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October 9th, 2018, 07:04 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 0  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 $$\partial_{rr}\phi\frac{n1}{r}\partial_r\phi=\Delta_x\phi$$ Thanks. Last edited by skipjack; October 9th, 2018 at 11:52 AM. 
October 9th, 2018, 08:20 AM  #2  
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 0 
This is the right version Quote:
Last edited by skipjack; October 9th, 2018 at 11:53 AM.  
October 9th, 2018, 09:18 AM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 473 Thanks: 262 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 11:53 AM. 

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