Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map $f:U\to\mathbb R$ that is weakly harmonic?
Weakly harmonic means that $$ \int_U \left\langle \nabla_g f,\nabla_g\varphi \right\rangle\mathrm d~\mathrm{vol}_g $$ vanishes for all test function $\varphi\in C^\infty_c(U)$. This requires $f$ to be in $H^1$ a priori. Can it be shown that $f\in C^{k+1}$? Many thanks!
I don't think you can get that $f \in C^{k + 1}$, the best you can hope for is $f \in C^{k,\alpha}$ for $0 < \alpha < 1$. If you write the Laplace-Beltrami operator in local coordinates, you see it depends on the derivative of the metric and so it is a second order linear elliptic operator with $C^{k-1}$ coefficients and in particular it has $C^{k-2,\alpha}_{\mathrm{loc}}$ coefficients for any $0 < \alpha < 1$. By usual regularity results for such operators (you can unravel the references of Theorem 1.4.3 of Joyce's Compact Manifolds with Special Holonomy or go directly to Gilbarg & Trudinger), you have $f \in C^{k,\alpha}_{\mathrm{loc}}$. Alternatively, Sobolev estimates give you that $f \in H^{k+1}_{\mathrm{loc}}$ (see Theorem 8.10 of Gilbarg & Trudinger).
If however you know that $g \in C^{k,\alpha}_{\mathrm{loc}}$ for $0 < \alpha < 1$, you can have $f\ \in C^{k+1,\alpha}_{\mathrm{loc}}$.