Seminar: Regularity Theory for Linear and Nonlinear Poisson-Type Equations

Bartomeu Garau Verger

Abstract: In this seminar, we will review classical results related to second-order divergence form elliptic partial differential equations, leading up to L^p regularity theory, with a particular focus on the Calderón-Zygmund inequality for the Poisson equation. We will explore this result from two perspectives: first, through interpolation of L^p spaces as done by Gilbarg and Trudinger; and second, via a geometric approach where the Hardy-Littlewood maximal function plays a critical role. This results will then be applied to study the regularity of nonlinear Poisson-type equations. Finally, we will provide a brief overview of the L^p regularity theory for fully nonlinear second-order divergence form equations.