On the oblique derivative problem for harmonic functions (Q1567890)

For a bounded domain \(D\) with smooth boundary \(\Gamma\), the oblique derivative problem is to find a harmonic function \(u\) regular in \(D\), continuously differentiable in \(D\cup \Gamma\), and satisfying the condition \(a \frac{\partial u}{\partial n}+ l(u)= b\) where \(a\) and \(b\) are given differentiable functions and \(l\) is a first-order differential operator with differentiable coefficients containing derivatives only in the directions tangent to \(\Gamma\). One formulates a similar problem for harmonic functions regular in the exterior of \(D\). The conditions for the solvability and the number of solutions to these problems are treated here and the following particular cases are considered as illustrations: Solve the oblique derivative problem (1) if \(D\) is the unit ball \(x^2+ y^2+ z^2< 1\), with the boundary conditions \(u_2=b\) or \(xu_x+ yu_y+ zy_z= b\), and (2) in an axially symmetric domain \(D\) with the boundary conditions \(2z (xu_x+ yu_y+ zu_z)- (x^2+ y^2+ z^2) u_z+ zu= b\) or \(xu_x+ yu_y+ (z-z_0) u_z= b\), where the solution depends on whether \((0,0,z_0)\) is inside or outside \(D\).