Elliptic Regularity Theory by Approximation Methods (London Mathematical Society Lecture Note Series)

Pimentel, Edgard A.

Cambridge University Press（2022/09発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 202 p.
• 言語 ENG
• 商品コード 9781009096669
• DDC分類 515.983

Full Description

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.

Contents

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.