Harmonic approximation on arcs (Q690357)

Let \(\Omega\) be a noncompact Riemannian manifold of dimension at least two. Let \(J:I \to \Omega\) be a continuous proper injection (called Jordan arc here), where \(I\) is an interval. The authors prove: If \(w:J \to \mathbb{R}\) and \(\varepsilon:J \to (0,1]\) are continuous functions, then there exists a harmonic function \(u\) on \(\Omega\) with \(| w-u |<\varepsilon\) on \(J\).