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 $\partial\Omega.$)
Best Answer
Hint: show that $\int_{\partial \Omega}{K(y,x)g(y)\mathrm{d}y}\geq 0$ for any nonnegative function $g(y)$ on $\partial \Omega$.