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

Format:

Book

Author/Creator:

Kozlov, Vladimir, 1954-

Contributor:

Mazʹi︠a︡, V. G.

Rossmann, J. (Jürgen), 1954-

Series:

Mathematical surveys and monographs ; no. 85.

Mathematical surveys and monographs, 0076-5376 ; v. 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:

ix, 436 pages : illustrations ; 27 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2001]

Contents:

Part 1. Singularities of solutions to equations of mathematical physics 7

Chapter 1. Prerequisites on operator pencils 9

1.1. Operator pencils 10

1.2. Operator pencils corresponding to sesquilinear forms 15

1.3. A variational principle for operator pencils 21

1.4. Elliptic boundary value problems in domains with conic points: some basic results 26

Chapter 2. Angle and conic singularities of harmonic functions 35

2.1. Boundary value problems for the Laplace operator in an angle 36

2.2. The Dirichlet problem for the Laplace operator in a cone 40

2.3. The Neumann problem for the Laplace operator in a cone 45

2.4. The problem with oblique derivative 49

2.5. Further results 52

2.6. Applications to boundary value problems for the Laplace equation 54

Chapter 3. The Dirichlet problem for the Lame system 61

3.1. The Dirichlet problem for the Lame system in a plane angle 64

3.2. The operator pencil generated by the Dirichlet problem in a cone 74

3.3. Properties of real eigenvalues 83

3.4. The set functions [Gamma] and F[subscript v] 88

3.5. A variational principle for real eigenvalues 91

3.6. Estimates for the width of the energy strip 93

3.7. Eigenvalues for circular cones 97

3.8. Applications 100

Chapter 4. Other boundary value problems for the Lame system 107

4.1. A mixed boundary value problem for the Lame system 108

4.2. The Neumann problem for the Lame system in a plane angle 120

4.3. The Neumann problem for the Lame system in a cone 125

4.4. Angular crack in an anisotropic elastic space 133

Chapter 5. The Dirichlet problem for the Stokes system 139

5.1. The Dirichlet problem for the Stokes system in an angle 142

5.2. The operator pencil generated by the Dirichlet problem in a cone 148

5.3. Properties of real eigenvalues 155

5.4. The eigenvalues [lambda]=1 and [lambda]=-2 159

5.5. A variational principle for real eigenvalues 168

5.6. Eigenvalues in the case of right circular cones 175

5.7. The Dirichlet problem for the Stokes system in a dihedron 178

5.8. Stokes and Navier-Stokes systems in domains with piecewise smooth boundaries 192

Chapter 6. Other boundary value problems for the Stokes system in a cone 199

6.1. A mixed boundary value problem for the Stokes system 200

6.2. Real eigenvalues of the pencil to the mixed problem 212

6.3. The Neumann problem for the Stokes system 223

Chapter 7. The Dirichlet problem for the biharmonic and polyharmonic equations 227

7.1. The Dirichlet problem for the biharmonic equation in an angle 229

7.2. The Dirichlet problem for the biharmonic equation in a cone 233

7.3. The polyharmonic operator 239

7.4. The Dirichlet problem for [Delta superscript 2] in domains with piecewise smooth boundaries 246

Part 2. Singularities of solutions to general elliptic equations and systems 251

Chapter 8. The Dirichlet problem for elliptic equations and systems in an angle 253

8.1. The operator pencil generated by the Dirichlet problem 254

8.2. An asymptotic formula for the eigenvalue close to m 263

8.3. Asymptotic formulas for the eigenvalues close to m - 1/2 265

8.4. The case of a convex angle 272

8.5. The case of a nonconvex angle 275

8.6. The Dirichlet problem for a second order system 283

8.7. Applications 286

Chapter 9. Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle 293

9.1. The operator pencil generated by a regular boundary value problem 293

9.2. Distribution of the eigenvalues 299

Chapter 10. The Dirichlet problem for strongly elliptic systems in particular cones 307

10.1. Basic properties of the operator pencil generated by the Dirichlet problem 308

10.2. Elliptic systems in R[superscript n] 313

10.3. The Dirichlet problem in the half-space 319

10.4. The Sobolev problem in the exterior of a ray 321

10.5. The Dirichlet problem in a dihedron 332

Chapter 11. The Dirichlet problem in a cone 345

11.1. The case of a "smooth" cone 346

11.2. The case of a nonsmooth cone 350

11.3. Second order systems 353

11.4. Second order systems in a polyhedral cone 365

11.5. Exterior of a thin cone 368

11.6. A cone close to the half-space 376

11.7. Nonrealness of eigenvalues 383

11.9. The Dirichlet problem in domains with conic vertices 386

Chapter 12. The Neumann problem in a cone 389

12.1. The operator pencil generated by the Neumann problem 391

12.2. The energy line 396

12.3. The energy strip 398

12.4. Applications to the Neumann problem in a bounded domain 411

12.5. The Neumann problem for anisotropic elasticity in an angle 414.

Notes:

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

ISBN:

0821827278

OCLC:

122978047