Moser regularity proof avoiding John-Nirenberg lemma I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely on the John-Nirenberg lemma. 
I was wondering if somebody can point out a reference for that proof, or a reason why the John-Nirenberg lemma cannot be truly avoided (or both!).
 A: Check this paper
Moser. On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math (1971) vol. 24 (5) pp. 727-740
The purpose of the above paper, is to avoid the use of the parabolic John-Nirenberg lemma.
A: Try this old book by Guido Stampacchia Équations elliptiques du second ordre à coefficients discontinus, Presses de l'Université de Montréal 1966.
If  you cannot find this try this clasic  by Olga Ladyzheskaya, Linear and quasilinear elliptic equations,  New York, Academic Press, 1968 
A: For the homogeneous equation, I have seen a proof of $C^{\alpha}$ regularity using an oscillation estimate based only on local boundedness and a Poincare-Sobolev inequality. Specifically:
Let u be a subsolution in $B_2$ satisfying $|(u \leq 0) \cap B_1| \geq \frac{1}{2}|B_1|$. Then $\sup_{B_{1/2}}u^{+} \leq \gamma \sup_{B_1}u^{+}$, where $\gamma < 1$ depends only on the ellipticity constants and $n$. (|.| denotes Lebesgue measure).
From there, one concludes that the oscillation of a solution decays by a fixed proportion each time we localize, which gives Holder regularity.
