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Full Description
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. This first volume is devoted to domains whose boundary is smooth in the neighborhood of finitely many conical points. In particular, the theory encompasses the important case of domains with small holes. The second volume, on the other hand, treats perturbations of the boundary in higher dimensions as well as nonlocal perturbations.
The core of this book consists of the solution of general elliptic boundary value problems by complete asymptotic expansion in powers of a small parameter that characterizes the perturbation of the domain. The construction of this method capitalizes on the theory of elliptic boundary value problems with nonsmooth boundary that has been developed in the past thirty years.
Much attention is paid to concrete problems in mathematical physics, for example in elasticity theory. In particular, a study of the asymptotic behavior of stress intensity factors, energy integrals and eigenvalues is presented.
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.
Contents
I Boundary Value Problems for the Laplace Operator in Domains Perturbed Near Isolated Singularities.- 1 Dirichlet and Neumann Problems for the Laplace Operator in Domains with Corners and Cone Vertices.- 2 Dirichlet and Neumann Problems in Domains with Singularly Perturbed Boundaries.- II General Elliptic Boundary Value Problems in Domains Perturbed Near Isolated Singularities of the Boundary.- 3 Elliptic Boundary Value Problems in Domains with Smooth Boundaries, in a Cylinder, and in Domains with Cone Vertices.- 4 Asymptotics of Solutions to General Elliptic Boundary Value Problems in Domains Perturbed Near Cone Vertices.- 5 Variants and Corollaries of the Asymptotic Theory.- III Asymptotic Behaviour of Functional on Solutions of Boundary Value Problems in Domains Perturbed Near Isolated Boundary Singularities.- 6 Asymptotic Behaviour of Intensity Factors for Vertices of Corners and Cones Coming Close.- 7 Asymptotic Behaviour of Energy Integrals for Small Perturbations of the BoundaryNear Corners and Isolated Points.- 8 Asymptotic Behaviour of Energy Integrals for Particular Problems of Mathematical Physics.- IV Asymptotic Behaviour of Eigenvalues of Boundary Value Problems in Domains with Small Holes.- 9 Asymptotic Expansions of Eigenvalues of Classic Boundary Value Problems.- 10 Homogeneous Solutions of Boundary Value Problems in the Exterior of a Thin Cone.- Comments on Parts I-IV.- Comments on Part I.- 1.- 2.- Comments on Part II.- 3.- 4.- 5.- Comments on Part III.- 6.- 7.- 8.- Comments on Part IV.- 9.- 10.- List of Symbols.- 1. Basic Symbols.- 2. Symbols for function spaces and related concepts.- 3. Symbols for functions, distributions and related concepts.- 4. Other symbols.- References.
