Elliptic Regularity Theory by Approximation Methods
Seiten
2022
Cambridge University Press (Verlag)
978-1-009-09666-9 (ISBN)
Cambridge University Press (Verlag)
978-1-009-09666-9 (ISBN)
An innovative approach to elliptic regularity, pushing forward important ideas in the area and tackling fundamental models in the theory. The book collects the basics of the field and seamlessly transitions to advanced, refined techniques in the context of genuinely challenging problems, providing a unifying perspective on the topic.
Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.
Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.
Edgard A. Pimentel is Research Scientist at the University of Coimbra and Assistant Professor of Mathematics at Pontifical Catholic University of Rio de Janeiro. He is researcher for the National Council of Science and Technology (CNPq-Brazil), a junior associate fellow of the International Centre for Theoretical Physics, and an affiliated member of the Brazilian Academy of Sciences.
Preface; 1. Elliptic partial differential equations; 2. Flat solutions are regular; 3. The recession strategy; 4. A regularity theory for the Isaacs equation; 5. Regularity theory for degenerate models; References; Index.
| Erscheinungsdatum | 29.09.2022 |
|---|---|
| Reihe/Serie | London Mathematical Society Lecture Note Series |
| Zusatzinfo | Worked examples or Exercises |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 152 x 228 mm |
| Gewicht | 300 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-009-09666-4 / 1009096664 |
| ISBN-13 | 978-1-009-09666-9 / 9781009096669 |
| Zustand | Neuware |
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