Dirichlet-type condition on Riemannian manifold

Question

Let $M$ be a Riemannian manifold and $S \subset M$ a compact submanifold of strictly lower dimension. Does every smooth function on $S$ extend to a harmonic function on a neighborhood of $S$?

Answer 1

This is not possible in general, for example because of constraints related to the analyticity of the extension.

For a concrete example, let $M = \mathbf{R}^2$, and $S \subset \mathbf{R}^2$ be a smooth, simple closed curve in the plane so that in the unit disc \begin{equation} S \cap D_1 = \{ x_2 = 0 \} \cap D_1 = (-1,1) \times \{ 0 \}. \end{equation}

On $S$ define a function $u_0$ so that on the portion lying inside $D_1$, \begin{equation} u_0: x_1 \in D_1 \cap S \mapsto \mathrm{e}^{-1/x_1^2}. \end{equation} This is not analytic, whereas a harmonic extension to a neighbourhood of $S$, say $u$ would be.

The existence of such a harmonic extension $u$ is therefore absurd. In conclusion: no matter how small $\delta > 0$ is chosen, there is no harmonic function $u: D_\delta \to \mathbf{R}$ extending $u_0$, let alone a function defined on a neighbourhood of $S$.