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.