(Edited)


**I need a reference to the following result:**
-----------------------------------------------


If $u \in H^2(B\_1^+) \cap {\rm Lip}(B\_1^+)$ satisfies
\begin{cases}
 {\rm div}(F(x,u,\nabla u)) = F\_0(x,u,\nabla u) \quad & {\rm in} \ B\_1^+ \\
 u = 0 & {\rm on} \ B\_1'
\end{cases}


where


$$F \in C^{1,\beta}(B\_1^+\times\mathbb{R}\times\mathbb{R}^{n+1};\mathbb{R}^{n+1}), \quad F\_0 \in C^{0,\beta}(B\_1^+\times\mathbb{R}\times\mathbb{R}^{n+1};\mathbb{R})$$


for some $0<\beta<1$, and


$$\langle D\_p F(x,u,p) \xi,\xi \rangle \ge \lambda(M) |\xi|^2$$


for some $0 < \lambda(M) < + \infty$, for every $x \in \overline{B\_1^+}$, $u \in \mathbb{R}$, and $|p| \le M$,


then $u \in C^{2,\alpha}(\overline{B\_{1/2}^+})$ for some $0<\alpha<1$.


**Notations:**
--------------


$$B\_1^+ = \{x = (x',x\_{n+1}) \in \mathbb{R}^{n+1} : |x| < 1, \, \, x\_{n+1} > 0\}$$
is the half-ball and
$$B\_1' = \{x = (x',0) \in \mathbb{R}^{n+1} : |x'| < 1\}$$
is the flat part of its boundary.   
 Also, we have $n \ge 1$.   
 $H^2$ denotes the Sobolev Space of functions with second order weak derivatives in $L^2$ and ${\rm Lip}$ is the space of Lipschitz-continuous funcions, whilst $C^{k,\alpha}$ is the space of functions whose $k$-th order classical derivatives are Hölder-continuous of exponent $\alpha$.



Reference to a Classical Regularity Theorem
The discussion from Section 13.1 in the book of Gilbarg and Trudinger shows that $u \in C^{1,\,\alpha}\left(B\_{3/4}^+\right)$. From here one can apply Schauder estimates for linear equations. For example, one can pass the divergence on the left hand side and view $u$ as a solution to a non-divergence form linear equation with Hölder continuous coefficients (namely $F^i\_j(\nabla u)$, in the case that $F$ depends only on $\nabla u$). For the relevant linear theory, see e.g. Section 5.5 from the book of Giaquinta and Martinazzi [here](https://link.springer.com/book/10.1007%2F978-88-7642-443-4).


