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I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as more "qualitative" methods, rather then approximations or techniques for solutions. I am looking for anything that might help me to study this, books, papers, surveys, etc.

What is your recommended road map for familiarizing myself with this subject?

Thank you.

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This probably should be a community wiki but I don't find the checkbox that makes it so. –  Blade Aug 25 '13 at 22:01
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Taylor's three book series on partial differential equations is almost certainly what you're looking for. –  Paul Siegel Aug 26 '13 at 0:36
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are you looking books on pde or on geometric analysis? –  Koushik Aug 26 '13 at 3:03
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@TomLeinster : and if you are not satisfied with the three term Taylor series, there's also a remainder term. :-) –  Willie Wong Aug 26 '13 at 21:57
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5 Answers 5

Lectures on Partial Differential Equations by V. Arnold.

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Since you claim that you know "nothing about PDEs", I think it would be very hard to appreciate the topological/geometric applications of PDEs without at least a basic familiarity with the theory of PDEs: i.e. methods used for proving existence, uniqueness, regularity and basic estimates in time/space. I would say start with the PDE lectures by V. Arnold, and read it along with Evans' PDE text.

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Micheal Taylor's series has already been mentioned. You might also look at Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry, which is a good guide for the use of nonlinear elliptic PDE in geometry. I took a reading course from this book when I was a Ph.D. student.

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I find your question too broad. I would recommend starting with a book that focuses on a particular question or area in differential geometry and presents the PDE theory needed. A very incomplete list of suggestions include the following:

There are other books on the Atiyah-Singer index theorem, harmonic maps, minimal surfaces, the complex Monge-Ampere equation, etc.

Books on elliptic PDE's used by many differential geometers include Gilbarg-Trudinger and Morrey.

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While I agree that Aubin's book is a very good way to learn the PDE theory required for the Yamabe problem, let me hasten to add that Aubin's book is more than just PDE theory for the Yamabe problem. It contains also a lot of information on basic PDE facts that are very useful for all sort of elliptic type problems in geometric analysis. –  Willie Wong Aug 26 '13 at 21:53
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Riemannian Geometry and Geometric Analysis by Jürgen Jost might be helpful.

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I have heard that this book is notorious for making mistakes –  Koushik Sep 9 '13 at 4:15
    
Would graduate courses in analysis, differential geometry suffice as prerequisites for reading this book ? –  Amr Jan 19 at 17:13
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