# Resolvent and Green Function?

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 $$[0,1]^2$$.