The laplace of the integral of Green's function Let $G(\mathbf{x}, \mathbf r)$ be the Green’s function of the Dirichlet problem in a bounded normal domain $\Omega$ . Set $$u(\mathbf r) = \int_{\Omega}
G(\mathbf x, \mathbf r) d^3x.$$Prove that $\Delta u(\mathbf r) = −1 $ for $\mathbf r ∈ \Omega$.
I think that I should choose a mall ball, but I don't know how to do it. Please help me.

Note that the Laplacian is with respect to $r$, while the integral is with respect to $x$, so the Laplacian moves straight in:
$$\Delta_r u =  \int_{\Omega} \Delta_rG(x,r)\,d^3x$$
Can you take it from there? What's the definition of the Green's function?
(As a side note, a word of warning: the definition of the Green's function that I assume you must be using is not universally standard; depending on conventions you might get +1 instead of -1.)
