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The names of Nash, Moser and De Giorgi are associated to elliptic and parabolic regularity theory.

But what are the differences in the approach between the three contributions? Can you briefly sketch the main points of the proof given by each of these authors?

In https://web.ma.utexas.edu/mediawiki/index.php/De_Giorgi-Nash-Moser_theorem? some idea is given without really distinguishing between the three.

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Here's a slightly more detailed explanation than what Willie Wong fit in the comments.

I will not go over the precise statements in detail, but at least some discussion is needed to appreciate the differences. Below I will have $u$ always be a (weak) solution of $\partial_i (a_{ij} \partial_j u) = 0$ on $B_1$, Here $a_{ij}$ is assumed symmetric and uniformly positive definite ($\lambda |\xi|^2 \leq a_{ij}\xi_i \xi_j \leq \Lambda |\xi|^2$ for some $\lambda, \Lambda$).

By the Holder estimate I refer to things along the lines of, there exist $\alpha, C$ depending on $n, \lambda, \Lambda$ (all constants below only depend on these unless noted) such that $$ \|u\|_{C^{0, \alpha}(B_{1/2})} \leq C \|u\|_{L^2(B_1)}. $$ By the Harnack inequality, I mean that for $u > 0$, $$ \sup_{B_{1/2}} u \leq C \inf_{B_{1/2}} u. $$ By the local maximum principle, I mean $$ \sup_{B_{1/2}} u \leq C \|u\|_{L^2(B_1)}, $$ possibly also for nonnegative subsolutions $u$.

De Giorgi

De Giorgi proved the Holder estimate, arguing in two steps. The first step actually proves the local maximum principle, via a nonlinear iteration argument playing off local energy estimates (basic $L^2 \rightarrow H^1$ estimates obtained by multiplying by $u$ and integrating by parts, which were well-known) against Sobolev embeddings. Actually, if you read the original proof you'll see him use weaker, more geometric isoperimetric-like inequalities instead of Sobolev embeddings, but that's the basic idea. This part was also new, and the local maximum principle now endures as an estimate interesting in its own right.

The second step was another iteration argument that shows that if $|u|\leq 1$, and $\{u > 0\}$ is large enough (has large measure), then $u \leq 1 - \delta$ on $B_{1/2}$ for some $\delta > 0$. This can then be iterated to show Holder continuity. This second step was extremely influential as an approach to showing Holder continuity in general, in completely different contexts.

I have no idea to what extent De Giorgi was motivated by Hilbert's program (to be clear, his estimate does solve the relevant Hilbert problem), but it is very clear he was specifically interested in applications to minimal surfaces. In work following this he uses the theorem (but more importantly, aspects of the proof) to show that hypersurfaces locally minimizing perimeter are smooth except on some lower-dimensional set.

It is worth noting that De Giorgi does not prove the Harnack inequality. His method can be used to prove it, but it's not completely trivial to adapt (maybe with modern knowledge, it could be considered a difficult exercise for graduate students).

Nash

Nash proves the Holder estimate. He worked directly with the parabolic problem, which is strictly more general than the elliptic one, phrasing the theorem as an estimate on the heat kernel. His proof also proceeds in steps, with one of them involving a nonlinear iteration. They are not directly comparable to De Giorgi's, as they are performed globally using Gaussian-like weights and entropy ($u\log u$) quantities natural to the parabolic setting's scaling.

Nash's proof is very original, and has been somewhat influential (though perhaps less so than the other two). The approach of working directly with the global heat kernel has not exactly caught on--it seems that local estimates are just easier most of the time--but has been revisited many times. However, heat kernel or Green's function estimates are now recognized as being of central importance, and arguably Nash's proof really highlights this point.

Nash also uses some weighted Sobolev/log-Sobolev embeddings and other parabolic techniques that have caught on, even if they are not always correctly attributed to him.

Nash does not prove the Harnack inequality. It is not that hard to recover it from his proof, however, along with essentially optimal information about the heat kernel.

Moser

Moser wrote several papers on this topic, all at least a couple of years after De Giorgi and Nash.

In his first paper, Moser proves the Holder estimate. Like De Giorgi, he proceeds in two steps, with the first being the local maximum principle. This part uses an iteration similar to De Giorgi's, using powers of $u$ instead of truncations to express the argument in terms of estimates in $L^p$. The second step is not analogous to De Giorgi's at all, instead using the PDE for $\log u$ and some tricks to close the argument.

In his second paper, Moser proves the Harnack inequality (he was the first to do so). The proof makes use of the local maximum principle, but then proceeds differently. First, using the PDE for $1/u$ and $u^\alpha$ one can show that for any $\epsilon > 0$, $$ \sup_{B_{1/2}} u \leq C \|u\|_{L^\epsilon(B_1)}, $$ an improved local maximum principle, and $$ \|u\|_{L^{-\epsilon}(B_{1})} \leq C \inf_{B_{1/2}} u, $$ basically this applied to $1/u$. This gives a proof of the Harnack inequality if one can show that $$ \|u\|_{L^{\epsilon}(B_{1/2})}\leq C\|u\|_{L^{-\epsilon}(B_{1/2})}, $$ something which Moser manages to do using an improvement of his argument with $\log u$ from the previous paper and an inequality of John and Nirenburg about the space $BMO$.

Moser then published an extension of this argument to parabolic equations.

This proof has also been extremely influential. Partly this is due to the fact that Moser's papers present a variety of ideas and methods for working with elliptic PDE in general. The use of the fact that convex functions of $u$ are subsolutions, the use of the PDE for other functions of $u$, the approach to $L^\infty$ estimates by iterating $L^p$ estimates, and the argument that the Harnack inequality implies the Holder estimate, are all ideas with a rich afterlife. The $BMO$ argument was extremely original and motivated developments in harmonic analysis involving spaces of weights, reverse Holder inequalities, and so on.

TL;DR

Three proofs were given for the Holder estimate in close succession. While their arguments for the local maximum principle are somewhat similar, the way they obtain the actual Holder estimate from that are quite different and individually influential.

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    $\begingroup$ Very nice explanations $\endgroup$
    – Deane Yang
    Jun 5, 2023 at 12:37
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    $\begingroup$ One thing that I thought was the case but couldn't check since I don't read Italian: De Giorgi's original energy estimates (leading to the local max principle) seems to rely on the fact that he is working with an energy minimizer; I am not sure how significant that reliance is, but Moser's method does not depend on that. My impression was that this is one of the reason popular textbooks (e.g. Gilbarg and Trudinger) usually include the Moser iteration method proof for the local max principle step. $\endgroup$ Jun 5, 2023 at 14:00
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    $\begingroup$ @WillieWong, De Giorgi's level set energy estimates can be obtained by using variants of $\max(u, 0)$ as test functions, and this is true even in contexts where you do not have minimality available. This is in fact quite similar to Moser, who essentially uses powers of $u$ as test functions. I think Moser's method is more common in books partly because Moser's original papers are very well-written and sell the argument well, partly because it avoids Sobolev space technicalities ($t^p$ is smooth, $t_+$ is Lipschitz), partly because it generalizes to other problems the authors like. $\endgroup$
    – user378654
    Jun 5, 2023 at 20:14

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