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Conical Refraction and Higher Microlocalization : Lecture Notes in Mathematics - Otto Liess

Conical Refraction and Higher Microlocalization

Lecture Notes in Mathematics

Paperback

Published: 29th September 1993
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The main topic of the book is higher analyticmicrolocalization and its application to problems ofpropagation of singularities. The part on highermicrolocalization could serve as an introduction to thesubject. The results on propagation refer to solutions oflinear partial differentialoperators with characteristicsof variable multiplicity and are of conical refraction type.The relation and interplay between these results and resultsor constructions from geometrical optics in crystal theoryis discussed with many details. The notes are writtenforemost for researchers working in microlocal analysis,but it is hoped that they can also be of interest formathematicians and physicists who work in propagationphenomena from a more classical point of view.

Introductionp. 1
Statement of the main resultsp. 1
An heuristic justificationp. 22
Discussion of theorem 1.1.6p. 24
Higher order wave front setsp. 33
Higher order wave front setsp. 34
[actual symbol not reproducible]-microlocalizationp. 44
Splitting and approximation of entire functionsp. 49
Higher microlocalization and dualityp. 55
Plurisubharmonic functions related to propagation phenomenap. 57
An improvement of proposition 2.5.10p. 66
Proof of theorem 2.1.12p. 70
Proof of theorem 2.1.13p. 73
Proof of proposition 2.1.15p. 74
A.1 Propagation of analyticity for partially holomorphic functionsp. 79
A.2 Proof of lemma 2.5.1p. 83
A.3 Proof of proposition 2.1.7p. 86
Pseudodifferential operatorsp. 95
Polyhomogeneous structures and polyconesp. 96
Polyhomogeneity and polyhomogeneous symbolsp. 102
Symbol classes and pseudodifferential operators in general [actual symbol not reproducible]-microlocalizationp. 107
Pseudodifferential operators in k-microlocalizationp. 110
Polyhomogeneity and symbolsp. 116
Finite sums of polyhomogeneous functions and principal parts. Relative Poisson bracketsp. 118
Successive localizationsp. 123
Regularity theorems Ip. 125
Regularity theorems II. Proof of theorem 1.1.6p. 127
Improvements in the case of constant coefficient operatorsp. 129
Proof of proposition 3.10.3p. 134
Proof of the propositions 1.1.9 and 1.1.12p. 137
Remarks on localization of polynomials. Proof of proposition 1.1.13p. 141
Bi-symplectic geometry and multihomogeneous mapsp. 145
Polyhomogeneous changes of coordinates in R[superscript n]p. 146
Polyhomogeneous changes of coordinates in R[superscript 2n]p. 150
Bihomogeneous approximations of homogeneous mapsp. 153
The bihomogeneous structure of N[Sigma]p. 157
The canonical foliation and the relative tangent space of N[Sigma]p. 161
The bisymplectic structure of N[Sigma]p. 163
The relative bicharacteristics foliation of [Lambda]p. 168
Phase functions and bihomogeneous approximations of canonical mapsp. 170
Construction of the phasep. 181
Reduction of p[subscript m] to model form, inequalities and a regularity theoremp. 186
Canonical transformations and estimates for symbolsp. 190
Fourier Integral Operatorsp. 193
Classical F.I.O.'s which leave [actual symbol not reproducible]= 0 invariantp. 194
Invariant definition of [actual symbol not reproducible]p. 204
F.I.O. associated with phase functions which live on bineighborhoodsp. 206
Phase functions associated with Lipschitzian weight functionsp. 212
Composition of F.I.O.p. 215
Invariant meaning and proof of theorem 1.1.3p. 222
Conical refraction, hyperbolicity and slowness surfacesp. 225
Influence domains and bicharacteristicsp. 226
The canonical form of operators when [actual symbol not reproducible] is second order and hyperbolicp. 232
The case when p[subscript m,k] is hyperbolicp. 235
Singular points on surfacesp. 254
Remarks on the velocity, the slowness and the wave surfacep. 262
Singular points on the slowness surface and conical refractionp. 266
Conical refraction in free space for the system of crystal opticsp. 270
Singular points on the slowness surface of the system of elasticity for cubic crystalsp. 273
Propagation of regularity up to the boundaryp. 279
Interior regularity when the traces are smoothp. 280
Extension across hypersurfaces of solutions of constant coefficient microlocally hyperbolic operatorsp. 281
The wave front set of a restrictionp. 286
Estimates for the Fourier transform of some surface densitiesp. 289
Boundary values and partial regularityp. 293
Regularity up to the boundaryp. 299
Proof of lemma 7.6.5p. 305
Extension of solutions from the interiorp. 307
Some results on transmission problemsp. 309
Formulation of the problemp. 310
Regularity of traces as a consequence of interior regularity, Ip. 312
Regularity of traces as a consequence of interior regularity, IIp. 321
Incoming and outgoing bicharacteristics and propagation conesp. 322
Conical refraction in transmission problemsp. 329
Conical reflection at the boundaryp. 333
The case of crystal opticsp. 335
Other phenomena: External conical refractionp. 342
Partial analyticity, higher microlocalization and sheavesp. 347
The [actual symbol not reproducible]-class of a partially analytic distributionp. 348
The first two wave front sets and partial analyticityp. 350
Proof of proposition 9.1.2p. 351
Review of hyperfunctionsp. 353
Hyperfunctions and real analytic functionalsp. 358
Partially analytic distributions and infinite order operatorsp. 361
Proof of proposition 9.1.3p. 362
The sheaf of partially analytic microfunctionsp. 366
Second microlocalization and sheavesp. 368
The Fourier transform of measures on C[superscript n] with supports concentrated on polyconesp. 371
Higher order wave front sets and boundary values of holomorphic functionsp. 374
Referencesp. 380
Notationsp. 385
Subject indexp. 387
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540571056
ISBN-10: 3540571051
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 398
Published: 29th September 1993
Publisher: SPRINGER VERLAG GMBH
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.13
Weight (kg): 0.57