# Showing harmonicity

1. Aug 30, 2008

### e12514

1. The problem statement, all variables and given/known data

Suppose u is harmonic on B\{0}, and that u is continuous in the closure of B.
How would we show that u is in fact harmonic on B?
(B = unit ball in R^n, n>=2)

2. Relevant equations

u is harmonic in Ω if Δu=0 in Ω.

u is weakly harmonic in Ω if integral over Ω of u(x)(Δψ(x))dx = 0.

u has the mean value property on Ω if for each x in Ω and r>0 such that the closed ball with centre x and radius r is contained in Ω, we have
u(x) = 1/ |B(x,r)| . integral over B(x,r) of u(y)dy

3. The attempt at a solution

One idea I had was showing u is weakly harmonic on B, and then correcting it at {0} to become harmonic.
So u is harmonic on B\{0} => u is weakly harmonic on B\{0}
=> integral over B\{0} of u(x)(Δψ(x))dx = 0 for all C^oo functions ψ supported in B\{0}
Now because {0} has measure 0, then
integral over B of u(x)(Δψ(x))dx = 0 = integral over B\{0} of u(x)(Δψ(x))dx
So u is weakly harmonic on B, and so we can correct it at {0} to make it harmonic.
But then something tells me that's too straightforward, can't be right, can it?

So then I thought an alternative (and probably safer) way is
to show u satisfies the mean value property on B, hence harmonic since it is continuous.
But then I have no idea how to show this...
Does that involve using that harmonic functions are smooth, or just some tricks at {0}, or...?