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!).
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.
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
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.
differentiating' the quantity which is usually treated discretely and iterated. A proof along the lines which Connor describes in his initial answer, i.e. for pure divergence form equations, using iteration but not John-Nirenberg existed right at the start' as it were - it is due essentially to De Giorgi himself. You can find it the book of Han and Lin, Elliptic PDE. The John-Nirenberg Lemma was used by Moser to prove his Harnack inequality, which itself was not crucial for regularity.
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