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Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations
V. A. Kozlov and V. G. Maz'ya, University of Linköping, Sweden, and J. Rossmann, University of Rostock, Germany
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Mathematical Surveys and Monographs
2001; 436 pp; hardcover
Volume: 85
ISBN-10: 0-8218-2727-8
ISBN-13: 978-0-8218-2727-7
List Price: US$110
Member Price: US$88
Order Code: SURV/85
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Elliptic Equations in Polyhedral Domains - Vladimir Maz'ya and Jurgen Rossmann

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.

Readership

Graduate students and research mathematicians interested in PDEs, spectral analysis, asymptotic methods and their applications, numerical analysis, mathematical elasticity, and hydrodynamics.

Table of Contents

  • Introduction
Singularities of solutions to equations of mathematical physics
  • Prerequisites on operator pencils
  • Angle and conic singularities of harmonic functions
  • The Dirichlet problem for the Lamé system
  • Other boundary value problems for the Lamé system
  • The Dirichlet problem for the Stokes system
  • Other boundary value problems for the Stokes system in a cone
  • The Dirichlet problem for the biharmonic and polyharmonic equations
Singularities of solutions to general elliptic equations and systems
  • The Dirichlet problem for elliptic equations and systems in an angle
  • Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle
  • The Dirichlet problem for strongly elliptic systems in particular cones
  • The Dirichlet problem in a cone
  • The Neumann problem in a cone
  • Bibliography
  • Index
  • List of symbols
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