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.
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 Asymptotics of the solutions at infinity and near the vertex of a cone for boundary value problems with coefficients independent of r.- 3.2.5 Boundary value problems for elliptic systems in a cone.- 3.2.6 Asymptotics of the solution for the right-hand side given by an asymptotic expansion.- 3.3 Boundary Value Problems in Domains with Cone Vertices.- 3.3.1 Statement of the problem.- 3.3.2 Asymptotics of the solution near a cone vertex.- 3.3.3 Formulas for coefficients in the asymptotics of solution (under simplified assumptions).- 3.3.4 Formula for coefficients in the asymptotics of solution (general case).- 3.3.5 Index of the boundary value problem.- 4 Asymptotics of Solutions to General Elliptic Boundary Value Problems in Domains Perturbed Near Cone Vertices.- 4.1 Formulation of the Boundary Value Problems and some Preliminary Considerations.- 4.1.1 The domains.- 4.1.2 Admissible scalar differential operators.- 4.1.3 Limit operators.- 4.1.4 Matrices of differential operators.- 4.1.5 Boundary value problems.- 4.1.6 Function spaces with norms depending on the parameter ?.- 4.2 Transformation of the Perturbed Boundary Value Problem into a System of Equations and a Theorem about the Index.- 4.2.1 The limit operator.- 4.2.2 Reduction of the problem to a system.- 4.2.3 Reconstruction of the original problem from the system.- 4.2.4 Fredholm property for the operator of the boundary value problem in a domain with singularly perturbed boundary.- 4.2.5 On the index of the original problem.- 4.3 Asymptotic Expansions of Data in the Boundary Value Problem.- 4.3.1 Asymptotic expansion of the coefficients and the right-hand sides.- 4.3.2 Asymptotic formulas for solutions of the limit problems.- 4.3.3 Asymptotic expansions of operators of the boundary value problem.- 4.3.4 Preliminary description of algorithm for construction of the asymptotics of solutions.- 4.3.5 The set of exponents in asymptotics of solutions of the limit problems.- 4.3.6 Formal expansion for the operator in powers of small parameter.- 4.4 Construction and Justification of the Asymptotics of Solution of the Boundary Value Problem.- 4.4.1 The problem in matrix notation.- 4.4.2 Auxiliary operators and their properties.- 4.4.3 Formal asymptotics of the solution in the case of uniquely solvable limit problems.- 4.4.4 A particular basis in the cokernel of the operator M0.- 4.4.5 Formal solution in the case of non-unique solvability of the limit problems.- 4.4.6 Asymptotics of the solution of the singularly perturbed problem.- 5 Variants and Corollaries of the Asymptotic Theory.- 5.1 Estimates of Solutions of the Dirichlet Problem for the Helmholtz Operator in a Domain with Boundary Smoothened Near a Corner.- 5.2 Sobolev Boundary Value Problems.- 5.3 General Boundary Value Problem in a Domain with Small Holes.- 5.4 Problems with Non-Smooth and Parameter Dependent Data.- 5.4.1 The case of a non-smooth domain.- 5.4.2 The case of parameter dependent auxiliary problems.- 5.4.3 The case of a parameter independent domain.- 5.5 Non-Local Perturbation of a Domain with Cone Vertices.- 5.5.1 Perturbations of a domain with smooth boundary.- 5.5.2 Regular perturbation of a domain with a corner.- 5.5.3 A non-local singular perturbation of a planar domain with a corner.- 5.6 Asymptotics of Solutions to Boundary Value Problems in Long Tubular Domains.- 5.6.1 The problem.- 5.6.2 Limit problems.- 5.6.3 Solvability of the original problem.- 5.6.4 Expansion of the right-hand sides and the set of exponents in the asymptotics.- 5.6.5 Redistribution of defects.- 5.6.6 Coefficients in the asymptotic series.- 5.6.7 Estimate of the remainder term.- 5.6.8 Example.- 5.7 Asymptotics of Solutions of a Quasi-Linear Equation in a Domain with Singularly Perturbed Boundary.- 5.7.1 A three-dimensional domain with a small gap.- 5.7.2 A planar domain with a small gap.- 5.7.3 A domain smoothened near a corner point.- 5.8 Bending of an Almost Polygonal Plate with Freely Supported Boundary.- 5.8.1 Boundary value problems in domains with corners.- 5.8.2 A singularly perturbed domain and limit problems.- 5.8.3 The principal term in the asymptotics.- 5.8.4 The principal term in the asymptotics (continued).- 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.- 6.1 Dirichlet’s Problem for Laplace’s Operator.- 6.1.1 Statement of the problem.- 6.1.2 Asymptotic behaviour of the coefficient C+03B5;.- 6.1.3 Justification of the asymptotic formula for the coefficient C+03B5;.- 6.1.4 The case g ? 0.- 6.1.5 The two-dimensional case.- 6.2 Neumann’s Problem for Laplace’s Operator.- 6.2.1 Statement of the problem.- 6.2.2 Boundary value problems.- 6.2.3 The case of disconnected boundary.- 6.2.4 The case of connected boundary.- 6.3 Intensity Factors for Bending of a Thin Plate with a Crack.- 6.3.1 Statement of the problem.- 6.3.2 Clamped cracks (The asymptotic behaviour near crack tips).- 6.3.3 Fixedly clamped cracks (Asymptotic behaviour of the intensity factors).- 6.3.4 Freely supported cracks.- 6.3.5 Free cracks (The asymptotic behaviour of solution near crack vertices).- 6.3.6 Free cracks (The asymptotic behaviour of intensity factors).- 6.4 Antiplanar and Planar Deformations of Domains with Cracks.- 6.4.1 Torsion of a bar with a longitudinal crack.- 6.4.2 The two-dimensional problem of the elasticity theory in a domain with collinear close cracks.- 7 Asymptotic Behaviour of Energy Integrals for Small Perturbations of the Boundary Near Corners and Isolated Points.- 7.1 Asymptotic Behaviour of Solutions of the Perturbed Problem.- 7.1.1 The unperturbed boundary value problem.- 7.1.2 Perturbed problem.- 7.1.3 The second limit problem.- 7.1.4 Asymptotic behaviour of solutions of the perturbed problem.- 7.1.5 The case of right-hand sides localized near a point.- 7.2 Asymptotic Behaviour of a Bilinear Form.- 7.2.1 The asymptotic behaviour of a bilinear form (the general case).- 7.2.2 Asymptotic behaviour of a bilinear form for right-hand sides localized near a point.- 7.2.3 Asymptotic behaviour of a quadratic form.- 7.3 Asymptotic Behaviour of a Quadratic Form for Problems in Regions with Small Holes.- 7.3.1 Statement of the problem.- 7.3.2 The case of uniquely solvable boundary problems.- 7.3.3 The case of the critical dimension.- 8 Asymptotic Behaviour of Energy Integrals for Particular Problems of Mathematical Physics.- 8.1 Dirichlet’s Problem for Laplace’s Operator.- 8.1.1 Perturbation of a domain near a corner or conic point.- 8.1.2 The case of right-hand, sides depending on ?.- 8.1.3 The case of right-hand sides depending on x and ?.- 8.1.4 Dirichlet’s problem for Laplace’s operator in a domain with a small hole.- 8.1.5 Refinement of the asymptotic behaviour.- 8.1.6 Two-dimensional domains with a small hole.- 8.1.7 Dirichlet’s problem for Laplace’s operator in domains with several small holes.- 8.2 Neumann’s Problem in Domains with one Small Hole.- 8.3 Dirichlet’s Problem for the Biharmonic Equation in a Domain with Small Holes.- 8.4 Variation of Energy Depending on the Length of Crack.- 8.4.1 The antiplanar deformation.- 8.4.2 A problem in the two-dimensional elasticity.- 8.5 Remarks on the Behaviour of Solutions of Problems in the Two-dimensional Elasticity Near Corner Points.- 8.5.1 Statement of problems.- 8.5.2 The asymptotic behaviour of solutions of the antiplanar deformation problem.- 8.5.3 Asymptotic behaviour of solutions of the planar deformation problem.- 8.5.4 Boundary value problems in unbounded domains.- 8.6 Derivation of Asymptotic Formulas for Energy.- 8.6.1 Statement of problems.- 8.6.2 Antiplanar deformation.- 8.6.3 Planar deformation.- 8.6.4 Refinement of the asymptotic formula for energy.- 8.6.5 Defect in the material near vertex of the crack.- IV Asymptotic Behaviour of Eigenvalues of Boundary Value Problems in Domains with Small Holes.- 9 Asymptotic Expansions of Eigenvalues of Classic Boundary Value Problems.- 9.1 Asymptotic Behaviour of the First Eigenvalue of a Mixed Boundary Value Problem.- 9.1.1 Statement of the problem.- 9.1.2 The three-dimensional case (formal asymptotic representation).- 9.1.3 The planar case (formal asymptotic representation).- 9.1.4 Justification of asymptotic expansions in the three-dimensional case.- 9.1.5 Justification of asymptotic expansions in the two-dimensional case.- 9.2 Asymptotic Expansions of Eigenvalues of Other Boundary Value Problems.- 9.2.1 Dirichlet’s 2m.- 10.1.3 The case n - 1 = 2m.- 10.2 Inversion of the Principal Part of an Operator Pencil on the Unit Sphere with a Small Hole. An Auxiliary Problem with Matrix Operator.- 10.2.1 “Nearly inverse” operator (the case 2m < n — 1).- 10.2.2 “Nearly inverse” operator (the case 2m = n — 1).- 10.2.3 Reduction to a problem with a matrix operator (the case 2m < n — 1).- 10.2.4 Reduction to a problem with a matrix operator (the case 2m = n — 1).- 10.3 Justification of the Asymptotic Behaviour of Eigenvalues (The Case 2m < n — 1).- 10.4 Justification of the Asymptotic Behaviour of Eigenvalues (The Case 2m = n — 1).- 10.5 Examples and Corollaries.- 10.5.1 A scalar operator.- 10.5.2 Lamé’s and Stokes’ systems.- 10.5.3 Continuity at the cone vertex of solution of Dirichlet’s problem.- 10.6 Examples of Discontinuous Solutions to Dirichlet’s Problem in Domains with a Conic Point.- 10.6.1 Equation of second order with discontinuous solutions.- 10.6.2 Dirichlet’s problem for an elliptic equation of the fourth order with real coefficients.- 10.7 Singularities of Solutions of Neumann’s Problem.- 10.7.1 Introduction.- 10.7.2 Formal asymptotic representation.- 10.8 Justification of the Asymptotic Formulas.- 10.8.1 Multiplicity of the spectrum near the point ? = 2.- 10.8.2 Nearly inverse operator for Neumann’s problem in G?.- 10.8.3 Justification of asymptotic representation of eigenvalues.- 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.
