Textbook for Partial Differential Equations with a viewpoint towards Geometry.

Question

Though similar questions have been asked at Good 1st PDE book for self study and Good reference texts for introduction to partial differential equation? but none of them really answer my query, so I am bounded to ask this.

I am basically interested in Differential and Riemannian Geometry and one of my Professors told me that it will be a good idea if I acquire a sound knowledge of PDE. I know about the basics of PDE (i.e., methods of solving PDE ) but I don't have any firm knowledge of the analysis which goes on in there.

So, my question is that what will be a good textbook to start learning PDE that could help in undrstanding the $\it{analysis}$ portion as well as with applications of PDE in Differential/Riemannian Geometry.

Background : I have studied Measure Theory, Functional Analysis, Complex Analysis and some Fourier Analysis (from Stein & Shakarchi's book on Fourier Analysis). I am currently studying Algebraic Topology, Differential and Riemannian Geometry (from Do Carmo's book).

Thanks!!

Answer 1

The standard PDE textbooks would serve you well as a start. For example Fritz John's Partial Differential Equations and LC Evans' Partial Differential Equations are both very good for just the analysis aspects.

In parallel you can consult Michael Taylor's 3 volume set titled, again, Partial Differential Equations. Taylor's books develops similar breadth and slightly more depth as Evans' book, but also with an eye toward geometry.

Once you get a few things under your belt, J Jost's Riemannian geometry and geometric analysis is a classic in the field.

For more advanced topics, particular related to geometry, two very good books have been written by Thierry Aubin: Nonlinear Analysis on Manifolds, Monge-Ampere Equations and Some Nonlinear Problems in Riemannian Geometry.

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If you particularly want to study elliptic type problems, you can also consider going from John's book to Fang-Hua Lin's Elliptic partial differential equations then to Gilbarg & Trudinger on the one hand, and Caffarelli & Cabré on the other for the analysis.