Phase Space Analysis of Partial Differential Equations (eBook)

Antonio Bove, Ferruccio Colombini, Daniele Del Santo (Herausgeber)

eBook Download: PDF

2007
329 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4521-2 (ISBN)

Infos

Covers phase space analysis methods, including microlocal analysis, and their applications to physics

Treats the linear and nonnlinear aspects of the theory of PDEs

Original articles are self-contained with full proofs; survey articles give a quick and direct introduction to selected topics evolving at a fast pace

Excellent reference and resource for grad students and researchers in PDEs and related fields

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This collection of original articles and surveys treats the linear and nonlinear aspects of the theory of partial differential equations. Original articles are self-contained with full proofs; survey articles give a quick and direct introduction to selected topics evolving at a fast pace. Phase space analysis methods, including microlocal analysis, have yielded striking results in past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theories. Graduate students at various levels as well as researchers in PDEs and related fields will find this an excellent resource.

Inhalt

Preface 7
Contents 9
List of Contributors 11
Trace theorem on the Heisenberg group on homogeneous hypersurfaces 15
1 Introduction 15
2 A Hardy type inequality 20
3 The proof of the trace and trace lifting theorem 24
4 Concluding remarks 28
References 28
Strong unique continuation and finite jet determination for Cauchy – Riemann mappings 30
1 Introduction 30
2 Local coordinates 31
3 Nondegeneracy conditions 33
4 Necessary conditions and su.cient conditions for finite jet determination 35
5 Lie group structures and jet parameterization 37
References 40
On the Cauchy problem for some hyperbolic operator with double characteristics 42
1 Introduction and statements 42
2 The model operator 44
3 Shibuya solutions 45
4 Stokes multipliers 47
5 Asymptotic analysis 49
6 Final steps in the proof of the necessary condition 55
References 57
On the differentiability class of the admissible square roots of regular nonnegative functions 58
1 Introduction 58
2 Regularity of well-chosen admissible roots 59
References 66
The Benjamin–Ono equation in energy space 67
1 Introduction 67
2 Bourgain spaces 69
3 A priori estimate on weak solutions 70
4 The gauge transformation 71
5 The existence and uniqueness result 73
References 74
Instabilities in Zakharov equations for laser propagation in a plasma 75
1 Introduction 75
2 The instability mechanism 78
3 Scheme of the proof 80
4 The linear instability 84
5 The linear equation 89
6 End of proofs 92
References 93
Symplectic strata and analytic hypoellipticity 94
1 Introduction 94
2 The symplectic case 95
3 The example of Baouendi–Goulaouic 95
4 Treves’ original conjecture 96
5 The Poisson stratification of S 97
6 Examples 98
7 Treves’ conjecture 98
8 Symplectic strata of codimension 2 99
9 Sketch of the proof 100
References 103
On the backward uniqueness property for a class of parabolic operators 106
1 Introduction, statements and remarks 106
2 Proof of Theorem 1.1 110
3 Proof of Theorem 1.2 114
References 116
Inverse problems for hyperbolic equations 117
1 Formulation of the problem and the main theorem 117
2 Hyperbolic systems with Yang–Mills potentials and domains with obstacles 120
3 A geometric optics approach 124
References 125
On the optimality of some observability inequalities for plate systems with potentials 127
1 Introduction 128
2 Preliminaries 132
3 The sharp observability estimate 135
4 Extension of Meshkov’s construction to the bi-Laplacian equation 138
5 Optimality of the observability constant for plate systems 139
6 Further remarks and open problems 141
References 141
Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach 143
Introduction 143
1 A general setting and a motivating example 144
2 Weak symplectic manifolds 149
3 Right invariant weak Riemannian metrics on Lie groups 157
4 The Hamiltonian approach 165
5 Vanishing H0-geodesic distance on groups of diffeomorphisms 171
6 The regular Lie group of rapidly decreasing diffeomorphisms 179
7 The diffeomorphism group of S1 or R, and Burgers’ hierarchy 188
8 The Virasoro–Bott group and the Korteweg–de Vries hierarchy 195
Appendix A Smooth calculus beyond Banach spaces 212
Appendix B Regular infinite-dimensional Lie groups 216
References 223
Non-effectively hyperbolic operators and bicharacteristics 226
1 Introduction 226
2 Non-effectively hyperbolic symbols, elementary decomposition and a priori estimates 227
3 Conditions for elementary decomposition 231
4 Behavior of bicharacteristics and elementary decomposition 240
5 Remarks 254
References 254
On the Fefferman–Phong inequality for systems of PDEs 256
1 Introduction 256
2 Background on the Weyl–Hörmander calculus 258
3 A proof by induction on the size of the system 260
References 274
Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations 276
1 Introduction 276
2 Resonances for time-periodic potentials 278
3 Strichartz estimates 282
4 Non-trapping moving obstacles 285
5 Trapping moving obstacles 289
References 293
An elementary proof of Fedili’s theorem and extensions 295
1 Introduction 295
2 Proof of the theorem 296
References 298
Outgoing parametrices and global Strichartz estimates for Schrödinger equations with variable coefficients 299
1 Introduction 299
2 Outline of the proofs 305
References 320
On the analyticity of solutions of sums of squares of vector fields 322
1 Global Poisson stratification 324
2 The analyticity conjectures 328
References 335

Produkt-Details

• Erscheint lt. Verlag: 28.12.2007
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Zusatzinfo: XIV, 329 p.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Optik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• Schlagworte: hyperbolic equation • microlocal analysis • partial differential equation • Partial differential equations • Potential • scattering theory • wave equation
• ISBN-10: 0-8176-4521-7 / 0817645217
• ISBN-13: 978-0-8176-4521-2 / 9780817645212