Given a bounded piecewise continuous function \(u:S^1\to {\mathbf{R}}\), is there a unique extension to a continuous harmonic function \(\tilde u: {\mathbb{D}}\to {\mathbf{R}}\)?
More generally, this is a boundary value problem for a region where the values of the function on the boundary are given. Compare to prescribing conditions on the normal vector on the boundary, which would be a Neumann BVP. Why these show up: a harmonic function on a simply connected region has a harmonic conjugate, and solutions of BVPs are always analytic functions with harmonic real/imaginary parts.
See section 27, example 1 in Brown and Churchill. On the strip \((x, y)\in (0, \pi) \times(0, \infty)\), set up the BVP for temperature on a thin plate with no sinks/sources: \begin{align*} \Delta T = 0 && T(0, y) = 0,\, T(\pi, y) = 0 \,\,\forall y \\ \\ T(x, 0) = \sin(x) && T(x, y) \overset{y\to\infty}\longrightarrow 0 .\end{align*}
Then the following function is harmonic on \({\mathbf{R}}^2\) and satisfies that Dirichlet problem: \begin{align*} T(x ,y) = e^{-y} \sin(x) = \Re(-ie^{iz}) = \Im(e^{iz}) .\end{align*}