Resource for $L^p$ regularity

Question

The $L^p$ theory for elliptic PDE tells us:

If u is a weak solution of $-\Delta u=f$, then we have $\| u\|_{W^{2,p}}\leq C(\|f\|_{L^p}+\|u\|_{L^{p}})$ for $1<p<\infty$.

Any recommandations of resources or books on this kind of results?

Thanks for anyone who could offer help.

Answer 1

The standard reference is Gilbarg and Trudinger's "Elliptic Partial Differential Equations of Second Order", Chapter 9. Caffarelli and Cabre deal with the nonlinear version of the theory in "Fully Nonlinear Elliptic Equations" and it's a nice reference since it provides a different approach from "singular integral" view presented in Gilbarg/Trudinger. Finally, I recommend the exercises in Mooney's notes

https://www.math.uci.edu/~mooneycr/EllipticPDE_BasicTheory.pdf

once you read the material, they will help you gain insight.

Hope it helps.