Let $B$ be the unit ball. $K = \{(0,0,x_3,...,x_n)|x_3^2 + ... +x_n^2 \leq 1/2\}$. Let $u$ be a harmonic function on $B\setminus K$, then $u$ can be extended to a harmonic function in $B$.
My attempt: Let $x' = (x_3,...x_n)$, pick $r$ small such that $B_r(0,x') \subset B$. Define $u(0,x')= \int_{B_r}u(y)dy$. The integral is well defined since $u$ is defined almost everywhere. Therefore, $u$ satisfies the mean value property everywhere, and therefore harmonic.
My questions:
- Is this proof correct?
- This statement generalize to $K$ being a compact codimension 2 manifold. I don't see how codimension 2 is important in this proof, as long as it is measure 0.
- Is there a proof using sub/sup harmonic functions? I can define $u(0,x) = \inf u$, and then the extended $u$ is a subharmonic function. Similary we can define super harmonic function, but it is not clear to me how to go from here.