Calderón-Zygmund estimates for the Laplacian

Abstract

[en] Regularity theory for Partial Differential Equations might be one of the most important topics in the field. With many applications, some of them in areas further away like Mathematical Physics, learning the basic regularity estimates for the Laplacian seems a crucial step into understanding more general results and solutions. This project intends to provide the tools and proofs of the CalderónZygmund estimates for the Laplacian equation $\Delta u=f$, with $f \in L^p$. We will separate in three distinct cases: $p=2, p \in(2, \infty)$ and $p=\infty$, each with a different proof. Further, using blow-up techniques introduced in [1] a new proof for the limiting case $p=\infty$ will be provided. Finally, we intend to remark a few points that could potentially lead towards a blow-up proof for the general $L^p$ case.