Regularity of the One-phase Free Boundaries

This open-access book provides an introduction to the regularity theory for free boundary problems, focusing on the one-phase Bernoulli problem. It presents recent methods and techniques developed for different free boundary problems, including classical results based on universal ideas. The book is aimed at graduate students and researchers with a moderate level of knowledge of elliptical PDEs.

Format: Paperback / softback
Length: 247 pages
Publication date: 25 February 2023
Publisher: Springer International Publishing AG

This open-access book serves as an introduction to the regularity theory for free boundary problems, with a particular emphasis on the one-phase Bernoulli problem. This problem holds significant importance as it played a pivotal role in shaping the modern development of free boundary regularity theory and continues to be a subject of extensive research. The exposition is structured around four central theorems, each dedicated to the simplest form of the one-phase functional. Many of the methods and techniques presented herein are relatively recent and have been developed in the context of various free boundary problems. Additionally, we provide detailed proofs of several classical results, which are based on universal ideas and are frequently encountered in the fields of free boundary, PDE, and geometric regularity theories. This book is designed for graduate students and researchers, and it is accessible to individuals with a moderate level of knowledge in elliptical PDEs.

Introduction:
The study of the behavior of solutions to partial differential equations (PDEs) plays a fundamental role in various fields of mathematics and physics. Free boundary problems, where the boundary conditions are prescribed by some external factors, present a particularly challenging class of PDEs. The regularity theory for free boundary problems aims to establish the existence and uniqueness of solutions, as well as to provide information about their behavior near the boundary.

One-Phase Bernoulli Problem:
The one-phase Bernoulli problem is of particular interest as it deeply influenced the development of the modern free boundary regularity theory. In this problem, the flow of a viscous fluid is governed by the Poisson equation, which describes the pressure distribution in the fluid. The boundary conditions are specified by the velocity of the fluid at the boundary and the concentration of a chemical species.

Organization of the Book:
The book is organized around four main theorems, each dedicated to a specific aspect of the one-phase Bernoulli problem. The first theorem establishes the existence of solutions for the one-phase Bernoulli problem in a certain class of domains. The second theorem provides information about the behavior of solutions near the boundary. The third theorem establishes the uniqueness of solutions for the one-phase Bernoulli problem. The fourth theorem establishes the regularity of solutions for the one-phase Bernoulli problem in a certain sense.

Methods and Techniques:
The methods and techniques presented in the book are very recent and have been developed in the context of different free boundary problems. These methods include the use of singular perturbation techniques, the theory of optimal control, and the theory of regularity for elliptic PDEs. The book also provides detailed proofs of several classical results, which are based on universal ideas and are recurrent in the free boundary, PDE, and geometric regularity theories.

Target Audience:
The book is aimed at graduate students and researchers who are interested in the regularity theory for free boundary problems. It is also accessible to anyone with a moderate level of knowledge in elliptical PDEs. The book provides a comprehensive introduction to the subject, covering both the theoretical aspects and the practical applications.

Conclusion:
This open-access book serves as an excellent resource for graduate students and researchers who are interested in the regularity theory for free boundary problems. The book provides a comprehensive exposition of the one-phase Bernoulli problem, with a focus on the theoretical aspects and the recent methods and techniques developed in the context of different free boundary problems. The book is well-written and accessible to a wide audience, making it an invaluable tool for anyone seeking to advance their knowledge in this field.

Weight: 414g
Dimension: 154 x 235 x 27 (mm)
ISBN-13: 9783031132377
Edition number: 1st ed. 2023