Let $A,Q$ be two regular bounded open subset of $\mathbb{R}^d$ such as $\overline{A}\subset Q$. Let $\varphi \in C^\infty(\overline{A})$ be fixed data.
We want to give a meaning to the following problem:
Find $u$ such as it is a (weak) solution of:
\begin{gather} \begin{aligned} -\Delta u &= 0 &&\text{in } Q{\setminus A}\\ u&= \varphi &&\text{on } \partial A\\ u &= 0 &&\text{on } \partial Q \end{aligned}\tag{$\star$}\label{eq:star} \end{gather}
If we want to give a meaning to this problem and find a solution, we take $\xi \in H^1_0(Q)$ such as (using Sobolev extension operator for example if it is possible):
- $\xi=\varphi$ in $A$
- $\|\nabla \xi\|_{L^2(Q)} \leq C \|\nabla \varphi\|_{L^2(A)}$
With this $\xi$, we can denote $v \in H^1_0(Q \setminus A)$ the solution of the following well-posed problem:
$$ -\Delta v = \Delta \xi \quad\text{in } Q \setminus A.\tag{$\star \star$}\label{eq:sstar} $$
Then we say that the function $u:= v + \xi \in H^1_0(Q \setminus A) + \xi$ is a solution of \eqref{eq:star} (extending $v$ to zero in $A$).
Is this the good way to define a weak solution of \eqref{eq:star}? How would you do it ? Also, can we prove something like:
There exists such a $\xi$ if and only if the problem \eqref{eq:star} is well-defined.
I feel like I'm not really understanding the problem \eqref{eq:star}. Any help or advice are welcomed. I'm also wondering if such a function $\xi$ can always be found, depending on the geometry of $A$.