This is is a homework problem I'm having trouble understanding. I am given the set $$Y=\{u\in \mathbb{C}^{\infty}(\bar{D_1})| \triangle u=0\}$$ and its closure with respect to the $L^2$ norm, $\bar{Y}$. I am asked to show that each $v\in \bar{Y}$ has a representative that is smooth on the interior of $D_1$ and that $\triangle v=0$ on the interior of $D_1$. Further, I am asked to describe the radial functions (functions of $r=\sqrt{x^2+y^2}$) orthogonal to $\bar{Y}$.
I have been advised to use the Poison kernel on the disk, although it's been a couple years since I've worked with that formula. $\bar{Y}$ is $Y$ together with its limit points in the $L^2$ norm, so it seems as if I should just consider $v$ that are limit points of sequences $\{u_n\}$ in $Y$. Also, from what I remember of measure theory, representatives of $v$ differ from it on a set of measure zero. At any rate, since this is a homework problem, could someone point me in the right direction?