# A PDE solution

#### mona123

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.

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#### mona123

This is the right version

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

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

Thanks.

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1 person

#### SDK

Ahh so we have graduated from doing your abstract algebra HW to doing your PDE HW. It's nice to be promoted.

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#### zylo

Ahh so we have graduated from doing your abstract algebra HW to doing your PDE HW. It's nice to be promoted.
Of course people are going to ask HW questions. If you don't know the answer don't reply.