Spectral problems associated with corner singularities of solutions to elliptic equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2001]
|
| Series: | Mathematical surveys and monographs
Volume 85 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009500044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | ix, 436 Seiten |
| ISBN: | 0821827278 9780821827277 |
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| 245 | 1 | 0 | |a Spectral problems associated with corner singularities of solutions to elliptic equations |c V. A. Kozlov ; V. G. Maz'ya ; J. Rossmann |
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| 490 | 1 | |a Mathematical surveys and monographs |v Volume 85 | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Boundary value problems |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Elliptic |x Numerical solutions | |
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Record in the Search Index
| DE-BY-TUM_call_number | 0102 MAT 358 2008 B 1613 |
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Contents
Introduction 1
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
1.5. Notes 31
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
2.7. Notes 57
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 T and Fv 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
3.9. Notes 105
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
4.5. Notes 138
Chapter 5. The Dirichlet problem for the Stokes system 139
vii
viii CONTENTS
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 A=l and A = 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
5.9. Notes 196
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
6.4. Notes 225
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 A2 in domains with piecewise smooth
boundaries 246
7.5. Notes 248
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 torn 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
8.8. Notes 291
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
9.3. Notes 305
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
CONTENTS ix
10.2. Elliptic systems in Rn 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
10.6. Notes 344
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
11.8. Further results 384
11.9. The Dirichlet problem in domains with conic vertices 386
11.10. Notes 387
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
12.6. Notes 415
Bibliography 417
Index 429
List of Symbols 433 |
| any_adam_object | 1 |
| author | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Roßmann, Jürgen 1954- |
| author_GND | (DE-588)103679014 (DE-588)121490602 (DE-588)122254856 |
| author_facet | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Roßmann, Jürgen 1954- |
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| id | DE-604.BV013886336 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-15T08:00:59Z |
| institution | BVB |
| isbn | 0821827278 9780821827277 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009500044 |
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| physical | ix, 436 Seiten |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spellingShingle | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Roßmann, Jürgen 1954- Spectral problems associated with corner singularities of solutions to elliptic equations Mathematical surveys and monographs Mathematische Physik Boundary value problems Numerical solutions Differential equations, Elliptic Numerical solutions Mathematical physics Singularities (Mathematics) Singularität Mathematik (DE-588)4077459-4 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4048395-2 (DE-588)4014485-9 (DE-588)4128130-5 |
| title | Spectral problems associated with corner singularities of solutions to elliptic equations |
| title_auth | Spectral problems associated with corner singularities of solutions to elliptic equations |
| title_exact_search | Spectral problems associated with corner singularities of solutions to elliptic equations |
| title_full | Spectral problems associated with corner singularities of solutions to elliptic equations V. A. Kozlov ; V. G. Maz'ya ; J. Rossmann |
| title_fullStr | Spectral problems associated with corner singularities of solutions to elliptic equations V. A. Kozlov ; V. G. Maz'ya ; J. Rossmann |
| title_full_unstemmed | Spectral problems associated with corner singularities of solutions to elliptic equations V. A. Kozlov ; V. G. Maz'ya ; J. Rossmann |
| title_short | Spectral problems associated with corner singularities of solutions to elliptic equations |
| title_sort | spectral problems associated with corner singularities of solutions to elliptic equations |
| topic | Mathematische Physik Boundary value problems Numerical solutions Differential equations, Elliptic Numerical solutions Mathematical physics Singularities (Mathematics) Singularität Mathematik (DE-588)4077459-4 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Mathematische Physik Boundary value problems Numerical solutions Differential equations, Elliptic Numerical solutions Mathematical physics Singularities (Mathematics) Singularität Mathematik Randwertproblem Elliptische Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009500044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT kozlovvladimir spectralproblemsassociatedwithcornersingularitiesofsolutionstoellipticequations AT mazʹjavladimirgilelevic spectralproblemsassociatedwithcornersingularitiesofsolutionstoellipticequations AT roßmannjurgen spectralproblemsassociatedwithcornersingularitiesofsolutionstoellipticequations |
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