How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

Question

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results:

Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in $\mathbb{R}^n$. Suppose that $u \in W^{1,2}(B_1)$, and for any $B_{\rho}(x_0) \subset B_1$, $$\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx \le C\rho^{\alpha},$$then $u \in C^{\alpha}(B_1)$.

He said that this result could be found in the Gilbarg and Trudinger in Chapter 7, but I didn't find it.

Can anyone show me how to prove this or give me a reference to read?

Thanks in advance!

Answer 1

It is something rather classical. You will find all the argument looking at the proof of the Sobolev embedding $W^{1,p}$ into $C^{0,\alpha}$, in the book "Sobolev spaces" of Adams.