# Integral representation of directional derivative

We consider the Dirichlet problem $$\begin{array}{c} \Delta\psi(x) + v(x)\psi(x) = 0, \;\;\; x \in D \\ \psi|_{\partial{D}} = f \end{array}$$ in some bounded region $D$ with smooth boundary $\partial D$. The operator $\Phi$ is defined by formula $$\Phi[f](x) = \frac{\partial \psi(x)}{\partial \nu}, \;\;\; x \in \partial D$$ where $\nu$ is an exterior normal to $D$ in $x$. I try to show constructively that $\Phi$ is an integral operator with Shwartz's kernel $\Phi(x,y), \; x,y \in \partial D$ and $$\Phi(x,y) = \frac{\partial G(x,y)}{\partial \nu_x \partial \nu_y} ,\;\;\; x,y \in \partial D$$ where $G(x,y)$ is a Green function for considered PDE: $$\begin{array}{c} (\Delta_x - v)G(x,y) = \delta(x-y), \;\;\; x,y \in D\\ G(x,y) = 0, \;\;\; x \in \partial D, \;\;\; y \in D \end{array}$$ What should I do to receive this representation? Maybe I can find the demonstration in some book?

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