Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Let $u\in C^2(\bar \Omega)$ be a solution of the Dirichlet problem for the non-linear equation \begin{eqnarray} F(x,u,\nabla u,\nabla^2 u)=0 \mbox{ in } \Omega,\\ u=\phi \mbox{ in } \partial \Omega. \end{eqnarray} Let the equation be elliptic with respect to $u$. Let us assume that $F,\phi$ are infinitely smooth.

QUESTION. Is it true that the assumption $u\in C^{2,\alpha}(\bar \Omega)$ for some $0<\alpha<1$ implies that $u\in C^\infty(\bar \Omega)$? A reference would be very helpful.

Remark. So far I was able to find in literature two special cases of this statement.

(1) In the above generality in follows that the solution $u\in C^\infty(\Omega)$ (i.e. smooth in the interior of $\Omega$, not including the boundary). This is Lemma 17.16 in the Gilbarg-Trudinger book.

(2) The question has positive answer (including the boundary) for $F$ of the form $$F(x,u,\nabla u,\nabla^2 u)=G(\nabla^2u)-f(x).$$ This is Prop. 5.1.10 in Qing Han's book.