It is know that the growth condition $$\sup_{x \in \Omega }\int_{B_r(x)\cap \Omega} |\nabla u|^2 \leq C r^{n-2+2\alpha} \text{ for all } r>0$$ would imply that $u \in C^{0, \alpha}(\Omega)$.
I have read the book Multiple Integrals in the Calculus of Variations by Morrey. Since the book are old and hard to read, I can not find a theorem about that. May I have a reference about that?
Actually, I want to know when can I obtain the boundary regularity that $u \in C^{0, \alpha}(\overline{\Omega})$. So I want to take a look of the proof.
I want to know that would $\sup_{x \in \Omega}|\nabla u|$ implies $u \in C^{0, 1}(\overline{\Omega})$. May I have the reference about that also? Thank you!