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Before the celebrated work of Nash and DiGiorgi, there were partial proofs of the regularity results known before, especially in 2-D.

I am looking for 2-D results of the regularity which predated Nash and DiGiorgi. Could someone let me know where these were proved?

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    Have you seen [this related Question at MathOverflow](http://mathoverflow.net/questions/203305/nashs-proof-of-de-giorgi-nash-moser-theorem)? It links to a pay-walled paper "A new proof of Moser's parabolic harnack inequality using the old ideas of Nash". Its references might give earlier references. – 2017-01-26
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    The Wikipedia article on [Hilbert's nineteenth problem](https://en.wikipedia.org/wiki/Hilbert's_nineteenth_problem#The_path_to_the_complete_solution) makes mention of a 2D result in the thesis of Sergei Bernstein (1904) and subsequent improvements to it. Is that the sort of thing you are looking for? – 2017-01-26
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    @hardmath, obtaining harnack inequality using Nash's idea is a beautiful paper which I have been aware of. Your other reference regarding the thesis of Bernstein is the sort of reference I am looking for. I am curious if there are any different ideas that existed in proving the digiorgi Nash moser theorem in 2D (something that is independent of the ideas of digiorgi Nash or moser). – 2017-01-27
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    The 2D case is much easier, and the ideas are sort of well-known in the PDE community, but I am not sure of the original work. I gave this as a homework assignment in a course I taught: http://www-users.math.umn.edu/~jwcalder/222BS16/hw3_sol.pdf – 2017-01-27
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    @Jeff, Your solution is really nice as it distills the main ideas. – 2017-01-30
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    Thanks. I should say I got the main ideas from a minicourse given by L.C. Evans. The hole-filling trick, in particular, is very nice. It is interesting to examine where the argument would fail for $n\geq 3$. – 2017-01-30
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    I believe the reason $n=2$ works in your proof is essentially because in (5) of your solution, the term $r^2$ in $r^{\lambda+2-n}$ comes because of the ellipticity condition is a quadratic form. This is where the $n=2$ makes life easy. Here is an alternate proof to see this more clearly, using the notion of weakly monotone function "http://link.springer.com/article/10.1007/BF02921588", continuity of the solution is trivial. To show H\"older continuity, all one needs to prove is a higher integrability result which is called Meyer's theorem (I seem to not remember the right reference). – 2017-01-31

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