Problem: Let $\Omega_1,\Omega_2\subseteq \mathbb{R}^2$ be two domains, define $\Omega:=\Omega_1\cup\Omega_2$ as their union as well as $\Gamma:=\partial\Omega_1 \cap \partial \Omega_2$ as their common boundary (and assume $\Gamma$ has some positive measure). Let $u \in H^1(\Omega)$ be the weak solution of the associated Laplace problem with inner and outer Dirichlet conditions $$ \begin{align} -\Delta u & = 0 \\ u|_{\partial \Omega} & = 0 \\ u|_{\Gamma} & = f \end{align} $$ where we assume that $\Omega$, $\,\Gamma$ and $f:\Gamma\rightarrow \mathbb{R}$ are "sufficiently smooth".
Question: Is there any regularity theory known that asserts $u\in H^{1+s}(\Omega)$ for some $s\in\mathbb{R}_{>0}$ when the boundary value $f$ is chosen "appropriately"?
To be honest, I got totally lost in the zoo of regularity theory. Maybe I just lost the overview or I maybe just don't know the keywords to search for. I didn't even have much luck when I looked for boundary regularity of "normal" Dirichlet-problems without an inner condition. Or is this problem maybe trivial? I'd appreciate any input from you that might help, let it be alone for the sake of improving my understanding of this topic.
The problem that I'm actually interested in is a little more complicated. Thus, if you know something that works even on higher order elliptic PDEs, I'd be really grateful.
In this section I want to collect information that might be useful for solving this problem. But it isn't necessarily needed.
Weak Formulation: To explain my thoughts on this, let's start with the standard weak formulation for $$ \begin{align} -\Delta u & = 0 \\ u|_{\partial\Omega} & = 0, \end{align} $$ which is given by searching for $u\in H_0^1(\Omega)$ with $$ \forall{v \in H_0^1(\Omega):} \int_{\Omega} \nabla u\cdot \nabla v \mathrm{d}x = 0. $$ In order to include the Dirichlet condition on $\Gamma$ we define the subspaces $$ \begin{align} H_{0,f}^1(\Omega) & := \{ v \in H_0^1(\Omega) \mid v|_{\Gamma} = f \} \\ H_{0,0}^1(\Omega) & := \{ v \in H_0^1(\Omega) \mid v|_{\Gamma} = 0 \}. \end{align} $$ For $\Gamma$ "appropriate" this is well-defined as $v|_{\Gamma} \in L^2(\Gamma)$ then exists by the trace theorem. The weak formulation for the initial problem then reads: We search $u\in H_{0,f}^1(\Omega)$ so that $$ \forall{v \in H_{0,0}^1(\Omega):} \int_{\Omega} \nabla u\cdot \nabla v \mathrm{d}x = 0. $$
Unique Existence of a Solution: The unique existence follows from Lax-Milgram. We have that the weak formulation is motivated by the bilinear operator $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v \mathrm{d}x, $$ which is known to be coercive on $H_0^1(\Omega)$ and continuous on $H^1(\Omega)$. In particular, $a$ is coercive and bounded on $H_{0,f}^1(\Omega)$. Since $H_{0,f}^1(\Omega)$ is a affine linear subspace we can apply Lax-Milgram and obtain the unique existence. (Note that the Lax-Milgram lemma usually is stated for non-affine linear spaces but there are also versions of it that work for affine linear spaces.)