Integro-Differential Elliptic Equations

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, Birkhäuser, 2024.

References in corpus (22)

• Regularity theory for fully nonlinear integro-differential equations
• Nonlocal diffusion and applications
• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Local behavior of fractional $p$-minimizers
• The influence of fractional diffusion in Fisher-KPP equations
• Some observations on the Green function for the ball in the fractional Laplace framework
• Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes
• Regularity Theory for Elliptic PDE
• Boundary Harnack inequality for Markov processes with jumps
• Regularity for parabolic integro-differential equations with very irregular kernels
• Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries
• Regularity estimates for elliptic nonlocal operators
• Direct epiperimetric inequalities for the thin obstacle problem and applications
• Min-max formulas for nonlocal elliptic operators on Euclidean space
• Fourier methods for fractional-order operators
• Obstacle problems for nonlocal operators with singular kernels
• Free boundary partial regularity in the thin obstacle problem
• Optimal regularity for nonlocal elliptic equations and free boundary problems
• Gradient regularity and first-order potential estimates for a class of nonlocal equations
• Regularity theory for nonlocal obstacle problems with critical and subcritical scaling
• Nonlocal elliptic and parabolic equations with general stable operators in weighted Sobolev spaces

Cited by in corpus (6)

• Symmetrization results for general nonlocal linear ellipitic and parabolic problems
• Nonlocal operators in divergence form and existence theory for integrable data
• Linear statistics of determinantal point processes and norm representations
• Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations
• A Brezis and Peletier type result for the fractional Robin function
• Minimizing Eigenvalues of the Fractional Laplacian