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 October 9th, 2018, 08:04 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 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.
October 9th, 2018, 09:20 AM   #2
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This is the right version

Quote:
 Originally Posted by 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 $$\partial_r\phi=\frac{r}{n\alpha(n)} \int_{B(0,1)}\Delta_xf(x+rz)dS(z)$$ 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 12:53 PM.

 October 9th, 2018, 10:18 AM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics 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.
November 2nd, 2018, 10:11 AM   #4
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Quote:
 Originally Posted by SDK 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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