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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    $\begingroup$ Have you seen this related Question at MathOverflow? 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. $\endgroup$
    – hardmath
    Jan 26, 2017 at 12:49
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    $\begingroup$ The Wikipedia article on Hilbert's nineteenth problem 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? $\endgroup$
    – hardmath
    Jan 26, 2017 at 12:58
  • $\begingroup$ @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). $\endgroup$
    – Adi
    Jan 27, 2017 at 13:51
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    $\begingroup$ 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: www-users.math.umn.edu/~jwcalder/222BS16/hw3_sol.pdf $\endgroup$
    – Jeff
    Jan 27, 2017 at 16:24
  • $\begingroup$ @Jeff, Your solution is really nice as it distills the main ideas. $\endgroup$
    – Adi
    Jan 30, 2017 at 5:06


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