Let $f\in C^2(\mathbb{R}^n)$.

We 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 $$\partial_r\phi=\frac{r}{n\alpha(n)} \int_{\partial B(0,1)}\Delta_xf(x+rz)dS(z)$$

Please help me to show that $$\partial_{rr}\phi-\frac{n-1}{r}\partial_r\phi=\Delta_x\phi$$

Thanks.

We 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 $$\partial_r\phi=\frac{r}{n\alpha(n)} \int_{\partial B(0,1)}\Delta_xf(x+rz)dS(z)$$

Please help me to show that $$\partial_{rr}\phi-\frac{n-1}{r}\partial_r\phi=\Delta_x\phi$$

Thanks.

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