A PDE solution

Jan 2015
104
1
usa
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.
 
Last edited by a moderator:
Jan 2015
104
1
usa
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.
 
Last edited by a moderator:
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SDK

Sep 2016
793
540
USA
Ahh so we have graduated from doing your abstract algebra HW to doing your PDE HW. It's nice to be promoted.
 
Last edited by a moderator:
Mar 2015
1,720
126
New Jersey
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.