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Set \begin{equation*} A:= - \Delta: \mathcal{D}(A) = H^2(\Omega) \bigcap H^1_0(\Omega) \subseteq L^2(\Omega) \to L^2(\Omega), \ \Omega \subseteq \mathbb{R}^2 \end{equation*} Its resolvent \begin{equation*} R_{\lambda}(A) := (\lambda I - A )^{-1}, \ \lambda \in \mathbb{C}\setminus \mathbb{R} . \end{equation*} We want to figure out the expression of \begin{equation*} R_{\lambda}(A) f := (\lambda I - A )^{-1} f, \ \textrm{where} \ f \in L^2(\Omega). \end{equation*} for some concrete choice of domain $ \Omega $, such as $ \Omega = B(0,1) $ or $ [0,1]^2$.

In order to acheive this goal, we utilize Green function method and rewrite the question in form of equation as follows.

Question: Find \begin{equation*} u = \int_\Omega G(x,y) f(y)dy \end{equation*} such that \begin{equation*} \Delta u + \lambda u = f, \ x \in \Omega \end{equation*} \begin{equation*} u|_{\partial \Omega} = 0 \end{equation*} where $\lambda \in \mathbb{C}\setminus \mathbb{R} $ and $ \Omega = B(0,1) $ or $ [0,1]^2$.

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