Название: Regularity of the One-phase Free Boundaries
Автор: Bozhidar Velichkov
Издательство: Springer
Год: 2023
Страниц: 249
Язык: английский
Формат: pdf (true), epub
Размер: 18.2 MB

This book is an introduction to the regularity theory for free boundary problems. The free boundary problems are a special type of boundary value problems, in which the domain, where the PDE is solved, depends on the solution of the boundary value problem. The focus is on the one-phase Bernoulli problem, which is of particular interest as it deeply influenced the development of the modern free boundary regularity theory and is still an object of intensive research.

In fact, the free boundary regularity theory stands on the crossroad of Calculus of Variations, PDEs and Geometric Analysis, and is characterized by the interaction between geometric and analytic objects, which is precisely what makes it so fascinating (and hard) field of Analysis.

The exposition is organized around four main theorems, which are dedicated to the one-phase functional in its simplest form. Many of the methods and the techniques presented here are very recent and were developed in the context of different free boundary problems. We also give the detailed proofs of several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories.

This book is aimed at graduate students and researches and is accessible to anyone with a moderate level of knowledge of elliptical PDEs.

Contents:
1. Introduction and Main Results
2. Existence of Solutions, Qualitative Properties and Examples
3. Lipschitz Continuity of the Minimizers
4. Non-degeneracy of the Local Minimizers
5. Measure and Dimension of the Free Boundary
6. Blow-Up Sequences and Blow-Up Limits
7. Improvement of Flatness
8. Regularity of the Flat Free Boundaries
9. The Weiss Monotonicity Formula and Its Consequences
10. Dimension of the Singular Set
11. Regularity of the Free Boundary for Measure Constrained Minimizers
12. An Epiperimetric Inequality Approach to the Regularity of the One-Phase Free Boundaries