Bove / Colombini / Del Santo Phase Space Analysis of Partial Differential Equations

1. Auflage 2007
ISBN: 978-0-8176-4521-2
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 69, 329 Seiten

Reihe: Progress in Nonlinear Differential Equations and Their Applications

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

Autoren/Hrsg.

Bove, Antonio

Colombini, Ferruccio

Del Santo, Daniele

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;List of Contributors;11
4;Trace theorem on the Heisenberg group on homogeneous hypersurfaces;15
4.1;1 Introduction;15
4.2;2 A Hardy type inequality;20
4.3;3 The proof of the trace and trace lifting theorem;24
4.4;4 Concluding remarks;28
4.5;References;28
5;Strong unique continuation and finite jet determination for Cauchy – Riemann mappings;30
5.1;1 Introduction;30
5.2;2 Local coordinates;31
5.3;3 Nondegeneracy conditions;33
5.4;4 Necessary conditions and su.cient conditions for finite jet determination;35
5.5;5 Lie group structures and jet parameterization;37
5.6;References;40
6;On the Cauchy problem for some hyperbolic operator with double characteristics;42
6.1;1 Introduction and statements;42
6.2;2 The model operator;44
6.3;3 Shibuya solutions;45
6.4;4 Stokes multipliers;47
6.5;5 Asymptotic analysis;49
6.6;6 Final steps in the proof of the necessary condition;55
6.7;References;57
7;On the differentiability class of the admissible square roots of regular nonnegative functions;58
7.1;1 Introduction;58
7.2;2 Regularity of well-chosen admissible roots;59
7.3;References;66
8;The Benjamin–Ono equation in energy space;67
8.1;1 Introduction;67
8.2;2 Bourgain spaces;69
8.3;3 A priori estimate on weak solutions;70
8.4;4 The gauge transformation;71
8.5;5 The existence and uniqueness result;73
8.6;References;74
9;Instabilities in Zakharov equations for laser propagation in a plasma;75
9.1;1 Introduction;75
9.2;2 The instability mechanism;78
9.3;3 Scheme of the proof;80
9.4;4 The linear instability;84
9.5;5 The linear equation;89
9.6;6 End of proofs;92
9.7;References;93
10;Symplectic strata and analytic hypoellipticity;94
10.1;1 Introduction;94
10.2;2 The symplectic case;95
10.3;3 The example of Baouendi–Goulaouic;95
10.4;4 Treves’ original conjecture;96
10.5;5 The Poisson stratification of S;97
10.6;6 Examples;98
10.7;7 Treves’ conjecture;98
10.8;8 Symplectic strata of codimension 2;99
10.9;9 Sketch of the proof;100
10.10;References;103
11;On the backward uniqueness property for a class of parabolic operators;106
11.1;1 Introduction, statements and remarks;106
11.2;2 Proof of Theorem 1.1;110
11.3;3 Proof of Theorem 1.2;114
11.4;References;116
12;Inverse problems for hyperbolic equations;117
12.1;1 Formulation of the problem and the main theorem;117
12.2;2 Hyperbolic systems with Yang–Mills potentials and domains with obstacles;120
12.3;3 A geometric optics approach;124
12.4;References;125
13;On the optimality of some observability inequalities for plate systems with potentials ;127
13.1;1 Introduction;128
13.2;2 Preliminaries;132
13.3;3 The sharp observability estimate;135
13.4;4 Extension of Meshkov’s construction to the bi-Laplacian equation;138
13.5;5 Optimality of the observability constant for plate systems;139
13.6;6 Further remarks and open problems;141
13.7;References;141
14;Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach;143
14.1;Introduction;143
14.2;1 A general setting and a motivating example;144
14.3;2 Weak symplectic manifolds;149
14.4;3 Right invariant weak Riemannian metrics on Lie groups;157
14.5;4 The Hamiltonian approach;165
14.6;5 Vanishing H0-geodesic distance on groups of diffeomorphisms;171
14.7;6 The regular Lie group of rapidly decreasing diffeomorphisms;179
14.8;7 The diffeomorphism group of S1 or R, and Burgers’ hierarchy;188
14.9;8 The Virasoro–Bott group and the Korteweg–de Vries hierarchy;195
14.10;Appendix A Smooth calculus beyond Banach spaces;212
14.11;Appendix B Regular infinite-dimensional Lie groups;216
14.12;References;223
15;Non-effectively hyperbolic operators and bicharacteristics;226
15.1;1 Introduction;226
15.2;2 Non-effectively hyperbolic symbols, elementary decomposition and a priori estimates;227
15.3;3 Conditions for elementary decomposition;231
15.4;4 Behavior of bicharacteristics and elementary decomposition;240
15.5;5 Remarks;254
15.6;References;254
16;On the Fefferman–Phong inequality for systems of PDEs;256
16.1;1 Introduction;256
16.2;2 Background on the Weyl–Hörmander calculus;258
16.3;3 A proof by induction on the size of the system;260
16.4;References;274
17;Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations;276
17.1;1 Introduction;276
17.2;2 Resonances for time-periodic potentials;278
17.3;3 Strichartz estimates;282
17.4;4 Non-trapping moving obstacles;285
17.5;5 Trapping moving obstacles;289
17.6;References;293
18;An elementary proof of Fedili’s theorem and extensions;295
18.1;1 Introduction;295
18.2;2 Proof of the theorem;296
18.3;References;298
19;Outgoing parametrices and global Strichartz estimates for Schrödinger equations with variable coefficients;299
19.1;1 Introduction;299
19.2;2 Outline of the proofs;305
19.3;References;320
20;On the analyticity of solutions of sums of squares of vector fields;322
20.1;1 Global Poisson stratification;324
20.2;2 The analyticity conjectures;328
20.3;References;335