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Regularity Techniques for Elliptic PDEs and the Fractional Laplacian
Pablo Raúl Stinga
- 334 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
Regularity Techniques for Elliptic PDEs and the Fractional Laplacian
Pablo Raúl Stinga
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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Table of contents
- 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