Spectral problems associated with corner singularities of solutions to elliptic equations.

*(English)*Zbl 0965.35003
Mathematical Surveys and Monographs. 85. Providence, RI: American Mathematical Society (AMS). ix, 436 p. (2001).

The authors of this monograph study singularities of solutions to classical problems of mathematical physics as well as to general elliptic equations and systems. Such singularities may be caused, in particular, by irregularities of the boundary. Most of the results of the elliptic theory for nonsmooth domains are conditional: the singularities of solutions are described in terms of spectral properties of operator pencils (i.e. operators polynomially depending on a spectral parameter) of boundary value problems on spherical domains. That’s why one of the main difficulties is the investigation of the distribution of the eigenvalues of the associated operator pencil.

The book is divided into two parts, the first being devoted to the power-logarithmic singularities of solutions to classical boundary value problems of mathematical physics, and the second deals with similar singularities for higher-order elliptic equations and systems.

The contents of the book: Part 1. Singularities of solutions to equations of mathematical physics. Chapter 1. Prerequisites on operator pencils. Chapter 2. Angle and conic singularities of harmonic functions. Chapter 3. The Dirichlet problem for the Lamé system. Chapter 4. Other boundary value problems for the Lamé system. Chapter 5. The Dirichlet problem for the Stokes system. Chapter 6. Other boundary value problems for the Stokes system in a cone. Chapter 7. The Dirichlet problem for the biharmonic and polyharmonic equations.

Part 2. Singularities of solutions to general elliptic equations and systems. Chapter 8. The Dirichlet problem for elliptic equations and systems in an angle. Chapter 9. Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle. Chapter 10. The Dirichlet problem for strongly elliptic systems in particular cones. Chapter 11. The Dirichlet problem in a cone. Chapter 12. The Neumann problem in a cone.

Bibliography: 274 titles.

This book will be interesting and useful for mathematicians who work in partial differential equations, spectral analysis, asymptotic methods and their applications. It will be of use also for those who are interested in numerical analysis, mathematical elasticity and hydrodynamics. Prerequisites for this book are undergraduate courses in partial differential equations and functional analysis.

The book is divided into two parts, the first being devoted to the power-logarithmic singularities of solutions to classical boundary value problems of mathematical physics, and the second deals with similar singularities for higher-order elliptic equations and systems.

The contents of the book: Part 1. Singularities of solutions to equations of mathematical physics. Chapter 1. Prerequisites on operator pencils. Chapter 2. Angle and conic singularities of harmonic functions. Chapter 3. The Dirichlet problem for the Lamé system. Chapter 4. Other boundary value problems for the Lamé system. Chapter 5. The Dirichlet problem for the Stokes system. Chapter 6. Other boundary value problems for the Stokes system in a cone. Chapter 7. The Dirichlet problem for the biharmonic and polyharmonic equations.

Part 2. Singularities of solutions to general elliptic equations and systems. Chapter 8. The Dirichlet problem for elliptic equations and systems in an angle. Chapter 9. Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle. Chapter 10. The Dirichlet problem for strongly elliptic systems in particular cones. Chapter 11. The Dirichlet problem in a cone. Chapter 12. The Neumann problem in a cone.

Bibliography: 274 titles.

This book will be interesting and useful for mathematicians who work in partial differential equations, spectral analysis, asymptotic methods and their applications. It will be of use also for those who are interested in numerical analysis, mathematical elasticity and hydrodynamics. Prerequisites for this book are undergraduate courses in partial differential equations and functional analysis.

Reviewer: Z.G.Sheftel’ (Göttingen)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J40 | Boundary value problems for higher-order elliptic equations |

35R05 | PDEs with low regular coefficients and/or low regular data |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

46N20 | Applications of functional analysis to differential and integral equations |