Elliptic Problems in Domains with Piecewise Smooth Boundaries

✍ Scribed by Sergey Nazarov; Boris A. Plamenevsky

• Publisher: De Gruyter
• Year: 1994
• Tongue: English
• Leaves: 536
• Series: De Gruyter Expositions in Mathematics; 13
• Category: Library

✦ Table of Contents

Introduction
Chapter 1. Elliptic boundary value problems in domains with smooth boundary
§1. Elliptic boundary value problems
§ 2. Elliptic problems with a parameter
Chapter 2. Dirichlet and Neumann problems for the Laplace operator in plane domains with corner points at the boundary
§ 1. Generalized solutions of the Dirichlet and Neumann problems in domains with corner points. Deviations from the "usual" properties of solutions
§ 2. Solvability and asymptotics of solutions of model problems in a strip and in an angle
§ 3. The Dirichlet problem in a bounded domain with a corner point
§4. The Neumann problem in a bounded domain with a corner point
Chapter 3. General elliptic boundary value problems in a cylinder and in a cone
§ 1. Solvability and asymptotics of solutions of boundary value problems in a cylinder with the coefficients constant along the axis
§ 2. Calculation of the coefficients cv(k,j) in the asymptotics of the solution
§ 3. Asymptotics of the solutions of boundary value problems in a cylinder with right-hand side of a special form
§4. Solvability and asymptotics of solutions of the boundary value problems with periodic coefficients
§ 5. Boundary value problems in a cone
§ 6. Estimates and asymptotics of solutions in Lp and Holder classes
§ 7. Fundamental solutions of the boundary value problem in a cone
§ 8. Two examples: The Dirichlet problem and the oblique derivative problem
Chapter 4. General elliptic problems in domains with conical points
§1. The Fredholm property
§2. Asymptotics of solutions near conical points
§ 3. Expressions for the coefficients in asymptotic formulas. Properties of the index
§4. Asymptotics of fundamental solutions
§ 5. Boundary value problems in spaces with nonhomogeneous norms
Chapter 5. Self-adjoint problems in domains with outlets to infinity
§ 1. Self-adjoint problems in domains with cylindrical outlets to infinity
§ 2. Special choices of Jordan chains of self-adjoint operator pencils
§ 3. Waves, scattering matrices and intrinsic radiation conditions
§4. Energetic solutions and the polarization matrix
§ 5. The extensions of the symmetric operator
§6. On a problem of the scattering of electro-magnetic waves
§ 7. Elasticity problems in domains having cylindrical outlets to infinity
§ 8. Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity
Chapter 6. Self-adjoint problems in domains with conical points. Applications
§ 1. Self-adjoint problems. Properties of their pencils
§ 2. The generalized Green formula, radiation conditions
§ 3. Self-adjoint extensions in spaces with weighted norms
§ 4. Boundary value problems in domains with compact complement. The polarization matrix
§ 5. Asymptotic analysis of the problem in domains with small holes
§ 6. Two problems in plates theory
Chapter 7. Applications to crack theory
§ 1. Stress singularities at a crack tip
§2. The rupture criterion
§ 3. The Cherepanov-Rice integral
§4. On the structure of the spectrum of the pencil generated by a self-adjoint problem in a plane domain with a crack
Chapter 8. Elliptic problems in domains with smooth edges
§ 1. Statement of the problem. Model problems
§ 2. Analysis of the model problems
§ 3. The Fredholm property of the elliptic problem in a bounded domain
§4. The Dirichlet and the Neumann problems for self-adjoint systems
§5. Lp-estimates of solutions
§ 6. Estimates of solutions in weighted Holder classes
Chapter 9. Elliptic problems on manifolds with intersecting edges
§1. Manifolds of the class D
§ 2. Differential operators on class D manifolds
§3. The function spaces
§4. Statement of the boundary problems. Model problems
§ 5. Boundary value problems on class D manifolds
§ 6. The Dirichlet and Neumann problems for self-adjoint systems
Chapter 10. Asymptotics of solutions of the Dirichlet problem for the Laplace operator in a three-dimensional domain having edges on the boundary
§ 1. Asymptotics of solution near smooth edges
§ 2. Asymptotics of solutions near the singular point having the type of the vertex of a polyhedron
§ 3. Formulas for coefficients in asymptotics of solutions near an edge
Chapter 11. The asymptotics of solutions of general problems near edges
§ 1. Guidelines for constructing asymptotics. Auxiliary assertions
§2. The asymptotics of solutions near an edge (in case of a smooth right-hand side)
§ 3. Asymptotics of solutions near an edge (general case)
§ 4. Corollaries of asymptotic formulas
Chapter 12. Self-adjoint problems with radiation conditions on the edges
§ 1. Setting of boundary value problems and information about the asymptotics of solutions
§ 2. The generalized Green formula. Fredholm property of the problem with radiation conditions
§ 3. Solvability conditions for the problem with radiation conditions in terms of the generalized Green formula
§4. Intrinsic radiation conditions on the edge
§ 5. Energetic radiation conditions. The polarization operator on the edge
§ 6. The scattering operator. The formula for the increment of the dimension of the kernel of the operator of the problem generated by the variation of the weight factor
§ 7. Sobolev problems
§ 8. The three-dimensional problem of crack mechanics
Bibliographical notes
References
Index
List of spaces