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-Assume $u$ is a harmonic function on $\mathbb{R}^3$, and assume u(x,y,z)=1+x when $x^2+y^2+z^2=1$. What is the value $u(0,0,0)$?

I am in doubt between $1$ and $0$, could it be both?

-Let $V(r)$ be a radial harmonic function in $\mathbb{R}^3$; which is the ODE for V?

My answer: $$V'' + \frac 2 r V' = 0, \quad \text{for} \quad r>0. $$

are they correct?

Cippa Lippa
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1 Answers1

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Your answer to the second part is correct. For the firwt part, the value of a harmonic function at the center of the surface is equal to the average of its value over that surface. By symmetry, that average value is 1, so that's your answer.

Paul
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