Let \(u\colon S^1\to\RR\) be bounded and piecewise continuous. The Dirichlet problem on \(\DD\) with boundary data \(u\) asks for a bounded harmonic function \(\tilde u\colon\DD\to\RR\) such that \[ \lim_{\substack{z\to\zeta\\ z\in\DD}}\tilde u(z)=u(\zeta) \] at every point \(\zeta\in S^1\) where \(u\) is continuous. A solution of the Dirichlet problem is such a function \(\tilde u\).
When \(u\) is continuous on \(S^1\), the boundary condition says that \(\tilde u\) extends to a continuous function on \(\overline{\DD}\) that equals \(u\) on \(S^1\).