Measure Theory & Functional Analysis for PDE's

Question

I'm looking to take an independent study on Partial Differential Equations. I will hopefully be keeping Walter Strauss' Partial Differential Equations: An Introduction and Lawrence Evans' Partial Differential Equations as references.

Can someone recommend a textbook that covers the pre-requisite material from measure theory, functional analysis (and maybe even vector calculus) that is used in PDE's. I know a bit of measure theory, and I'll be taking a class on it as well. I haven't studied functional analysis.

It'd be great if someone could recommend a textbook that covers the said pre-requisite material required for PDE's at the level appropriate for the aforementioned books.

Answer 1

Lang's Real and Functional Analysis covers all three topics you mentioned (vector calculus being in the form of differential forms on manifolds). A bonus is that it covers integration of Banach-valued functions.

Brézis's Functional Analysis, Sobolev Spaces and Partial Differential Equations is tailored to provide the functional analysis background needed for PDEs, but assumes prior knowledge of Lebesgue integration.

Also check the appendices in Evans's book, as he has his own references for the prerequisite material.

Answer 2

I was specifically looking for a book (on PDE's) that covers the pre-requisite material on measure theory and functional analysis in a self contained manner.

The book Partial Differential Equations by Mikhailov contains all the requisite materials from measure theory and functional analysis in its second and third chapters. However, its quite old.

Instead, if you are willing to learn measure theory elsewhere, the book by Renardy and Rogers is worth considering. The authors develop the required results from functional analysis when required and has a lot of material on PDEs.