Bove Advances in Phase Space Analysis of Partial Differential Equations

In Honor of Ferruccio Colombini's 60th Birthday
1. Auflage 2009
ISBN: 978-0-8176-4861-9
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 292 Seiten

This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as 'sums of squares' of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.

Autoren/Hrsg.

Bove, Antonio

Inhaltsverzeichnis

1;Preface;8
2;Contents;10
3;List of Contributors;12
4;Tangent Halfspaces to Sets of Finite Perimeterin Carnot Groups;14
4.1;1 Introduction;14
4.2;2 Differentiability and rectifiability;14
4.2.1;2.1 Sets of finite perimeter in Euclidean spaces;15
4.3;3 Motivations;16
4.3.1;3.1 Generalized differentiability;16
4.4;4 Carnot groups, differentiability of Lipschitz functions and sets of finite perimeter;17
4.4.1;4.1 The Heisenberg groups;18
4.4.2;4.2 Dilations and Carnot--Carathéodory distance;18
4.4.3;4.3 Pansu differentiability theorem;18
4.4.4;4.4 X-derivative and sets of finite perimeter;19
4.4.5;4.5 Generalized inner normal to sets of finite perimeter;20
4.5;5 Rectifiability in Euclidean spaces and in step 2 groups;21
4.5.1;5.1 Measure-theoretic properties of |D0=x"011FE|;21
4.5.2;5.2 De Giorgi's rectifiability proof;22
4.5.3;5.3 De Giorgi's argument in Carnot groups;22
4.6;6 Our main results, and open problems;24
4.7;7 Some ideas from the proof;25
4.7.1;7.1 Regular directions become invariant;26
4.7.2;7.2 New regular directions;26
4.8;References;27
5;The Heat Kernel and Frequency Localized Functionson the Heisenberg Group;30
5.1;1 Introduction;30
5.1.1;1.1 The Heisenberg group Hd;31
5.1.2;1.2 Statement of the results;32
5.2;2 Elements of Littlewood--Paley theory on the Heisenberg group;34
5.2.1;2.1 The Fourier transform on the Heisenberg group;34
5.2.2;2.2 Littlewood--Paley theory on the Heisenberg group;39
5.2.3;2.3 Frequency localized functions and Bernstein inequalities on the Heisenberg group;40
5.3;3 Proof of Theorem 1.4;41
5.4;4 Proofs of Lemma 1.3 and Theorems 1.6 and 1.8;42
5.4.1;4.1 Proof of Lemma 1.3 ;43
5.4.2;4.2 Proof of Theorem 1.6;45
5.4.3;4.3 Proof of Theorem 1.8;47
5.5;References;48
6;A Generalization of the Rudin--Carleson Theorem;49
6.1;1 Introduction;49
6.2;2 Preliminaries and statement of the main result;50
6.3;3 Proof of Theorem 2.1;52
6.3.1;3.1 Case 1;52
6.3.2;3.2 Case 2;58
6.4;4 Some examples;59
6.5;5 A local version of the Rudin--Carleson property;62
6.6;6 A link with the F. and M. Riesz theorem;66
6.7;References;68
7;Evolution Equations and Generalized Fourier Integral Operators;70
7.1;1 Weyl--Hörmander calculus of pseudodifferential operators;72
7.1.1;1.1 Quantization;72
7.1.2;1.2 Admissible metrics;73
7.1.3;1.3 Weights and symbols;74
7.1.4;1.4 Characterization of pseudodifferential operators;75
7.2;2 Generalized Fourier integral operators;76
7.2.1;2.1 Principal symbol of Fourier integral operators;77
7.3;3 Evolution equations;78
7.4;4 Proof of Theorem 3.1;79
7.5;5 Proof of Theorem 3.2;82
7.6;References;83
8;The Solvability and Subellipticity of Systems of Pseudodifferential Operators;84
8.1;1 Introduction;84
8.2;2 Solvability of systems;86
8.3;3 Subellipticity of systems;95
8.4;References;105
9;Uniform Exponential Decay for Viscous Damped Systems;106
9.1;1 Introduction;107
9.2;2 Proof of Theorem 1.1;109
9.3;3 Variants of Theorem 1.1;112
9.3.1;3.1 General viscosity operators;112
9.3.2;3.2 Wave-type systems;113
9.4;4 Applications;118
9.4.1;4.1 The viscous Schrödinger equation;118
9.4.2;4.2 The viscous damped wave equation;120
9.5;5 Further comments;121
9.6;References;122
10;The Hyperbolic Symmetrizer: Theory and Applications;124
10.1;1 Introduction;124
10.2;2 The standard symmetrizer;129
10.2.1;2.1 Definition and elementary properties;129
10.2.2;2.2 The standard symmetrizer and the Bezout matrix;132
10.2.3;2.3 How symmetrizer checks hyperbolicity;134
10.2.4;2.4 A few examples;135
10.3;3 The quasi--symmetrizer;136
10.3.1;3.1 Sketch proof of Theorem 2;136
10.3.2;3.2 Going deep into quasi--symmetrizer;138
10.3.3;3.3 The proof of Theorem 5;141
10.3.4;3.4 An example;142
10.4;4 Hyperbolic symmetrizer and weakly hyperbolic equations;142
10.4.1;4.1 Sketch proof of C well--posedness of (4.1);143
10.4.2;4.2 Toward more general homogeneous equations;144
10.5;References;149
11;Time Global Solutions to the Cauchy Problem for Multidimensional Kirchhoff Equations;151
11.1;1 Introduction;151
11.2;2 Proof of Theorem 1.1;152
11.3;3 Proof of Theorem 1.2;160
11.4;References;163
12;The Order of Accuracy of Quadrature Formulaefor Periodic Functions;164
12.1;1 Upper bound on the error;165
12.2;2 A lower bound and comparison with best trigonometric approximation;167
12.3;3 Conclusion;168
12.4;References;168
13;A Note on the Oseen Kernels;169
13.1;1 Introduction;169
13.2;2 The action of the Leray projector on Gaussian functions;171
13.3;3 Appendix;175
13.4;References;178
14;Instability Behavior and Loss of Regularity;179
14.1;1 Introduction;179
14.2;2 Proof of Theorem 1.4;183
14.3;3 Optimality of conditions for infinite loss of regularity;189
14.4;4 Optimality of conditions for finite loss of regularity;201
14.5;5 Concluding remarks;206
14.6;References;207
15;Decay Estimates for Variable Coefficient Wave Equations in Exterior Domains;209
15.1;1 Introduction;209
15.2;2 The localized energy estimates;216
15.2.1;2.1 Analysis near and classical Morawetz-type estimates;216
15.2.2;2.2 Analysis near and frequency localized estimates;217
15.2.3;2.3 Proof of Theorem 1.1;219
15.3;3 The Strichartz estimates;220
15.4;References;223
16;On Gevrey Well-Posedness of the Cauchy Problem for Some Noneffectively Hyperbolic Operators;225
16.1;1 Introduction;225
16.2;2 Asymptotic solution;227
16.3;3 Lemmas;230
16.4;4 A priori estimate;235
16.5;5 Proof of Theorem 1.1;237
16.6;6 Proof of Propositions 1.3 and 1.4;239
16.7;References;241
17;Singularities of the Scattering Kernel Relatedto Trapping Rays;242
17.1;1 Introduction;242
17.2;2 Scattering kernel;244
17.3;3 Trapping obstacles;248
17.4;4 Trapping rays and estimates of the scattering amplitude;252
17.5;References;257
18;Analytic Hypoellipticity for a Sum of Squares of Vector Fields in R3 Whose Poisson Stratification Consists of a Single Symplectic Stratum of Codimension Four;259
18.1;1 Introduction and statement of theorems;259
18.2;2 The proof in the case of Theorem 1;260
18.3;3 The proof in the case of Theorem 2;266
18.4;References;267
19;Multidimensional Soliton Integrodifferential Systems;268
19.1;1 Basic facts of noncommutative KdV theory;268
19.1.1;1.1 Noncommutative setup;269
19.1.2;1.2 Fundamental properties of the noncommutative KdV equation;272
19.1.3;1.3 Traveling wave solutions in the abstract noncommutative setup;275
19.2;2 Finite-dimensional systems;275
19.2.1;2.1 Matrix systems;275
19.2.2;2.2 Constants of motion and absence of isospectrality;276
19.3;3 Noncommmutative KdV hierarchy based on Schwartz space;278
19.3.1;3.1 Schwartz space and its bounded linear operators;278
19.3.2;3.2 Differential subalgebras and KdV equation;279
19.4;4 The constant coefficients case;281
19.4.1;4.1 Existence of solutions;281
19.4.2;4.2 Traveling wave solutions;282
19.5;5 KdV equation based on the harmonic oscillator;284
19.5.1;5.1 KdV equation with L=Dx2+x2;284
19.5.2;5.2 Global Cauchy problem in an algebra of upper-triangular matrices;285
19.5.3;5.3 Traveling wave solutions;288
19.6;6 Appendix: Hermite functions expansion;291
19.7;References;294
20;Selected Lectures in Microlocal Analysis;295
20.1;1 Decomposition in plane waves: analytic wave front set;295
20.2;2 Operators/symbols: symplectic transformations;298
20.3;3 The theorem of elliptic regularity;299
20.4;4 Propagation in the interior: real simply-characteristic operators;301
20.5;5 Propagation at the boundary: reflection and diffraction of the light;302
20.6;6 Propagation at the boundary: transversal ellipticity and non-microcharacteristicity;304
20.7;References;305