eBook - ePub

Regularity Techniques for Elliptic PDEs and the Fractional Laplacian

Name: Regularity Techniques for Elliptic PDEs and the Fractional Laplacian
Author: Pablo Raúl Stinga

Pablo Raúl Stinga

334 pages
English
ePUB (mobile friendly)
Available on iOS & Android

eBook - ePub

Regularity Techniques for Elliptic PDEs and the Fractional Laplacian

Pablo Raúl Stinga

Book details

Table of contents

Citations

About This Book

Regularity Techniques for Elliptic PDEs and the Fractional Laplacian presents important analytic and geometric techniques to prove regularity estimates for solutions to second order elliptic equations, both in divergence and nondivergence form, and to nonlocal equations driven by the fractional Laplacian. The emphasis is placed on ideas and the development of intuition, while at the same time being completely rigorous. The reader should keep in mind that this text is about how analysis can be applied to regularity estimates. Many methods are nonlinear in nature, but the focus is on linear equations without lower order terms, thus avoiding bulky computations. The philosophy underpinning the book is that ideas must be flushed out in the cleanest and simplest ways, showing all the details and always maintaining rigor. Features

Self-contained treatment of the topic
Bridges the gap between upper undergraduate textbooks and advanced monographs to offer a useful, accessible reference for students and researchers.
Replete with useful references.

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Information

Publisher

Chapman and Hall/CRC

Year

2024

ISBN

9781040041574

Edition

Topic

Mathematics

Subtopic

Differential Equations

Index

Mathematics

Cover Page
Half-Title Page
Title Page
Copyright Page
Dedication Page
Contents
Foreword
Preface
Author Bio
Chapter 1 ◾ Introduction
Section I The Laplacian
Chapter 2 ◾ Harmonic functions
Chapter 3 ◾ The Schauder estimates for the Laplacian
Chapter 4 ◾ The Calderón–Zygmund estimates for the Laplacian
Section II Divergence form Equations
Chapter 5 ◾ The De Giorgi theorem
Chapter 6 ◾ The Moser theorem
Chapter 7 ◾ Perturbation theory for divergence form equations
Section III Nondivergence form equations
Chapter 8 ◾ Viscosity solutions and the ABP estimate
Chapter 9 ◾ The Krylov–Safonov Harnack inequality
Chapter 10 ◾ Savin's method of sliding paraboloids
Chapter 11 ◾ Perturbation theory for nondivergence form equations
Section IV The fractional Laplacian
Chapter 12 ◾ Basic properties of the fractional Laplacian
Chapter 13 ◾ Hölder and Schauder estimates
Chapter 14 ◾ The Caffarelli–Silvestre extension problem
Bibliography
Index