Singularly Perturbed Boundary Value Problems: A Functional Analytic Approach

This book is about the analysis of the Laplace equation in singularly perturbed domains. It illustrates the Functional Analytic Approach, which describes the dependence of solutions upon a singular perturbation parameter in terms of analytic functions. The book covers the theoretical material and its applications to a series of problems, from simple illustrative examples to more involved research results. It is designed to serve various purposes and readerships, with an extensive introductory part that can be used as a reference textbook for graduate courses on classical Potential Theory and its applications to boundary value problems.

Format: Hardback
Length: 672 pages
Publication date: 05 October 2021
Publisher: Springer Nature Switzerland AG

This comprehensive book delves into the analysis of fundamental boundary value problems for the Laplace equation in singularly perturbed domains. Its primary objective is to showcase the Functional Analytic Approach, a powerful method for describing the dependence of solutions on singular perturbation parameters in terms of analytic functions. With a focus on domains featuring small holes and considering the perturbation parameter as the size of these holes, the book offers a comprehensive introduction to this topic. It encompasses both theoretical material and its practical applications to a diverse range of problems, spanning from simple illustrative examples to intricate research findings.

The Functional Analytic Approach is a versatile tool that employs the integral representation method, Potential Theory, the Theory of Analytic Functions, both in finite and infinite dimensions, and Nonlinear Functional Analysis. This approach is designed to cater to a wide range of purposes and readerships. The extensive introductory section, spanning Chapters 1 to 7, serves as a valuable reference textbook for graduate courses on classical Potential Theory and its applications to boundary value problems. Additionally, the early chapters contain results that are rarely presented in the literature, making it appealing to both novice and expert readers alike.

The exposition then proceeds to introduce the Functional Analytic Approach, providing simple illustrative examples specifically tailored for those seeking a quick introduction to the method. For more advanced readers, a comprehensive presentation of the Functional Analytic Approach is offered, allowing for a comparison with more classical expansion methods such as Asymptotic Analysis. It also provides insights into the unique features of the approach, making it a valuable resource for researchers and practitioners in the field.

In conclusion, this book is a valuable contribution to the field of mathematical analysis, offering a comprehensive and insightful exploration of boundary value problems for the Laplace equation in singularly perturbed domains. Through the Functional Analytic Approach, it provides a powerful tool for describing the behavior of solutions and their dependence on perturbation parameters, making it accessible to a wide range of readers. Whether you are a student, researcher, or professional seeking to advance your knowledge in this area, this book is an essential resource that will greatly assist you in your endeavors.

Weight: 1190g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030762582
Edition number: 1st ed. 2021