Spectral problems associated with corner singularities of solutions of elliptic equations / by V.A. Kozlov, V.G. Maz'ya and J. Rossmann.

Format:

Book

Author/Creator:

Kozlov, Vladimir, 1954- author.

Mazʹi︠a︡, V. G., author.

Rossmann, J. (Jürgen), 1954- author.

Series:

Mathematical surveys and monographs ; no. 85.

Mathematical surveys and monographs, 0076-5376 ; volume 85

Language:

English

Subjects (All):

Differential equations, Elliptic--Numerical solutions.

Differential equations, Elliptic.

Boundary value problems--Numerical solutions.

Boundary value problems.

Singularities (Mathematics).

Mathematical physics.

Physical Description:

1 online resource (449 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2001]

Language Note:

English

Summary:

This book focuses on the analysis of eigenvalues and eigenfunctions that describe singularities of solutions to elliptic boundary value problems in domains with corners and edges. The authors treat both classical problems of mathematical physics and general elliptic boundary value problems. The volume is divided into two parts: The first is devoted to the power-logarithmic singularities of solutions to classical boundary value problems of mathematical physics. The second deals with similar singularities for higher order elliptic equations and systems. Chapter 1 collects basic facts concerning operator pencils acting in a pair of Hilbert spaces. Related properties of ordinary differential equations with constant operator coefficients are discussed and connections with the theory of general elliptic boundary value problems in domains with conic vertices are outlined. New results are presented. Chapter 2 treats the Laplace operator as a starting point and a model for the subsequent study of angular and conic singularities of solutions. Chapter 3 considers the Dirichlet boundary condition beginning with the plane case and turning to the space problems. Chapter 4 investigates some mixed boundary conditions. The Stokes system is discussed in Chapters 5 and 6, and Chapter 7 concludes with the Dirichlet problem for the polyharmonic operator. Chapter 8 studies the Dirichlet problem for general elliptic differential equations of order 2m in an angle. In Chapter 9, an asymptotic formula for the distribution of eigenvalues of operator pencils corresponding to general elliptic boundary value problems in an angle is obtained. Chapters 10 and 11 discuss the Dirichlet problem for elliptic systems of differential equations of order 2 in an n-dimensional cone. Chapter 12 studies the Neumann problem for general elliptic systems, in particular with eigenvalues of the corresponding operator pencil in the strip $\mid {\Re} \lambda - m + /2n \mid \leq 1/2$. It is shown that only integer numbers contained in this strip are eigenvalues. Applications are placed within chapter introductions and as special sections at the end of chapters. Prerequisites include standard PDE and functional analysis courses.

Contents:

""Contents""; ""Introduction""; ""Part 1. Singularities of solutions to equations of mathematical physics""; ""Chapter 1. Prerequisites on operator pencils""; ""1.1. Operator pencils""; ""1.2. Operator pencils corresponding to sesquilinear forms""; ""1.3. A variational principle for operator pencils""; ""1.4. Elliptic boundary value problems in domains with conic points: some basic results""; ""1.5. Notes""; ""Chapter 2. Angle and conic singularities of harmonic functions""; ""2.1. Boundary value problems for the Laplace operator in an angle""

""5.1. The Dirichlet problem for the Stokes system in an angle""""5.2. The operator pencil generated by the Dirichlet problem in a cone""; ""5.3. Properties of real eigenvalues""; ""5.4. The eigenvalues λ=1 and λ=â€?2""; ""5.5. A variational principle for real eigenvalues""; ""5.6. Eigenvalues in the case of right circular cones""; ""5.7. The Dirichlet problem for the Stokes system in a dihedron""; ""5.8. Stokes and Navierâ€?Stokes systems in domains with piecewise smooth boundaries""; ""5.9. Notes""; ""Chapter 6. Other boundary value problems for the Stokes system in a cone""

""6.1. A mixed boundary value problem for the Stokes system""""6.2. Real eigenvalues of the pencil to the mixed problem""; ""6.3. The Neumann problem for the Stokes system""; ""6.4. Notes""; ""Chapter 7. The Dirichlet problem for the biharmonic and polyharmonic equations""; ""7.1. The Dirichlet problem for the biharmonic equation in an angle""; ""7.2. The Dirichlet problem for the biharmonic equation in a cone""; ""7.3. The polyharmonic operator""; ""7.4. The Dirichlet problem for Δ[sup(2)] in domains with piecewise smooth boundaries""; ""7.5. Notes""

""Part 2. Singularities of solutions to general elliptic equations and systems""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 417-428) and index.

Description based on print version record.

ISBN:

1-4704-1312-4