Reference on Sobolev spaces $W^{k,p}(\Omega;\mathbb{R}^n)$

Question

While reading about calculus of variations I stumbled upon the Sobolev spaces $W^{k,p}(\Omega;\mathbb{R}^n)$ of order $k$ weakly differentiable functions with $p$ integrable derivatives, and codomain $\mathbb{R}^n$ instead of $\mathbb{R}$. Do you have any reference where definitions and properties about those spaces are provided?

Answer 1

One of the best books for a whole course about Sobolev Spaces is “Giovanni Leoni - A first course in Sobolev Spaces”

Another book is “Adams - Sobolev Spaces”.

If you are interested in Sobolev Spaces from the point of view of PDEs then:

• Brezis’ book Functional Analysis, Sobolev Spaces and Partial Differential Equations (mainly chapter 8 of the second edition)

• Evans’ book Partial Differential Equations (mainly chapter 5 of the second edition).

Just since you told me you are interested in Sobolev spaces from the calculus of variations (dark) side, let me point out we all can't wait - no joke - for the publication of Fonseca and Leoni's new book, Modern Methods in the Calculus of Variations: Sobolev Spaces, in preparation, accepted for publication by Springer.

Answer 2

You may also have a look into

E. Lieb, M. Loss: Analysis