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