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). 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!
De Giorgi solved Hilbert's 19th problem (http://en.wikipedia.org/wiki/Hilbert's_nineteenth_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).