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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume I - Softcover
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Inhaltsangabe
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.- 1.1 Boundary Value Problems for the Laplace Operator in a Strip.- 1.1.1 The Dirichlet problem.- 1.1.2 The complex Fourier transform.- 1.1.3 Asymptotics of solution of the Dirichlet problem.- 1.1.4 The Neumann problem.- 1.1.5 Final remarks.- 1.2 Boundary Value Problems for the Laplace Operator in a Sector.- 1.2.1 Relationship between the boundary value problems in a sector and a strip.- 1.2.2 The Dirichlet problem.- 1.2.3 The Neumann problem.- 1.3 The Dirichlet Problem in a Bounded Domain with Corner.- 1.3.1 Solvability of the boundary value problem.- 1.3.2 Particular solutions of the homogeneous problem.- 1.3.3 Asymptotics of solution.- 1.3.4 A domain with a corner outlet to infinity.- 1.3.5 Asymptotics of the solutions for particular right-hand sides.- 1.3.6 The Dirichlet problem for the operator ? - 1.- 1.3.7 The Dirichlet problem in a domain with piecewise smooth boundary.- 1.4 The Neumann Problem in a Bounded Domain with a Corner.- 1.5 Boundary Value Problems for the Laplace Operator in a Punctured Domain and the Exterior of a Bounded Planar Domain.- 1.5.1 Dirichlet and Neumann problems in a punctured planar domain.- 1.5.2 Boundary value problems in the exterior of a bounded domain.- 1.6 Boundary Value Problems in Multi-Dimensional Domains.- 1.6.1 A domain with a conical point.- 1.6.2 A punctured domain.- 1.6.3 Boundary value problems in the exterior of a bounded domain.- 2 Dirichlet and Neumann Problems in Domains with Singularly Perturbed Boundaries.- 2.1 The Dirichlet Problem for the Laplace Operator in a Three-Dimensional Domain with Small Hole.- 2.1.1 Domains and boundary value problems.- 2.1.2 Asymptotics of the solution. The method of compound expansions.- 2.1.3 Asymptotics of the solution. The method of matched expansions.- 2.1.4 Comparison of asymptotic representations.- 2.2 The Dirichlet Problem for the Operator ? - 1 in a Three-Dimensional Domain with a Small Hole.- 2.3 Mixed Boundary Value Problems for the Laplace Operator in a Three-Dimensional Domain with a Small Hole.- 2.3.1 The boundary value problem with Dirichlet condition at the boundary of the hole.- 2.3.2 First version of the construction of asymptotics.- 2.3.3 Second version of the construction of asymptotics.- 2.3.4 The boundary value problem with the Neumann condition at the boundary of the gap.- 2.4 Boundary Value Problems for the Laplace Operator in a Planar Domain with a Small Hole.- 2.4.1 Dirichlet problem.- 2.4.2 Mixed boundary value problems.- 2.5 The Dirichlet Problem for the Operator ? - 1 in a Domain Perturbed Near a Vertex.- 2.5.1 Formulation of the problem.- 2.5.2 The first terms of the asymptotics.- 2.5.3 Admissible series.- 2.5.4 Redistribution of discrepancies.- 2.5.5 The set of exponents in the powers of ?, r, and ?.- 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.- 3.1 Boundary Value Problems in Domains with Smooth Boundaries.- 3.1.1 The operator of an elliptic boundary value problem.- 3.1.2 Elliptic boundary value problems in Sobolev and Hölder spaces.- 3.1.3 The adjoint boundary value problem (the case of normal boundary conditions).- 3.1.4 Adjoint operator in spaces of distributions.- 3.1.5 Elliptic boundary value problems depending on a complex parameter.- 3.1.6 Boundary value problems for elliptic systems.- 3.2 Boundary value problems in cylinders and cones.- 3.2.1 Solvability of boundary value problems in cylinders: the case of coefficients independent of t.- 3.2.2 Asymptotics at infinity of solutions to boundary value problems in cylinders with coefficients independent of t.- 3.2.3 Solvability of boundary value problems in a cone.- 3.2.4 Asymptot
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