Integro-Differential Elliptic Equations

TL;DR

This book provides a comprehensive, self-contained introduction to the regularity theory of integro-differential elliptic equations, covering basic to advanced topics, with new proofs and open problems, mainly developed in the 21st century.

Contribution

It offers the first unified presentation of the regularity theory for integro-differential elliptic equations, including new proofs and a focus on main ideas rather than full generality.

Findings

Development of regularity results for integro-differential equations

Introduction of viscosity solutions for nonlinear cases

Analysis of obstacle problems and free boundary regularity

Abstract

This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of equations often arises in analysis, probability theory, mathematical physics, and in several contexts in applied sciences. The authors give a detailed presentation of all the necessary techniques, primarily focusing on the main ideas rather than proving all results in their greatest generality. The book starts from the very basics, studying the square root of the Laplacian and weak solutions to linear equations. Then, the authors develop the theory of viscosity solutions to nonlinear equations and prove the main known results in this context. Finally, they study obstacle problems for integro-differential operators and establish the regularity of solutions and free boundaries. Almost all the covered material appears in book form for the first time, and several proofs are different (and shorter) than those in the original papers. Moreover, several open problems are listed throughout the book. This is a draft of the book ''Integro-Differential Elliptic Equations''. The final version has been published in Progress in Mathematics, vol. 350, Birkh\"auser, 2024.