, a priori estimates, linear problems, boundary value problems
AMS Subject Classification: 35J25, 35A05, 35B65