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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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    Ah, [18-155](http://ocw.mit.edu/courses/mathematics/18-155-differential-analysis-spring-2004/) and [18-156](http://ocw.mit.edu/courses/mathematics/18-156-differential-analysis-spring-2004/) on OCW@MIT seem good supplements.2011-06-13
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    Haim Brezis' Functional Analysis has some facts about $L^p$ spaces, although maybe not really what you're looking for.2011-06-13
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    $L^p$ 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.2011-06-13
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    @ziyuang: The first link in your first comment does not work.2011-07-19
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    @Jack: Oops... [18-155](http://ocw.mit.edu/courses/mathematics/18-155-differential-analysis-fall-2004/)2011-07-20

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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...