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.
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.
