For the first time in the mathematical literature, this two-volume work introduces a unified and general approach to the subject. To a large extent, the book is based on the authors’ work, and has no significant overlap with other books on the theory of elliptic boundary value problems

For the first time in the mathematical literature, this two-volume work introduces a unified and general approach to the subject. To a large extent, the book is based on the authors’ work, and has no significant overlap with other books on the theory of elliptic boundary value problems.

For the first time in the mathematical literature, this two-volume work introduces a unified and general approach to the subject. To a large extent, the book is based on the authors’ work, and has no significant overlap with other books on the theory of elliptic boundary value problems

For the first time in the mathematical literature this two-volume work introduces a unified and general approach to the asymptotic analysis of elliptic boundary value problems in singularly perturbed domains. While the first volume is devoted to perturbations of the boundary near isolated singular points, the second volume treats singularities of the boundary in higher dimensions as well as nonlocal perturbations. At the core of this work are solutions of elliptic boundary value problems by asymptotic expansion in powers of a small parameter that characterizes the perturbation of the domain. In particular, it treats the important special cases of thin domains, domains with small cavities, inclusions or ligaments, rounded corners and edges, and problems with rapid oscillations of the boundary or the coefficients of the differential operator. The methods presented here capitalize on the theory of elliptic boundary value problems with nonsmooth boundary that has been developed in the past thirty years. Moreover, a study on the homogenization of differential and difference equations on periodic grids and lattices is given. Much attention is paid to concrete problems in mathematical physics, particularly elasticity theory and electrostatics. To a large extent the work is based on the authors' work and has no significant overlap with other books on the theory of elliptic 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 1–7 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.

The book presents new results and applications of the topological derivative method in control theory, topology optimization and inverse problems. It also introduces the theory in singularly perturbed geometrical domains using selected examples. Recognized as a robust numerical technique in engineering applications, such as topology optimization, inverse problems, imaging processing, multi-scale material design and mechanical modeling including damage and fracture evolution phenomena, the topological derivative method is based on the asymptotic approximations of solutions to elliptic boundary value problems combined with mathematical programming tools. The book presents the first order topology design algorithm and its applications in topology optimization, and introduces the second order Newton-type reconstruction algorithm based on higher order topological derivatives for solving inverse reconstruction problems. It is intended for researchers and students in applied mathematics and computational mechanics interested in the mathematical aspects of the topological derivative method as well as its applications in computational mechanics.

This contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Thirteenth International Conference on Integral Methods in Science and Engineering, held July 21–25, 2014, in Karlsruhe, Germany. A broad range of topics is addressed, from problems of existence and uniqueness for singular integral equations on domain boundaries to numerical integration via finite and boundary elements, conservation laws, hybrid methods, and other quadrature-related approaches. This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines and other professionals for whom integration is an essential tool.

For the first time in the mathematical literature this two-volume work introduces a unified and general approach to the asymptotic analysis of elliptic boundary value problems in singularly perturbed domains. While the first volume is devoted to perturbations of the boundary near isolated singular points, the second volume treats singularities of the boundary in higher dimensions as well as nonlocal perturbations. At the core of this work are solutions of elliptic boundary value problems by asymptotic expansion in powers of a small parameter that characterizes the perturbation of the domain. In particular, it treats the important special cases of thin domains, domains with small cavities, inclusions or ligaments, rounded corners and edges, and problems with rapid oscillations of the boundary or the coefficients of the differential operator. The methods presented here capitalize on the theory of elliptic boundary value problems with nonsmooth boundary that has been developed in the past thirty years. Moreover, a study on the homogenization of differential and difference equations on periodic grids and lattices is given. Much attention is paid to concrete problems in mathematical physics, particularly elasticity theory and electrostatics. To a large extent the work is based on the authors' work and has no significant overlap with other books on the theory of elliptic boundary value problems.