Partial Differential Equations: Time-Periodic Solutions

ISBN-13: 9789400976740 / Angielski / Miękka / 2011 / 358 str.

Otto Vejvoda

Opis

As far as the number of new results and quoted papers is concerned the present book may be considered a monograph. However, it also has some features of a textbook. Firstly, it proceeds from concrete problems to abstract ones, and secondly, all considerations and procedures are presented in much detail when met for the first time (such very elementary expositions can be found especially at the beginning of Chapters III and V). Finally, the authors focus their attention on elementary problems which can be dealt with by relatively simple methods. The authors hope that all this will make it possible also for an applied or technical research worker with some mathematical training to read this book. Naturally, the reader is supposed to be familiar with some basic notions from mathematical analysis, functional analysis and theory of partial differential equations. Also, the arguments and procedures which are repeated in the book are presented more briefly when met again, the reader being expected to become gradually more thoroughly acquainted with them. The authors have tried to provide a complete bibliography of all relevant publications (their number reaches about 500) from the theory of time-periodic solutions to non-linear partial and abstract differential equations whose origin may be put in the early thirties of this century.

Spis treści

I Preliminaries from Functional Analysis.- § 1. Linear Spaces and Operators.- 1.1. Basic Notation.- 1.2. Banach Spaces.- 1.3. Hilbert Spaces.- 1.4. Fourier Series.- 1.5. Self-Adjoint Operators.- 1.6. Spectral Properties of Operators; Compact Operators.- 1.7. Embeddings; Negative Norms.- 1.8. Differentials.- §2. Function Spaces.- 2.1. Spaces of Smooth Functions.- 2.2. Spaces of Integrable Functions.- 2.3. Sobolev Spaces. (Spaces of Differentiable Functions).- 2.4. Periodic Functions.- 2.5. Representation of Vector-Valued Functions.- 2.6. Periodic Functions of Two Arguments.- 2.7. Embedding Theorems.- 2.8. Traces of Functions.- 2.9. The Substitution Operators.- § 3. Existence Theorems for Operator Equations.- 3.1. Theorems of a “Metric” Character.- 3.2. Theorems of a Topological Character.- 3.3. Equations with Monotone Operators.- 3.4. Equations Depending on a Parameter; Local Theorems.- 3.5. Equations Depending on a Parameter; Global Theorems.- II Preliminaries from the Theory of Differential Equations.- § 1. Boundary-Value and Eigenvalue Problems for Elliptic and Ordinary Differential Operators.- 1.1. Elliptic operators.- 1.2. Boundary-value problems for ordinary differential equations.- 1.3. Eigenvalue problems for elliptic operators.- § 2. The Wave and the Telegraph Equations.- 2.1. The Cauchy Problem for the Wave Equation in R.- 2.2. The Initial Boundary-Value Problem with the Dirichlet Boundary Conditions.- 2.3. The Telegraph Equation.- 2.4. The Fourier Method.- § 3. The Heat Equation.- 3.1. The Cauchy Problem.- 3.2. The Initial Boundary-Value Problems.- 3.3. The Fourier Method.- III The Heat Equation.- § 1. The (t, s)-Fourier Method.- 1.1. The Linear Ease.- 1.2. The weakly Non-Linear Case.- 1.3. The n-dimensional Case.- § 2. The t-Fourier and s-Fourier Methods.- 2.1. The t-Fourier Method: 0 ? x ? ?.- 2.2. The t-Fourier Method: ? ? < x < ?.- 2.3. The s-Fourier Method.- § 3. The Poincaré Method.- 3.1. The Linear Case.- 3.2. The Weakly Non-Linear Case.- § 4. Supplements and Comments on the Linear and Weakly Non-Linear Heat Equation.- 4.1. The Adjoint Problem Method.- 4.2. Comments on other Results.- § 5. Comments on Strongly Non-Linear Parabolic Equations.- 5.1. Results of Prodi, Vaghi and Bange. (Differential Inequality Techniques).- 5.2. Results of Kolesov, Klimov, Amann, Tsai, Deuel and Hess. (Use of Upper and Lower Solutions).- 5.3. Results of Šmulev, Fife, Kusano, Kružkov, Gaines and Walter. (A priori estimates techniques).- 5.4. Results of Šmulev, Mal’cev, Walter and Knolle. (Methods of discretizations).- 5.5. Results of Palmieri, Nakao, Nanbu, Biroli and Zecca. (Problems with Special Type of Non-Linearity).- 5.6. Results of Brézis and Nirenberg, Š?astnová and Fu?ík. (Critical Cases).- 5.7. Results of Klimov, Krasnosel’skií and Sobolevskií. (Eigenvalue Problem).- 5.8. Results of other Authors.- 5.9. Comments on Papers of an Applied Character.- § 6. Comments on the Navier-Stokes Equations and Related Problems.- 6.1. The Non-Autonomous Navier-Stokes Equations.- 6.2. The Autonomous Navier-Stokes Equations.- 6.3. Related Problems.- IV The Telegraph Equation.- § 1. The Fourier Methods.- 1.1. The (t, s)-Fourier Method; the Linear Case.- 1.2. The Weakly Non-Linear Case.- 1.3. The t-Fourier Method.- 1.4. The s-Fourier Method.- § 2. The Poincaré Method.- 2.1. The Linear Case: ? ? gt; x gt; ?.- 2.2. The linear case: 0 ? x ? ?.- 2.3. The weakly Non-Linear Case.- § 3. Singularly Perturbed Problems.- 3.1. General Considerations.- 3.2. The Main Theorems.- § 4. Supplements and Comments on Linear and Weakly Non-Linear Problems.- 4.1. The Adjoint Problem Method.- 4.2. The Ficken-Fleishman Method.- 4.3. Results of Rabinowitz.- 4.4. The Telegraph Equation with a = 0.- § 5. Comments on Strongly Non-Linear Problems.- 5.1. Results of Prodi, Prouse, Krylová, Buzzetti, Lions, TouSck, Biroli and Nakao. (The Wave Equation with Strongly Non-Linear Damping).- 5.2. Results of Mawhin, Fucík, Brézis, Nirenberg, Biroli, HoráSek and Zecca. (Problems with Non-Linear Forcing Term).- 5.3. Results of v. Wahl, Clements, Kakita and Sowunmi. (Problems with Non-Linear Elliptic Part and other Problems).- V The Wave Equation.- § 1. The Dirichlet Boundary Conditions; the Poincaré Method.- 1.1. The Linear Case; General Considerations.- 1.2. The case ? = 2?n.- 1.3. The case ? = 2?p/q.- 1.4. The Non-Linear Case.- § 2. The Dirichlet boundary conditions; the Günzlcr method.- 2.1. The Linear Case; General Considerations.- 2.2. The Case ? = 2 ?.- 2.3. The case ? = 2 ? p/q.- 2.4. The Non-Linear Case.- 2.5. The Case of Monotone Perturbations.- §3. Examples.- 3.1. The Problem (P?2?) with F(u, ?) (t, x) = h(t, x) + ? u(t, x) + + ? u3(t, x) (the Poincaré Method).- 3.2. The Problem (P?2?) with F(u, ?) (t, x) = h(t, x) + ? u(t, x) + + ? u3(t, x) (the Günzler Method).- § 4. The Newton and Combined Boundary Conditions; the Günzler Method.- 4.1. General Considerations.- 4.2. The case ? = 1, ? = 2?.- 4.3. The case ? = 0, ?1(t) = 1, ? = 2 ?.- § 5. Entrainment of Frequency.- 5.1. Introduction.- 5.2. The existence of (2? + ??)-Periodic Solutions.- 5.3. Equations with Monotone Right-Hand Sides.- § 6. The Fourier Method.- 6.1. Introductory Remarks.- 6.2. The case ? = 2?p/q.- 6.3. The case ? = 2??.- 6.4. The Weakly Non-Linear Case.- 6.5. The (t, s)-Fourier Method; the Newton and Combined Boundary Conditions.- 6.6. Problems with Variable Coefficients and Problems in Several Variables.- § 7. The Wave Equation in an Unbounded Domain.- 7.1. A general Existence Theorem.- 7.2. Applications.- § 8. Supplements and Comments on Non-Autonomous Hyperbolic Equations.- 8.1. The Adjoint Problem Method.- 8.2. Averaging Methods.- 8.3. Comments on other Papers.- 8.4. Comments on Papers of an Applied Character.- § 9. Comments on Autonomous Hyperbolic Equations.- 9.1. Comments on Periodic Solutions of Boundary-Value Problems.- 9.2. Comments on Periodic Solutions in Unbounded Domains.- 9.3. Comments on Papers of an Applied Character.- VI The Beam Equation and Related Problems.- § 1. The Equations of a Beam and of a Thin Plate.- 1.1. Equations without Damping.- 1.2. Equations with a Damping Term.- § 2. Supplements and Comments.- 2.1. The Adjoint Problem Method.- 2.2. Results of Krylová, Vejvoda, Kopá?ková, Solov’ev, Karimov, Mitrjakov and Filip. (Results Based on Fourier Methods).- 2.3. Results of Kurzweil, Hall, Petzeltová and Nakao. (Results Based on Monotonicity of Perturbations).- 2.4. Results of Št?drý, Krylová and Krej?í. (Problems with Damping).- § 3. The Dynamic von Kármán Equations of Thin Plates Involving Rotational Inertia and Damping.- 3.1. Preliminaries.- 3.2. The Existence Theorem.- 3.3. Comments on Papers of an Applied Character.- VII The Abstract Equations.- § 1. The t-Fourier method: Preliminaries.- 1.1. The Setting of the Problem.- 1.2. Auxiliary Notions and Results.- § 2. The t-Fourier Method: Spectral Properties of Periodic Operators.- 2.1. Main Results.- 2.2. Examples.- § 3. The t-Fourier Method: Weakly Non-Linear Problems.- 3.1. General Scheme.- 3.2. A Special Problem.- § 4. Comments on Papers Using Direct Methods.- 4.1. Results of Lions and Magenes, and Ton.- 4.2. Results of Taam and Cend.- 4.3. Results of Da Prato and Barbu.- 4.4. Results of Gajewski, Gröger and Zacharias.- 4.5. Results of Sova.- 4.6. Results of Dubinski?.- 4.7. Results of Herrmann.- 4.8. Results of Straškraba.- 4.9. Results of Moke??ev and Kopá?ková.- 4.10. Results of Crandall and Rabinowitz.- 4.11. Results of Dezin.- 4.12. Results of Comincioli and Gaultier.- 4.13. Results of Borisovié.- § 5. Comments on Papers using Indirect Methods.- 5.1. Results of Browder.- 5.2. Results of Brézis, Benilan, Biroli, Crandall, Pazy, Pavel and Prouse.- 5.3. Results of Vejvoda, Straškraba, Krylová, Sobolevski? and Pogore-lenko.- 5.4. Results of Simon?nko.- 5.5. Results of Fink, Hall and Hausrath.- 5.6. Results of Amann.- 5.7. Other Results.- Bibliography to Chapter I.- Bibliography to Chapter II.- Bibliography to Chapter III.- Bibliography to Chapter IV.- Bibliography to Chapter V.- Bibliography to Chapter VI.- Bibliography to Chapter VII.- Bibliography of papers on related topics.- Addenda to bibliography.- List of Symbols.