Green's function of a bounded domain is strictly negative

Question

I have been struggling with this proof for half a day. I am really baffled. The result of my proof is close but not exactly the same as the proposal.

Here is the question:

Here is my proof:

The proposal is $G<0$, but what my proof arrives at $G\le0$. Where did I go wrong? Thanks so much!

Answer 1

Details may depend on the definition of Green's function you are using, but here are the key points:

1. Fix $y$. The function $h(x) = G(\cdot,y)$ is harmonic in $\Omega\setminus \{y\}$

2. Since $G(x,y)\to-\infty$ as $x\to y$, there is $r>0$ such that $h(x)<0$ when $|x-y|\le r$.

3. Apply the strong form of the maximum principle to $h$ on $\Omega \setminus \{x: |x-y|\le r\}$. Since $h\le 0$ on the boundary, and is not constant (being strictly negative on the circle $|x-y|=r$), it follows that $h<0$ on $\Omega \setminus \{x: |x-y|\le r\}$