Floquet Theory for Partial Differential Equations

ISBN-13: 9783034896863 / Angielski / Miękka / 2012 / 354 str.

P. a. Kuchment

Opis

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].

Spis treści

1. Holomorphic Fredholm Operator Functions.- 1.1. Lifting and open mapping theorems.- 1.2. Some classes of linear operators.- 1.3. Banach vector bundles.- 1.4. Fredholm operators that depend continuously on a parameter.- 1.5. Some information from complex analysis.- A. Interpolation of entire functions of finite order.- B. Some information from the complex analysis in several variables.- C. Some problems of infinite-dimensional complex analysis.- 1.6. Fredholm operators that depend holomorphically on a parameter.- 1.7. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections.- 1.8. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds.- 1.9. Comments and references.- 2. Spaces, Operators and Transforms.- 2.1. Basic spaces and operators.- 2.2. Fourier transform on the group of periods.- 2.3. Comments and references.- 3. Floquet Theory for Hypoelliptic Equations and Systems in the Whole Space.- 3.1. Floquet — Bloch solutions. Quasimomentums and Floquet exponents.- 3.2. Floquet expansion of solutions of exponential growth.- 3.3. Completeness of Floquet solutions in a class of solutions of faster growth.- 3.4. Other classes of equations.- A. Elliptic systems.- B. Hypoelliptic equations and systems.- C. Pseudodifferential equations.- D. Smoothness of coefficients.- 3.5. Comments and references.- 4. Properties of Solutions of Periodic Equations.- 4.1. Distribution of quasimomentums and decreasing solutions.- 4.2. Solvability of non-homogeneous equations.- 4.3. Bloch property.- 4.4. Quasimomentum dispersion relation. Bloch variety.- 4.5. Some problems of spectral theory.- 4.6. Positive solutions.- 4.7. Comments and references.- 5. Evolution Equations.- 5.1. Abstract hypoelliptic evolution equations on the whole axis.- 5.2. Some degenerate cases.- 5.3. Cauchy problem for abstract parabolic equations.- 5.4. Elliptic and parabolic boundary value problems in a cylinder.- A. Elliptic problems.- B. Parabolic problems.- 5.5. Comments and references.- 6. Other Classes of Problems.- 6.1. Equations with deviating arguments.- 6.2. Equations with coefficients that do not depend on some arguments.- 6.3. Invariant differential equations on Riemannian symmetric spaces of non-compact type.- 6.4. Comments and references.- Index of symbols.