I will preface this question by referring to a similar question here which did not give a full solution. I found this question on the January 2017 Applied Mathematics Qualifying Exam from the University of Wisconsin-Madison, specifically question 5 of this document.
We wish to solve Laplace's equation $u_{xx} + u_{zz} = 0$ on a 2D rectangular domain with $x \in \left[ 0,\ L\right)$ and $z \in \left[0,\ H\right]$ with periodic boundary conditions in $x$ and the following Neumann boundary conditions at $z = 0$ and $z = H$:
\begin{alignat*}{2} u_z(x,\ 0) &= \theta_{-},\quad 0 \leq x < l,\quad& u_z(x,\ 0) &= \theta_{+},\quad l \leq x < L,\\ u_z(x,\ H) &= \theta_{+},\quad 0 \leq x < L - l,\quad& u_z(x,\ H) &= \theta_{-},\quad L - l \leq x < L. \end{alignat*}
To make the solution unique, the condition $u(0,\ 0) = 0$ must also be satisfied.
There are two hints given with the problem:
(1) Rather than considering Green's functions, separation of variables, etc., think about simple solutions of $u_{xx} + u_{zz} = 0$.
(2) Draw a picture, consider the solution in different parts of the domain, and derive jump conditions to connect the different parts together.
My Attempt
My initial attempt was to take the hints to heart and consider piecewise linear functions of $z$ alone (with the branches dependent on $x$), since these would trivially satisfy Laplace's equation. We can define such a function to match the desired Neumann boundary conditions, but without $x$ dependence I don't believe the periodic boundary conditions can be satisfied.
Another thought I had was to take the Fourier transform in $x$, leading to the ODE $$\hat{u}_{zz} = k^2\,\hat{u},$$ which has the solution $$\hat{u} = \hat{f}(k)\,e^{k\,z} + \hat{g}(k)\,e^{-k\,z}.$$ The trouble I ran into with this attempt was finding the Fourier transform of the Neumann boundary conditions (and interpreting what the periodic boundary conditions would be in Fourier space).