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Let $G(y,x)$ be the Green's function for the Dirichlet problem for Laplace equation on domain $\Omega$ with smooth boundary. Show that $$K(y,x):=\frac{\partial}{\partial \textbf{n}_y}G(y,x)\geq 0, \forall y \in \partial \Omega, x \in \Omega$$ In which $\textbf{n}_y $ is the outer unit normal.

Recall that Green's function for Laplace equation with Dirichlet boundary condition saisfies $$\begin{cases}\Delta_y G(y,x)=\delta (y-x), &\forall y \in \Omega\\ G(y,x)=0, &\forall y\in \partial \Omega\end{cases}$$ Remark: I know $\frac{\partial}{\partial \textbf{n}_y}G(y,x)$ is harmonic in $x$ over $\Omega$ (for each fixed $y\neq x $) and $\int_{\partial \Omega}\frac{\partial}{\partial \textbf{n}_y}G(y,x)dS_y=1.$ But they seem not to be directly related to $K(y,x)\geq0.$ (I want to apply maximum principle to $K$, but there is little information about $K$ on $\partial\Omega.$)

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  • $\begingroup$ What is $K(.,.)$? $\endgroup$ – mvw Nov 15 '14 at 20:51
  • $\begingroup$ Check Evans, "Partial differential equations", Chapter II, the part on Green functions and Poisson kernels. There you'll find all about this. $\endgroup$ – Qwertuy Nov 15 '14 at 21:03
  • $\begingroup$ I checked it, but it only inroduced Poisson kernel for the upper half plane and the ball. In my problem the $\Omega$ is an arbitrary region. $\endgroup$ – user31899 Nov 16 '14 at 0:11
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Hint: show that $\int_{\partial \Omega}{K(y,x)g(y)\mathrm{d}y}\geq 0$ for any nonnegative function $g(y)$ on $\partial \Omega$.

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