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I need a book covering $L^p$ theory (is it?) on PDE. Stuff should include: De Giorgi-Nash-Moser’s iteration, Harnack inequalities and Schauder estimates on elliptic/parabolic homogeneous/heterogeneous equations, together with their divergence forms.

I've found Jürgen Jost's Partial Differential Equations, whose second half provides more or less I need. Can you recommend some other books providing full details on those topics for me? Thank you~

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    $\begingroup$ Ah, 18-155 and 18-156 on OCW@MIT seem good supplements. $\endgroup$ – ziyuang Jun 13 '11 at 18:06
  • $\begingroup$ Haim Brezis' Functional Analysis has some facts about $L^p$ spaces, although maybe not really what you're looking for. $\endgroup$ – Beni Bogosel Jun 13 '11 at 19:08
  • $\begingroup$ $L^p$ theory of PDE typically cope with the properties of the solution in $L^p$ space (and Sobolev, Hölder...) of some PDEs, rather than what we learn at undergraduate courses, where solutions are smooth. $\endgroup$ – ziyuang Jun 13 '11 at 19:29
  • $\begingroup$ @ziyuang: The first link in your first comment does not work. $\endgroup$ – Jack Jul 19 '11 at 18:21
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    $\begingroup$ @Jack: Oops... 18-155 $\endgroup$ – ziyuang Jul 20 '11 at 1:51
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A reference that comes close is also

  • Michael E. Taylor: Partial Differential Equations III: Nonlinear Equations. (2nd edition)

See here: ZMATH

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For the elliptic case the classic reference (and the best I can think of) is "Elliptic Partial Differential Equations of Second Order" by Gilbarg and Trudinger. You'll find a very detailed exposition of Schauder's theory, Harnak inequality and maximum principles, Calderon-Zygmund and so on and so forth. For a good understanding of $L^p$ spaces in general, I agree with Beni Bogosel with the book of Brezis and also recommend Folland's "Real Analysis". It's developed in a slightly more general (and abstract) setting, because instead of $\mathbb{R}^n$ he studies arbitrary measure spaces, but the book is an excellent reference in real analysis. Hope I've been helpful...

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