Floquet Theory for Partial Differential Equations

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Springer Science & Business Media, Jul 1, 1993 - Science - 354 pages
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Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389]. They arise in many physical and technical applications [177, 178, 272]. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for integration of nonlinear differential equations. This has led to significant progress in this traditional area [27, 71, 72, 111 119, 250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393, 403, 404]. At the same time, many theoretical and applied problems lead to periodic partial differential equations. We can mention, for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107, 123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417], hydrodynamics [179, 180], elasticity theory [395], the theory of guided waves [87-89, 208, 300], homogenization theory [29, 41, 348], direct and inverse scattering [175, 206, 216, 314, 388, 406-408], parametric resonance theory [122, 178], and spectral theory and spectral geometry [103 105, 381, 382, 389]. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations. The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267].
 

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Contents

HOLOMORPHIC FREDHOLM OPERATOR FUNCTIONS
1
12 Some classes of linear operators
5
13 Banach vector bundles
21
14 Fredholm operators that depend continuously on a parameter
28
15 Some information from complex analysis
32
A Interpolation of entire functions of finite order
34
B Some information from the complex analysis in several variables
54
C Some problems of infinitedimensional complex analysis
60
35 Comments and references
121
PROPERTIES OF SOLUTIONS OF PERIODIC EQUATIONS
125
42 Solvability of nonhomogeneous equations
147
43 Bloch property
150
44 Quasimomentum dispersion relation Bloch variety
151
45 Some problems of spectral theory
158
46 Positive solutions
172
47 Comments and references
183

16 Fredholm operators that depend holomorphically on a parameter
66
17 Image and cokernel of a Fredholm morphism in spaces of holomorphic sections
74
18 Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds
76
19 Comments and references
86
SPACES OPERATORS AND TRANSFORMATIONS
91
22 Fourier transform on the group of periods
93
23 Comments and references
102
FLOQUET THEORY FOR HYPOELLIPTIC EQUATIONS AND SYSTEMS IN THE WHOLE SPACE
103
32 Floquet expansions of solutions of exponential growth
110
33 Completeness of Floquet solutions in a class of solutions of faster growth
112
34 Other classes of equations
115
B Hypoelliptic equations and systems
116
C Pseudodifferential equation Let LxD be a pseudodifferential operator of
117
D Smoothness of coefficients
118
EVOLUTION EQUATIONS
187
52 Some degenerate cases
209
53 Cauchy problem for abstract parabolic equations
224
54 Elliptic and parabolic boundary value problems in a cylinder
244
B Parabolic Problems
255
55 Comments and references
259
OTHER CLASSES OF PROBLEMS
263
62 Equations with coefficients that do not depend on some variables
268
63 Invariant differential equations on Riemannian symmetric spaces of noncompact type
283
64 Comments and references
297
Bibliography
303
Index of symbols
345
Index
349
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Page 321 - Geometrical background for the perturbation theory of the polyharmonic operator with periodic potentials.

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