1

I am currently an undergraduate studying mathematics and have been mode on the pure side. Now I would like to get started with PDEs. There have been some questions from other people about book recommendations in this area. However they were looking for something that focuses on application. I would like to study the subject for its own sake and more proof based. Does anyone know some book that really goes into the theory of it. For context: I have done multivariable calculus, measure theory and functional analysis. But I don't know anything about differential geometry.

struggling_student
  • 15
  • 3

2 Answers2

1

The standard graduate textbook for PDEs is Partial differential equations by Evans. The first half of the book discusses explicit model examples by treating them rigorously.

The next half deals with the general theory of PDEs, including Sobolev spaces, followed by standard linear elliptic, parabolic, and hyperbolic equations. In the end, he also discusses some nonlinear examples.

The book is very readable and has a comprehensive set of appendices that contain any material you may not know but based on what you have mentioned in your question you should be alright.

almosteverywhere
  • 2,080
0

For my engineering degree I used Engineering Mathematics Volume 1 by A.J.M. Spencer and others (total of 9 authors) (Van Nostrand, 1977), which is a solid workmanlike text which covers the basics of the following areas: ODEs, Fourier series, Laplace transforms, PDEs with applications (wave equation, Laplace's equation, heat equation), multiple integrals, vector analysis, linear algebra, numerical analysis, and some basic statistics.

It's very much from an applied mathematics point of view.

I remember working through numerous examples of solutions of PDEs of the abovementioned types, and by the time the exam came along I was pretty well au fait with solving them.

Not sure whether it's still in print, but it crops up every so often on the 2nd hand market. You may find a copy by shopping online.

I also have Arne Broman's Introduction to Partial Differential Equations on my bookshelf (Dover, 1970) but haven't really got round to cracking it open since I got it. It covers Fourier, Laplace and Bessel, and then the Wave, Heat and Laplace equations are covered at the end of the book in a final chapter.

Prime Mover
  • 5,005
  • 1
    I think the OP is looking for something on the more theoretical side :) – Severin Schraven Mar 15 '24 at 17:31
  • @SeverinSchraven I stand by my answer. – Prime Mover Mar 15 '24 at 17:32
  • You are entitle to! I am just saying that if the question is specifically not about applications, then the answer might benefit from a word or two why they still might want to check your recommendation out. Personally I feel that knowing a real-life phenomenon modeled by the PDE helps to get some intuition, but that is my own bias. – Severin Schraven Mar 15 '24 at 17:38
  • @SeverinSchraven IMO you really can't study PDEs without at least some idea of the applications. You can't make any sense of what's going on otherwise. – Prime Mover Mar 15 '24 at 17:40
  • I feel the same about this, but I have met PDE people who think differently about this. – Severin Schraven Mar 15 '24 at 17:45
  • 1
    @PrimeMover I do not agree with that. It is possible to study PDEs without any reference to applications. It's just that for historical reasons that is not the way it is usually taught. You are interested in zero sets of a differential operator, the study of which is as pure/applied as that of the zero sets of a polynomial, say. It just so happens that a lot of PDEs are connected to physics but there are also quite a few of them which are not for e.g. the Ricci flow. – almosteverywhere Mar 15 '24 at 21:04
  • 1
    And while physical intuition may sometimes help in developing techniques to treat/analyse PDEs, more often than not the techniques are completely devoid of any physical content. – almosteverywhere Mar 15 '24 at 21:05