ISBN-10:
9024727723
ISBN-13:
9789024727728
Pub. Date:
11/16/1982
Publisher:
Springer Netherlands
Partial differential equations: time-periodic solutions / Edition 1

Partial differential equations: time-periodic solutions / Edition 1

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Product Details

ISBN-13: 9789024727728
Publisher: Springer Netherlands
Publication date: 11/16/1982
Edition description: 1982
Pages: 358
Product dimensions: 6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

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.

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