I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{ {{\nabla ^2}\Phi (x,y) = 0,} \hfill & { - a \le x \le a} \hfill & { - b \le y \le b} \hfill \cr {{{\partial \Phi } \over {\partial y}}(a,y) = f(y)} \hfill & {} \hfill & {} \hfill \cr {{{\partial \Phi } \over {\partial y}}( - a,y) = f(y)} \hfill & {} \hfill & {} \hfill \cr {{{\partial \Phi } \over {\partial x}}(x,b) = 0} \hfill & {} \hfill & {} \hfill \cr {{{\partial \Phi } \over {\partial x}}(x, - b) = 0} \hfill & {} \hfill & {} \hfill \cr } $$
where $f(-y)=-f(y)$. We have prescribed the tangential derivatives of $\Phi(x,y)$ on the boundary instead of the function value.
The more general form of the problem in 2D or 3D can be written as
$$\matrix{ {{\nabla ^2}\Phi ({\bf{x}}) = 0,} \hfill & {{\bf{x}} \in \Omega } \hfill \cr {\nabla \Phi ({\bf{x}}).{\bf{t}}({\bf{x}}) = f({\bf{x}})} \hfill & {{\bf{x}} \in \partial \Omega } \hfill \cr } $$
where $\Omega$ is the domain of interest and ${\partial \Omega }$ is its boundary. Also, ${{\bf{t}}({\bf{x}})}$ is the tangent unit vector to the boundary at point $\bf{x}$ on the boundary.
Questions
1) Is the solution to this BVP unique? If NO, what is the degree of non-uniqueness?
2) Is there a relation between the solution to this BVP and the one with Dirichlet boundary conditions, i.e., when we determine the function value on the boundary?
My Thought
I don't think that the solution is unique so I was thinking to relate this in some manners to the solution of the Laplace BVP with Dirichlet boundary conditions where the function value is prescribed over the boundary since this BVP has a unique solution.