Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering the spherical average $M_u(0, r) =\frac{1}{n w_n} \int_{|\xi|=1} u(r \xi) \, dS_\xi$ show that the problem has no solution.
My attempt:
We have $\lim_{r \to 1} M_u(0,r)=0$ since $u=0$ on the boundary $|x|=1$ and $u$ must be continuous on $\Omega$. However, $u(0)=1$ so the Gauss Mean Value Theorem does not hold, and u is not harmonic at the origin.
I am uncomfortable with this argument. The origin is not even in the domain $\Omega$, so why should the mean value theorem hold when $x=0$? Why should we expect $u$ to be harmonic at $x=0$?
I can't think of any other proof for this problem, especially given that we are told to use the average over a sphere...