I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium (air on top for $i=a$ and liquid at the bottom for $i=b$). We need to solve Laplace equation (3) subject to Dirichlet B.C. at $z=0$ and $z=a+1$ and Neumann B.C. at $x=0$ and $x=\pi/k$ (Eq.4-7). At the interface $z=\xi(x)$ we have the usual matching conditions for electric field and potentials. This is just a standard problem of Laplace equation if $z=\xi(x)$ is known. However, $z=\xi(x)$ is unknown, and it has to satisfy an addition equation (10). $\tau_e$ is the electric stress which depends on $\phi_i$. $K$ is a scalar unknown which needs to be determined as well. $z=\xi(x)$ is subject to Neumann B.C. (eq. 13) and an integral constraint (eq.12). So in total we need to solve for $\phi_i(x,z)$, $\xi(x)$ and $K$. The paper just said it is solved by FEM, but I have no idea how to set it up. Are there any general strategy for solving these kind of Laplace equation where the interface is also an unknown?
Update: added a diagram for better illustration of the problem