Carleman inequalities: an introduction and more
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2019]
|
| Series: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
volume 353 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxvii, 557 Seiten Diagramme |
| ISBN: | 9783030159924 |
Staff View
MARC
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| 100 | 1 | |a Lerner, Nicolas |d 1953- |0 (DE-588)140393692 |4 aut | |
| 245 | 1 | 0 | |a Carleman inequalities |b an introduction and more |c Nicolas Lerner |
| 264 | 1 | |a Cham |b Springer |c [2019] | |
| 264 | 4 | |c © 2019 | |
| 300 | |a xxvii, 557 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |v volume 353 | |
| 650 | 4 | |a Inequalities (Mathematics) | |
| 650 | 4 | |a Carleman theorem | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-030-15993-1 |
| 830 | 0 | |a Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |v volume 353 |w (DE-604)BV000000395 |9 353 | |
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Record in the Search Index
| _version_ | 1804180165817794560 |
|---|---|
| adam_text | Contents
1 Prolegomena 1
1 1 Preliminaries 1
1 2 Hyperbolicity, the Energy Method and Well-Posedness 2
1 3 The Lax-Mizohata Theorems 10
131 Strictly Hyperbolic Operators 10
132 Ill-Posedness Examples 14
1 4 Holmgren’s Uniqueness Theorems 17
1 5 Carleman’s Method Displayed on a Simple Example 18
151 The 9 Equation 18
152 The Laplace Equation 21
2 A Toolbox for Carleman Inequalities 27
2 1 Weighted Inequalities 27
2 2 Conjugation 33
2 3 Sobolev Spaces with Parameter 35
2 4 The Symbol of the Conjugate 39
2 5 Choice of the Weight 45
3 Operators with Simple Characteristics: Calderon’s Theorems 47
3 1 Introduction 47
3 2 Inequalities for Symbols 51
33A Carleman Inequality 53
3 4 Examples 56
341 Second-Order Real Elliptic Operators 56
342 Strictly Hyperbolic Operators 58
343 Products 59
344 Generalizations of Calderon’s Theorems 61
3 5 Cutting the Regularity Requirements 62
xxiii
XXIV
Contents
4 Pseudo-convexity: Hömiander’s Theorems 69
4 1 Introduction 69
4 2 Inequalities for Symbols 73
4 3 Pseudo-convexity 79
431 Carleman Inequality, Definition 79
432 Invariance Properties of Strong Pseudo-convexity 80
433 Unique Continuation 82
4 4 Examples 83
441 Pseudoconvexity for Real Second-Order Operators 84
442 The Tricomi Operator 84
443 Constant Coefficients 85
444 The Characteristic Case 87
4 5 Remarks and Open Problems 88
451 Stability Under Perturbations 88
452 Higher Order Tangential Bicharacteristics 89
453A Direct Method for Proving Carleman Estimates? 91
5 Complex Coefficients and Principal Normality 93
5 1 Introduction 93
511 Complex-Valued Symbols 93
512 Principal Normality 94
513 Our Strategy for the Proof 97
5 2 Pseudo-convexity and Principal Normality 99
521 Pseudo-Convexity for Principally Normal Operators 99
522 Inequalities for Symbols 102
523 Inequalities for Elliptic Symbols 106
5 3 Unique Continuation via Pseudo-convexity 108
5 4 Unique Continuation for Complex Vector Fields 109
541 Warm-Up: Studying a Simple Model 109
542 Carleman Estimates in Two Dimensions 113
543 Unique Continuation in Two Dimensions 119
544 Unique Continuation Under Condition (P) 122
5 5 Counterexamples for Complex Vector Fields 125
551 Main Result 125
552 Explaining the Counterexample 134
553 Comments 136
6 On the Edge of Pseudo-convexity 137
6 1 Preliminaries 137
611 Real Geometrical Optics 137
612 Complex Geometrical Optics 141
6 2 The Alinhac-Baouendi Non-uniqueness Result 144
621 Statement of the Result 144
622 Proof of Theorem 6 6 145
Contents
xxv
6 3 Non-uniqueness for Analytic Non-linear Systems 167
631 Preliminaries 167
632 Proof of Theorem 6 27 169
6 4 Compact Uniqueness Results 173
641 Preliminaries 173
642 The Result 174
643 The Proof 174
6 5 Remarks, Open Problems and Conjectures 187
651 Finite Type Conditions for Actual Uniqueness 187
652 Ill-Posed Problems with Real-Valued Solutions 192
7 Operators with Partially Analytic Coefficients 195
7 1 Preliminaries 195
711 Motivations 195
712 Between Holmgren’s and Hörmander’s Theorems 196
713 Some Invariant Assumptions 197
7 2 Operators with Real Coefficients 198
73A Modification of Carleman’s Method 199
731 Gaussian Mollifiers and Supports 199
732 Conjugation 200
733 Some Technical Lemmas 201
734 Conormal Pseudo-convexity and Carleman
Estimates 204
735 Proof of Theorem 7 2 206
7 4 An Improvement of Theorem 7 2 212
7 5 Transversally Elliptic Operators 218
751 Statement of the Result 218
752 More Technical Lemmas 218
753 Inequalities for Transversally Elliptic Symbols 224
754 Modified Transversally Elliptic Symbols 226
755 Proof of Theorem 7 26 231
8 Strong Unique Continuation Properties for Elliptic Operators 237
8 1 Radial Potentials 237
811 Preliminaries 237
812 Radial Potentials x ~2, |x|_l 238
813 Proofs 240
814 Kato Potentials 270
815 Additional Remarks on Radial Potentials 276
8 2 Laplace Operator, Ln!2 Potential 277
821 Statement of the Results 277
822 Proof of the Main Result 278
823 Extensions and Remarks 283
XXVI
Contents
8 3 The Dirac Operator, Square Root of the Laplace Operator 284
831A Counterexample for the Dirac Operator 284
832 On the Scalar Square-Root of the Laplace Operator 289
8 4 On Wolff’s Modification of Carleman’s Method 297
841 Introduction 297
842 Wolff’s Measure-Theoretic Lemma 298
8 5 Carleman-Type Inequalities and Unique Continuation 331
851 Some Inequalities 331
852 Weak Unique Continuation Results 336
853 Continuation with Respect to Sets of Positive
Measure 337
854 Proof of Theorem 8 89 338
855 Complementary Remarks 353
9 Carleman Estimates via Brenner’s Theorem and Strichartz
Estimates 355
9 1 Preliminaries 355
9 2 Strichartz Estimates for Real Principal-Type Operators 356
921 Classical Pseudo-differential Operators 356
922 Strichartz Estimates 358
923 Proof of Theorem 9 10 359
9 3 Preliminaries for a Unique Continuation Theorem 370
931 General Setting 370
932 Factorization Arguments 372
9 4 Unique Continuation Results 373
941 Statement of the Results 373
942 The Strictly Hyperbolic Case 376
9 5 Comments and Additional Results 378
951 Complex Roots, Positive Elliptic Imaginary Part 378
952 Complex Roots, Negative Elliptic Imaginary Part 378
10 Elliptic Operators with Jumps; Conditional Pseudo-convexity 381
10 1 Introduction to Elliptic Operators with Jumps 381
10 1 1 Preliminaries 381
10 1 2 Jump Discontinuities 381
10 1 3 Framework 382
10 2 A Carleman Estimate for Elliptic Operators with Jumps 384
10 2 1 Proof for a Model Case 385
10 3 Comments 395
10 3 1 Condition (¥) 395
10 3 2 Quasi-mode Construction 402
Contents xxvii
10 4 Open Problems 404
10 41A BV Elliptic Matrix 404
10 4 2 An Elliptic Matrix with Infinitely Many Jumps 405
10 4 3 Strong Unique Continuation 405
10 5 Conditional Pseudo-convexity 406
10 5 1 The Result 406
10 52A More General Result 407
10 5 3 Proof of Theorem 10 20 408
10 5 4 Comments 413
10 5 5 The Lorentzian Geometry Setting 413
11 Perspectives and Developments 415
11 1 Parabolic Equations 415
11 1 1 On Tychonoff’s Example 415
11 1 2 Backward Parabolic Equations 417
11 2 Control Theory 423
11 2 1 The Heat Equation 423
11 2 2 The F John and H Bahouri Method 426
11 3 Inverse Problems 429
11 4 Spectral Theory 433
11 41A Global Carleman Estimate 433
11 4 2 Absence of Embedded Eigenvalues 434
11 4 3 Absence of Embedded Eigenvalues, Continued 436
11 5 Fluid Mechanics 437
11 5 1 Regularity Results for the Navier-Stokes System 437
11 5 2 Unique Continuation for the Stokes System 438
Appendix A: Elements of Fourier Analysis 443
Appendix B: Miscellanea 491
References 545
|
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| id | DE-604.BV045954078 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:31:26Z |
| institution | BVB |
| isbn | 9783030159924 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-031336103 |
| oclc_num | 1107354346 |
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| owner_facet | DE-20 DE-188 DE-11 DE-83 DE-19 DE-BY-UBM |
| physical | xxvii, 557 Seiten Diagramme |
| publishDate | 2019 |
| publishDateSearch | 2019 |
| publishDateSort | 2019 |
| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Lerner, Nicolas 1953- (DE-588)140393692 aut Carleman inequalities an introduction and more Nicolas Lerner Cham Springer [2019] © 2019 xxvii, 557 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics volume 353 Inequalities (Mathematics) Carleman theorem Erscheint auch als Online-Ausgabe 978-3-030-15993-1 Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics volume 353 (DE-604)BV000000395 353 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031336103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lerner, Nicolas 1953- Carleman inequalities an introduction and more Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics Inequalities (Mathematics) Carleman theorem |
| title | Carleman inequalities an introduction and more |
| title_auth | Carleman inequalities an introduction and more |
| title_exact_search | Carleman inequalities an introduction and more |
| title_full | Carleman inequalities an introduction and more Nicolas Lerner |
| title_fullStr | Carleman inequalities an introduction and more Nicolas Lerner |
| title_full_unstemmed | Carleman inequalities an introduction and more Nicolas Lerner |
| title_short | Carleman inequalities |
| title_sort | carleman inequalities an introduction and more |
| title_sub | an introduction and more |
| topic | Inequalities (Mathematics) Carleman theorem |
| topic_facet | Inequalities (Mathematics) Carleman theorem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031336103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT lernernicolas carlemaninequalitiesanintroductionandmore |