# Local L^p regularity theory for elliptic operators

I've been studying the elliptic regularity theory using $H^s$ spaces as done in Folland's "Introduction to Partial Differential Equations".

At the end of section C, chapter 6, Folland affirms that we have similar theorems using $W^{m,p}$ spaces for $p\in (1,+\infty)$, even though their proofs are more involved.

Unfortunately, he does not provide any reference.

Can you point me to a book where these topics are covered?

• The standard reference for those things is the book by Gilbarg and Trudinger. (However, it seems to me that the methods they use are rather different from Folland's ones - "more integration by parts, less Fourier transform", so to speak). – Giuseppe Negro Dec 7 '16 at 12:01
• See Sec. 10.3 of these lectures www3.nd.edu/~lnicolae/Lectures.pdf It's not a complete presentation, but you'll fet the main ideas and appropriate rederences. – Liviu Nicolaescu Dec 7 '16 at 14:34

There are two different parts in elliptic regularity theory.

The first and easier is interior regularity, which can be proven for $p\in (1,+\infty)$ essentially by the same method as for $p=2$, using the continuity properties of Calderon's singular integrals on $L^p$.

The second part is regularity up to the boundary for some elliptic boundary value problem such as the Dirichlet or Neumann problem. Then, if the boundary is smooth, you can also replicate for $p$ the $L^2$ argument, with $L^p$ continuity properties of boundary integrals.

See for example

'Second Order Elliptic Equations and Elliptic Systems' by Lan-Cheng Wu and Ya-Zhe Chen

and/or

Mariano Giaquinta and Luca Martinazzi 'An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs'

for detailed presentations of the $L^p$ regularity theory.