I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958, https://doi.org/2372841). The author there establishes some a priori bounds on solutions of parabolic and elliptic equations. I would be interested in the following: What was the significance and impact of Nash's results in this paper? In which way was it important for the development of the field? Also I would like to know, why this paper is special, i.e. where there any new methods used to prove a priori bounds? Every answer will be appreciated!
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2$\begingroup$ If my memory serves me, The Nash inequality was developed in this paper. This inequality controls $\|u'\|_{L^2}$ in terms of $\|u\|_{L^2}$ and $\|u\|_{L^1}$ and hence allows the basic energy estimate for the heat equation to close. (Note that we are on the whole line here so there is no Poincare inequality). In this paper I also so for the first time the trick of obtaining $L^\infty$ bounds on Green's functions from $L^2$ bounds on the same by writing the semigroup from 0 to t as the product of semigroups from 0 to $t/2$ and from $t/2$ to $t$. $\endgroup$– Aaron HoffmanFeb 7, 2013 at 18:52
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1$\begingroup$ this is now known as the De Giorgi-Nash theorem, and Moser gave a different proof. If you google De Giorgi-Nash, you'll find lots of stuff about it. I would say that the impact of this theorem has been quite extensive. $\endgroup$– Deane YangFeb 7, 2013 at 20:22
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De Giorgi solved Hilbert's 19th problem.
Nash independently and almost simultaneously obtained the parabolic version of the same result. Nash's result implies that all quasilinear parabolic equations, under some very reasonable assumptions, have smooth solutions.
Both De Giorgi's proof and Nash's proof are very original and develop brand new methods. Pretty much everything in regularity theory for elliptic and parabolic equations that came afterwards was influenced by these two papers.
Both Nash and De Giorgi were under 40 at that time. People usually speculate that neither one got the Fields medal because they cancelled each other out. One could even argue that these papers are the most important result in the history of PDE (or at least in ellitic PDE to avoid too many complaints).
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$\begingroup$ ...possibly along the two fundamental papers of Agmon-Douglis-Nirenberg. $\endgroup$ Feb 8, 2013 at 8:46
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1$\begingroup$ ...possibly along with Leray's fundamental paper :) . $\endgroup$ Feb 27, 2022 at 10:02