Let $D\_1$ be a domain with smooth boundary and assume that $D\_1$ is a proper subset of $D\_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D\_2\setminus D\_1$ is connected. We write $L^2(D\_2\setminus D\_1)$ for the set of functions in the space
$$\{f \in L^2(D\_2)\,:\,\textrm{supp}(f)\subset D\_2\setminus \overline{D\_1}\}$$
let us define the mapping
$$S: L^2(D\_2\setminus \overline{D\_1})\mapsto H^{\frac{3}{2}}(\partial D\_1),$$
through
$$ Sf:= u|\_{\partial D\_1},$$
where $u \in H^2(D\_2)$ is the unique solution to the equation
$$ \Delta u =f \quad \text{on $D\_2$},$$
subject to $u|\_{\partial D\_2}=0$. Is it true that the image of $S$ is dense in $H^{\frac{1}{2}}(\partial D\_1)$?



Density of traces of solutions to an elliptic equation
The answer is yes: take any smooth function $g\_0$ on $\partial D\_1$ and solve the Dirichlet problem
$$
\begin{cases}
\Delta g = 0 & \text{ on } D\_1\\
g = g\_0 & \text{ on } \partial D\_1.
\end{cases}
$$
Now extend $g$ to a smooth function on $\mathbb{R}^n$. Multiply by a smooth cutoff function $\eta$ which is $1$ on $D\_1$ and compactly supported on $D\_2$.


Then $f = \Delta (\eta g)$ is smooth and supported on $D\_2 \setminus \bar{D}\_1$, so in particular lies in the given $L^2$ space. This shows that $g\_0$ is in the image of your operator $S$, and smooth functions are dense in $H^s$.


