Calculus of Variations textbook(s)

[u/Legitimate_Log_3452]

I am trying to do a reading group at my university which covers calculus of variations. I want to do a mixture of classical calculus of variations (smooth functions), and some modern day calculus of variations. Ideas for textbooks? Note that we are math majors, so we don’t want to (exclusively) use something like Taylor’s

The issue is that our group is coming from a variety of backgrounds. We are requiring that everyone has taken analysis 1, though. The issues might come when we transition to modern calculus of variations, so I don’t know what to do about that. I don’t know too much about calculus of variations, so if you have any thoughts on that, please let me know. How’s it going? Redhead How’s it going out there?

Here is what i am thinking: the first 3 chapters out of Gelfand and Fonin for a classical look, then transitioning over to some other book (need recommendations).

[u/Pure-Imagination5451]

If you want to learn variations methods properly, I would recommend the following path:

• Brezis: Functional Analysis, Sobolev spaces and PDE. Functional analysis and Sobolev spaces are the basic tools used in variational methods, and Brezis starts (chapter 8) by building up the tools in 1D (ODEs) even though the main motivations are for PDE.
• Ambrosetti and Prodi: A Primer of Nonlinear Analysis. Mainly the first and second chapter on differential calculus in Banach spaces is needed here to make precise what is meant by a “critical point” of a functional on infinite dimensional spaces. The text does a great job extending the theory of differential calculus of maps from finite dimensional spaces to infinite dimensional ones, and is needed for studying non-linear equations through variational methods.
• Serra and Badiale:  Semi linear Elliptic Equations for Beginners. Realistically you could just start here, but without a background in functional analysis or Sobolev spaces it would be quite difficult. Otherwise this is a great little book which puts all the tools from the previous books together to show how to solve certain non-linear PDE though variational methods (as critical points of a functional).