**Mathematical Surveys and Monographs**

Volume: 85;
2001;
436 pp;
Hardcover

MSC: Primary 31; 35; 47; 74;

**Print ISBN: 978-0-8218-2727-7
Product Code: SURV/85**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-1312-5
Product Code: SURV/85.E**

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# Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations

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*V. A. Kozlov; V. G. Maz’ya; J. 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

# Table of Contents

## Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations

- Contents vii8 free
- Introduction 112 free
- Part 1. Singularities of solutions to equations of mathematical physics 718 free
- Chapter 1. Prerequisites on operator pencils 920
- Chapter 2. Angle and conic singularities of harmonic functions 3546
- 2.1. Boundary value problems for the Laplace operator in an angle 3647
- 2.2. The Dirichlet problem for the Laplace operator in a cone 4051
- 2.3. The Neumann problem for the Laplace operator in a cone 4556
- 2.4. The problem with oblique derivative 4960
- 2.5. Further results 5263
- 2.6. Applications to boundary value problems for the Laplace equation 5465
- 2.7. Notes 5768

- Chapter 3. The Dirichlet problem for the Lamé system 6172
- 3.1. The Dirichlet problem for the Lamé system in a plane angle 6475
- 3.2. The operator pencil generated by the Dirichlet problem in a cone 7485
- 3.3. Properties of real eigenvalues 8394
- 3.4. The set functions Γ and F[sub(v)] 8899
- 3.5. A variational principle for real eigenvalues 91102
- 3.6. Estimates for the width of the energy strip 93104
- 3.7. Eigenvalues for circular cones 97108
- 3.8. Applications 100111
- 3.9. Notes 105116

- Chapter 4. Other boundary value problems for the Lamé system 107118
- Chapter 5. The Dirichlet problem for the Stokes system 139150
- 5.1. The Dirichlet problem for the Stokes system in an angle 142153
- 5.2. The operator pencil generated by the Dirichlet problem in a cone 148159
- 5.3. Properties of real eigenvalues 155166
- 5.4. The eigenvalues λ=1 and λ=…2 159170
- 5.5. A variational principle for real eigenvalues 168179
- 5.6. Eigenvalues in the case of right circular cones 175186
- 5.7. The Dirichlet problem for the Stokes system in a dihedron 178189
- 5.8. Stokes and Navier–Stokes systems in domains with piecewise smooth boundaries 192203
- 5.9. Notes 196207

- Chapter 6. Other boundary value problems for the Stokes system in a cone 199210
- Chapter 7. The Dirichlet problem for the biharmonic and polyharmonic equations 227238

- Part 2. Singularities of solutions to general elliptic equations and systems 251262
- Chapter 8. The Dirichlet problem for elliptic equations and systems in an angle 253264
- 8.1. The operator pencil generated by the Dirichlet problem 254265
- 8.2. An asymptotic formula for the eigenvalue close to m 263274
- 8.3. Asymptotic formulas for the eigenvalues close to m – 1/2 265276
- 8.4. The case of a convex angle 272283
- 8.5. The case of a nonconvex angle 275286
- 8.6. The Dirichlet problem for a second order system 283294
- 8.7. Applications 286297
- 8.8. Notes 291302

- Chapter 9. Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle 293304
- Chapter 10. The Dirichlet problem for strongly elliptic systems in particular cones 307318
- 10.1. Basic properties of the operator pencil generated by the Dirichlet problem 308319
- 10.2. Elliptic systems in R[sup(n)] 313324
- 10.3. The Dirichlet problem in the half-space 319330
- 10.4. The Sobolev problem in the exterior of a ray 321332
- 10.5. The Dirichlet problem in a dihedron 332343
- 10.6. Notes 344355

- Chapter 11. The Dirichlet problem in a cone 345356
- 11.1. The case of a "smooth" cone 346357
- 11.2. The case of a nonsmooth cone 350361
- 11.3. Second order systems 353364
- 11.4. Second order systems in a polyhedral cone 365376
- 11.5. Exterior of a thin cone 368379
- 11.6. A cone close to the half-space 376387
- 11.7. Nonrealness of eigenvalues 383394
- 11.8. Further results 384395
- 11.9. The Dirichlet problem in domains with conic vertices 386397
- 11.10. Notes 387398

- Chapter 12. The Neumann problem in a cone 389400

- Bibliography 417428
- Index 429440
- List of Symbols 433444