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.

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.