Consider $f\in C^2(\mathbb{R}^n)$.
Define $$\phi(x,r)=\frac{1}{n\alpha(n)}\int_{\partial B(0,1)}f(x+rz)dS(z)$$ where $\alpha(n)$ is the volume of $B(0,1)$.
I calculated $$\partial_r\phi=\frac{r}{n\alpha(n)}\int_{ B(0,1)}\Delta_xf(x+rz)dS(z)$$
I want to show that $$\partial_{rr}\phi-\frac{n-1}{r}\partial_r\phi=\Delta_x\phi$$
Please help me to do so. Thanks.