Hi!
Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball
$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.
Let $f\in C^{0,\alpha}(B_1)$ and suppose i have a function $u\in C^{4,\gamma}(B_{1})$ with $\gamma<\alpha$ s.t.
$$Lu+ N(\nabla^{2} u,\nabla^{3}u,\nabla^{4}u)=f$$
Moreover i know that
$$ \left\|N(\nabla^{2} u,\nabla^{3}u,\nabla^{4}u)\right\|_{C^{0,\gamma}} <\left\|Lu\right\|_{C^{0,\gamma}}$$
does it follow that $u\in C^{4,\alpha}(B_{1/2})$?
