In Physics (specifically electrostatics) it is often encountered that one must solve the Dirichlet problem for Laplace's equation
\begin{equation} \nabla^2 \varphi(x_1,x_2,x_3) = 0 \end{equation}
between two surfaces of a constant coordinate, on each of which $\varphi$ is a different constant value. For example:
Two concentric spheres with radii $R_1$ and $R_2$ ($R_1 \ne R_2$) where $\varphi$ is given to be two different constant values on these surfaces. In spherical coordinates, at the surface of the plates, $\varphi$ is independent of $\theta$ and $\phi$, and it just so happens that the solution of Laplace's equation for $\varphi$ between the spheres is also independent of $\theta$ and $\phi$.
Two parallel-infinite planes separated by a distance where $\varphi$ is given to be two different constant values on these surfaces. In cartesian coordinates, at the surface of the plates, $\varphi$ is independent of $x$ and $y$, and it just so happens that the solution of Laplace's equation for $\varphi$ between the plates is also independent of $x$ and $y$.
Two intersecting planes separated by an angular distance where $\varphi$ is given to be two different constant values on these surfaces. In cylindrical coordinates (with the z-axis defined along the line where the planes intersect), at the surface of the plates, $\varphi$ is independent of $r$ and $z$, and it just so happens that the solution of Laplace's equation for $\varphi$ between the plates is also independent of $r$ and $z$.
Is there a formal theorem that says something similar to:
In a region bounded by two different surfaces of a constant coordinate, if $\nabla^2\varphi=0$ within the region and $\varphi$ is constant over both surfaces, then $\varphi$ within the region can only be a function of the coordinate used to define the constant surfaces.
?