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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!).

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3 Answers

up vote 6 down vote accepted

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.

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hmm ok, I will check it. Is it the same as the elliptic case? –  Mircea Mar 22 '12 at 12:06
    
The parabolic implies the elliptic –  Bynne Mar 22 '12 at 12:38
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well, that settles the question! Actually the elliptic case was done by Bombieri-Giusti, and it's what the above article refers to (the reference is "harnack inequality for ellpitic equaitons on minimal submanifolds"). Since both articles are short, it's a quite nice passtime reading them! Thanks! –  Mircea Mar 22 '12 at 18:01
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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

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ah is it proved without john-nirenberg in Ladyzhenskaya? I'll look in the book tomorrow. –  Mircea Mar 22 '12 at 18:16
    
actually that would be strange since the article of moser cited above came after the book.. I immagine that the book would prove it DeGiorgi-way, since it fits better to good old russian style. but I'm curious, I'll let you know what I find. –  Mircea Mar 22 '12 at 18:24
    
How about this paper by Stampacchia archive.numdam.org/ARCHIVE/SJL/SJL_1963-1964___3/… –  Liviu Nicolaescu Mar 22 '12 at 21:41
    
they use John-Nirenberg (theorem 8.2) –  Mircea Mar 24 '12 at 5:57
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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.

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Thank you, this is a nice proof indeed, it is due to P. Tilli, I think ("Remarks on the Hölder continuity of solutions to elliptic equations in divergence form",Calculus of Variations and Partial Differential Equations, Vol. 25, Number 3, 395-401, DOI: 10.1007/s00526-005-0348-3) –  Mircea Mar 25 '12 at 18:25
    
..was that what you were referring to? –  Mircea Mar 25 '12 at 18:27
    
I hadn't seen the paper before, but that's exactly the proof I was thinking of. Thanks for the reference! –  Connor Mooney Mar 26 '12 at 15:20
    
The new point about this paper Mircea seems to be that they avoid the iteration altogether, by basically 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. –  Spencer May 13 '12 at 23:31
    
To clarify, the rest of Tilli's paper - i.e. the non-so-new stuff - looks, at a glance, to be the same as the old argument that's in Han and Lin. So it seems a perfectly good source. –  Spencer May 13 '12 at 23:34
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