I have questions about the proof of Theorem $14$ (poisson's formula for half-space) in Page $38$.
we easily verify as well $u \in C^\infty (\mathbb{R_+^n})$, with $$\Delta u(x) = \int_{\partial\mathbb{R_+^n}}\Delta_x K(x,y) g(y) dy = 0$$
I look up the post. I understand why $\Delta u(x) = \int_{\partial\mathbb{R_+^n}}\Delta_x K(x,y) g(y) dy$. The only thing confuses me is that $ \int_{\partial\mathbb{R_+^n}}\Delta_x K(x,y) g(y) dy = 0.$
So how can I prove that $ \int_{\partial\mathbb{R_+^n}}\Delta_x K(x,y) g(y) dy = 0$ then.
My question is why $ \int_{\partial\mathbb{R_+^n}}\Delta_x K(x,y) g(y) dy = 0$.