Singularly Perturbed Boundary Value Problems

This book examines fundamental boundary value problems for the Laplace equation in domains that undergo a singular perturbation. Its central aim is to present the Functional Analytic Approach, a technique for describing how solutions depend on a singular perturbation parameter through analytic functions. The perturbation studied here is the size of small holes in the domain.

As the first book on this subject, it covers both the underlying theory and its applications, moving from straightforward illustrative cases to more demanding research results. The method relies on the integral representation of solutions to boundary value problems, potential theory, the theory of analytic functions in both finite and infinite dimensions, and nonlinear functional analysis.

The opening seven chapters are written to serve several audiences. They can function as a reference text for graduate courses on classical potential theory and its use in boundary value problems, and they include results seldom found elsewhere in the literature that may interest specialists. Readers who want a quick introduction to the method will find simple illustrative examples designed for that purpose.

More advanced readers will find a thorough treatment of the Functional Analytic Approach, including a comparison with the classical expansion techniques of asymptotic analysis, an account of the method's distinctive features, and its application to both linear and nonlinear boundary value problems.