Floquet Theory for Partial Differential Equations

Erscheinungsjahr 1993
ISBN: 978-3-7643-2901-3
Verlag: Springer

Buch, Englisch, Band 60, 354 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1540 g

Reihe: Operator Theory: Advances and Applications

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].

Zielgruppe

Research

Autoren/Hrsg.

Kuchment, P.A.

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Experimentalphysik
• Naturwissenschaften Physik Physik Allgemein Geschichte der Physik
• Naturwissenschaften Biowissenschaften Angewandte Biologie Biomathematik
• Interdisziplinäres Wissenschaften Wissenschaften Interdisziplinär Naturwissenschaften, Technik, Medizin
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Interdisziplinäres Wissenschaften Wissenschaft und Gesellschaft | Kulturwissenschaften

Inhaltsverzeichnis

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.- 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.- 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.- 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 somearguments.- 6.3. Invariant differential equations on Riemannian symmetric spaces of non-compact type.- 6.4. Comments and references.- Index of symbols.