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October 9th, 2018, 08:04 AM   #1
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A PDE solution

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 skipjack; October 9th, 2018 at 12:52 PM.
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October 9th, 2018, 09:20 AM   #2
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This is the right version

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Originally Posted by mona123 View Post
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.
Thanks from topsquark

Last edited by skipjack; October 9th, 2018 at 12:53 PM.
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October 9th, 2018, 10:18 AM   #3
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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 skipjack; October 9th, 2018 at 12:53 PM.
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November 2nd, 2018, 10:11 AM   #4
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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.
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