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(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 $\alpha$.

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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.

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