Consider the subset $\Omega$ of $\mathbb{R}^2$ which is bounded by the vertical lines $x = 0$ and $x = L$, the horizontal line $z = -h$ and the curve $(x, \eta(x))$. In the figure below, $\Omega$ is shown as the shaded region. We will assume the function $\eta(x)$ to be smooth and that it has the property $\eta(0) = \eta(L)$.
Assume that the function $\Phi : \Omega \rightarrow \mathbb{R}$ solves the Laplace equation $\nabla^2 \Phi = 0$ in the interior of $\Omega$ with the following boundary conditions:
1) $\Phi(x, \eta(x)) = \Phi_s(x)$ where $\Phi_s(x)$ is a smooth function with the property $\Phi_s(0) = \Phi_s(L)$.
2) $\frac{\partial \Phi}{\partial z}(x,-h) = 0$ for all $x \in [0, L]$.
3) $\Phi(0,z) = \Phi(L, z)$ for all $z \in [-h, \eta(0)]$.
In connection with this setup I want to ask the following question:
- Is it possible to prove/disprove that if $\Phi$ is continued to the region above the curve $(x, \eta(x))$, then the continuation is a smooth function? Furthermore, if the continuation is not smooth, can anything be said about the distribution of singularities of $\Phi$ above $(x, \eta(x))$ (optionally in the complex $z$-plane)?
I have not been able to come up with any answer to this question myself, simply because I don't know where to start. Suggestions where I can read about problems of this type are therefore also very welcome.
I stumbled over this question while trying to develop a numerical method for the time evolution of water waves. The method heavily depends on the assumption that $\Phi$ is smooth above $(x, \eta(x))$ so it would be nice if this assumption could in fact be substantiated.
