Let $B\subset \mathbb{R}^2$ be the unit-circle and suppose that $f:\partial B\to \mathbb{R}$ is a continuous function. Using the Poisson Integral, it can be shown that
$\Delta u=0$ on $B$, $u=f$ on $\partial B$
has a solution which is of class $C^0(\bar{B})\cap C^2(B)$ (cf. Gilbarg Trudinger chapter 2.5). However, the book that I am reading states that the solution of this problem is also of class $W_2^1(B)$. Could someone provide me with a reference for that statement?