Are harmonic functions always real analytic?

Question

Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb{D}^n$. Let $f$ be a harmonic function w.r.t $g$.

Is it true that $f$ must be real-analytic?

I think that this is true if we assume that $g$ is real-analytic, but I am not sure. Is it true in that case? I would like to find a reference.

This should be related to whether or not the Riemannian laplacian $\Delta_g$ is "analytically hypoelliptic".

Answer 1

Not if $g$ is just smooth: take any nonconstant harmonic function $f_0$ on $\mathbb{D}^n$ (in the usual sense, i.e. with respect to the Euclidean metric) and let $T:\mathbb{D}^n\to\mathbb{D}^n$ be a diffeomorphism such that $f=f_0\circ T$ is not real-analytic. Transporting the Euclidean metric along $T$, we get a smooth metric $g$ such that $f$ is harmonic with respect to $g$.

I don't know anything about what you can say if $g$ must be real-analytic.

Answer 2

When $g$ is real-analytic, (local) analyticity of harmonic functions is just a special case of Cauchy–Kovalevskaya theorem.

Edit: You are right, CKT does not suffice. In case if you still care, here are references that do the job:

In your setting, the Laplacian is uniformly (or strongly) elliptic. Now, apply

C. Morrey, On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior. Amer. J. Math. 80 (1958), 198–218.

(for the interior, a textbook reference is Hormander's book on Linear PDEs, Theorem 7.5.1)

and then at the boundary:

C. Morrey, On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. II. Analyticity at the boundary. Amer. J. Math. 80 (1958), 219–237.

I am not sure if the boundary analyticity has a textbook treatment. The standard references that I have (Hormander, Gilbarg-Trudinger, Evans, Taylor) don't do it.