Abstract Question
Do you know results on consistency conditions on boundary conditions for PDEs in order to keep/gain differentiability when combining strong solutions of a PDE on different domains that share a boundary?
A Concrete Question
Because this formulation is too general and a little ambiguous, I'd like to ask this question for a specific problem. However, hints on general ways to solve such problems still are very welcome, too. Here it is:
Let $\Omega_1 = B(1) \subseteq \mathbb{R}^2$ (two-dimensional unit circle), $\Gamma_1 = \partial \Omega_1$, $\,f,g \in C^\infty(\Gamma_1)$ and $u_1 \in H^2(\Omega)$ the unique weak solution of $$ \Delta^2 u = 0, \quad u|_{\Gamma_1} = f, \quad \partial_n u|_{\Gamma_1} = g $$ where $\Delta^2 u = \Delta \Delta u$ denotes the Bi-Laplacian.
Furthermore, let $\Omega_2 = B(2) \setminus \Omega_1$ (annulus), $\Gamma_2 = \partial B(2)$ where $u_2 \in H^2(\Omega_2)$ denotes the unique weak solution of $$ \Delta^2 u = 0, \quad u|_{\Gamma_1} = f, \quad \partial_n u|_{\Gamma_1} = g, \quad u|_{\Gamma_2} = 0, \quad \partial_n u|_{\Gamma_2} = 0. $$
Define $\Omega = B(2)$ and $$ u\colon \Omega \longrightarrow \mathbb{R}~~~~~~~~~~~~~~~~~~ \\ ~~~~~~~~~~~~~~~~~x \longmapsto \begin{cases} u_1(x) \text{ for } x \in \Omega_1 \\ u_2(x) \text{ for } x\in \Omega_2.\end{cases} $$ We have $u_1 \in H^4(\Omega_1)$ and $u_2 \in H^4(\Omega_2)$. Under what conditions on $f,g$ is $u \in H^3(\Omega)$? Or, under what conditions on $f,g$ can't hold $u\in H^3(\Omega)$?
Some Thoughts on the Problem
I encountered this problem when dealing with solutions of PDEs where the respective domains share some boundary. At first, I thought that I could easily combine these two solutions to a big one, but this is only partially true. To illustrate this, in the given example above we have that $u\in H_0^2(\Omega)$ (because of the boundary conditions) and that $u$ satisfies $$ \Delta^2 u = 0, \quad u|_{\Gamma_1} = f, \quad \partial_n u|_{\Gamma_1} = g $$ in a weak sense. So combining weak solutions works. However, while you can show that $u_1 \in H^4(\Omega_1)$ and $u_2 \in H^4(\Omega_2)$ from regularity theory in order to get strong solutions (in a sense), you will in general have $u \notin H^4(\Omega)$ because the boundary conditions do not necessarily match.
If we ask for conditions on $f,g$ which imply $u\in H^4(\Omega)$ the answer maybe is kind of trivial because you probably end up with $f=g=0$ (although I did not show this yet, so I might be wrong there). But for some theory I might still want that at least $u\in H^3(\Omega)$ holds (or $u\in H^{\frac{7}{2}}(\Omega)$ would be awesome). I already experimented a little with simpler situations but I couldn't find a pattern yet. Has anybody an idea or knows how to treat this problem? Has this maybe been done already and somebody knows where to find it?