Integro-Differential Elliptic Equations

verfasst von: Xavier Fernández-Real, Xavier Ros-Oton

Verlag: Springer Nature Switzerland

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathematical physics, and in several contexts in the applied sciences. The work gives a detailed presentation of all the necessary techniques, with a primary focus on the main ideas rather than on proving all the results in their greatest generality.
The basic building blocks are presented first, with the study of the square root of the Laplacian, and weak solutions to linear equations. Subsequently, the theory of viscosity solutions to nonlinear equations is developed, and proofs are provided for the main known results in this context. The analysis finishes with the investigation of obstacle problems for integro-differential operators and establishes the regularity of solutions and free boundaries.
A distinctive feature of this work lies in its presentation of nearly all covered material in a monographic format for the first time, and several proofs streamline, and often simplify, those in the original papers. Furthermore, various open problems are listed throughout the chapters.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Square Root of the Laplacian

Abstract

In this chapter, we study the simplest integro-differential elliptic operator: the square-root of the Laplacian, \(\sqrt {-\varDelta }\). We start by establishing its basic properties, including the harmonic extension representation and the corresponding heat kernel and fundamental solution. We then prove the comparison principle, compute its Poisson kernel in a ball and find the corresponding mean value property, deduce the Harnack inequality, and establish interior regularity estimates. Finally, we construct some explicit solutions and develop the analogous results for the fractional Laplacian \((-\Delta )^s\), with \(s \in (0, 1)\).

Xavier Fernández-Real, Xavier Ros-Oton

Chapter 2. Linear Integro-differential Equations

In this chapter, we study general elliptic operators of order 2s, which are linear and translation invariant. We start with their probabilistic motivation and then move on to establishing their basic properties and different notions of solution (strong, weak, and distributional). After that, we develop the interior regularity theory for these operators and then extend it to the more general setting of x-dependent operators, both in divergence and in non-divergence form. Finally, we study the Hölder regularity of solutions up to the boundary and conclude the chapter by presenting some further results and open problems.

Xavier Fernández-Real, Xavier Ros-Oton

Chapter 3. Fully Nonlinear Equations

In this chapter, we develop the theory of fully nonlinear nonlocal elliptic equations. We begin with the definition of viscosity solutions and their existence and then prove the Harnack inequality and Hölder estimates for equations with “bounded measurable coefficients.” After that, we establish the main interior regularity estimates in both the general and the concave/convex case and conclude with a brief discussion of some open problems.

Xavier Fernández-Real, Xavier Ros-Oton

Chapter 4. Obstacle Problems

Abstract

In this chapter, we study obstacle problems for nonlocal elliptic operators. In order to do so, we first prove the boundary Harnack inequality for such operators in Lipschitz domains, a crucial tool in the regularity theory for free boundary problems. After that, we establish the smoothness of free boundaries near regular points and prove the optimal regularity of solutions. We finish the chapter with a brief discussion of some further results and open problems.

Xavier Fernández-Real, Xavier Ros-Oton

Backmatter

Titel: Integro-Differential Elliptic Equations
verfasst von: Xavier Fernández-Real
Xavier Ros-Oton
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-54242-8
Print ISBN: 978-3-031-54241-1
DOI: https://doi.org/10.1007/978-3-031-54242-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Square Root of the Laplacian

Chapter 2. Linear Integro-differential Equations

Chapter 3. Fully Nonlinear Equations

Chapter 4. Obstacle Problems

Backmatter

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