I am reading the book *"Regularity theory for elliptic PDE"* by Xavier Fernández-Real
and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $u = g$ on $\Omega$.

If $\Omega$ is a $C^{1,\alpha}$ domain then solutions are $C^{1,\alpha}(\overline{\Omega})$

They cited two sources

- C. Kenig,
*Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems*, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994 - N. V. Krylov,
*Lectures on Elliptic and Parabolic Equations in Hölder spaces*, American Mathematical Society, 1996.

However I could not find the exact result in either of these books. Can anyone provide me a reference? Or roughly the idea on how to prove this result (at least for $f = 0$)?

Thank you very much.