Elliptic regularity of Dirichlet problem

Question

Suppose $\Omega\subset\mathbb{C}$ is a simply connected domain with $C^\infty$ boundary. Consider the following Dirichlet problem
$$\Delta u |_{\Omega}=0 $$ $$u|_{\partial\Omega}=f$$ Under what condition on $f$ or perhaps $\partial\Omega$ we have local regularity up to the boundary?Also what are norm estimates for derivatives of $u$? Can you refer me to a right place to study the regularity matter of PDEs?