Singularly Perturbed Boundary Value Problems

This book is devoted to the analysis of the basic boundary value problems for the Laplace equation in singularly perturbed domains. The main purpose is to illustrate a method called Functional Analytic Approach, to describe the dependence of the solutions upon a singular perturbation parameter in terms of analytic functions. Here the focus is on domains with small holes and the perturbation parameter is the size of the holes. The book is the first introduction to the topic and covers the theoretical material and its applications to a series of problems that range from simple illustrative examples to more involved research results. The Functional Analytic Approach makes constant use of the integral representation method for the solutions of boundary value problems, of Potential Theory, of the Theory of Analytic Functions both in finite and infinite dimension, and of Nonlinear Functional Analysis.

Designed to serve various purposes and readerships, the extensive introductory part spanning Chapters 17 can be used as a reference textbook for graduate courses on classical Potential Theory and its applications to boundary value problems. The early chapters also contain results that are rarely presented in the literature and may also, therefore, attract the interest of more expert readers. The exposition moves on to introduce the Functional Analytic Approach. A reader looking for a quick introduction to the method can find simple illustrative examples specifically designed for this purpose. More expert readers will find a comprehensive presentation of the Functional Analytic Approach, which allows a comparison between the approach of the book and the more classical expansion methods of Asymptotic Analysis and offers insights on the specific features of the approach and its applications to linear and nonlinear boundary value problems.

Presents the effect of perturbations in terms of analytic functions instead of in terms of the more classical asymptotic expansions Shows a powerful tool for the analysis of nonlinear and non-variational boundary value problems Presents a step-by-step exposition from the theoretical foundations of the Functional Analytic Approach to the implementation in challenging problems Provides an effective tool in the analysis of specific perturbation problems arising in continuum mechanics and material sciences Introductory style is accessible to a wide readership

Auteur
Matteo Dalla Riva is professor at College of Engineering and Natural Science in The University of Tulsa. Massimo Lanza de Cristoforis is professor at Dipartimento di Matematica Universita` degli Studi di Padova. Paolo Musolino is professor at Dipartimento di Scienze Molecolari e Nanosistemi Università Ca' Foscari Venezia.

Résumé
"The monograph is a carefully written presentation one of the deep approaches developing our knowledge on the theory of partial differential equation. It can be recommended to the experts in Analysis, Partial Differential Equations and Applications." (Sergei V. Rogosin, zbMATH 1481.35005, 2022)

Contenu

1. Introduction.- 2. Preliminaries.- 3. Preliminaries on Harmonic Functions.- 4. Green Identities and Layer Potentials.- 5. Preliminaries on the Fredholm Alternative Principle .- 6. Boundary Value Problems and Boundary Integral Operators.- 7. Poisson Equation and Volume Potentials.- 8. A Dirichlet Problem in a Domain with a Small Hole.- 9. Other Problems with Linear Boundary Conditions in a Domain with a Small Hole.- 10. A Dirichlet Problem in a Domain with Two Small Holes.- 11. Nonlinear Boundary Value Problems in Domains with a Small Hole.- 12. Boundary Value Problems in Periodic Domains, A Potential Theoretic Approach.- 13. Singular Perturbation Problems in Periodic Domains.- Appendix.- References.- Index.