Dirichlet problem on the unit disc (Tag D-XR64P)
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 \(S^1\).
In a Dirichlet problem the values of the function on the boundary are prescribed; in a Neumann problem its normal derivative on the boundary is prescribed. On a simply connected domain every real harmonic function is the real part of a holomorphic function, so solutions can be sought among the functions \(\Re g\) and \(\Im g\) with \(g\) holomorphic.
(Dirichlet problem on a half-strip)
On the half-strip \(\ts{(x, y) \st 0 < x < \pi,\ y > 0}\), consider the problem \[ \begin{aligned} \laplacian T &= 0, & T(0, y) &= T(\pi, y) = 0 \text{ for all } y > 0, \\ T(x, 0) &= \sin(x), & T(x, y) &\to 0 \text{ as } y\to\infty. \end{aligned} \] The function \[ T(x ,y) = e^{-y} \sin(x) = \Im\qty{e^{iz}} = \Re\qty{-ie^{iz}}, \qquad z = x+iy, \] is harmonic on \(\RR^2\) as the imaginary part of the entire function \(e^{iz}\), and it satisfies each boundary condition.