Maximum Principles For The Hill's Equation by Alberto Cabada

Maximum Principles For The Hill's Equation

byAlberto Cabada, Jose Cid, Lucia Somoza

Paperback | October 1, 2017

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The equation is crucial for the understanding of many physical models, and particularly in non-linear applications. The "maximum principles" or equivalently the constant sign of the Green's function are the key for the development of nonlinear methods such as the lower and upper solution technique, monotone iterations or fixed points theorems for positive operators. The authors will focus on the application of these methods to nonlinear equations with singularities e.g. Brillouin-bem focusing equation, Ermakov-Pinney, and and for problems with parametric dependence. For the homogeneous problem, the spectral problem will be treated, along with the oscillation of the solutions and their stability. For the non-homogeneous problem, the authors will consider comparison principles for the Hill's equation. The authors will discuss the properties of the related Green's functions coupled with different boundary value conditions. They will establish the equations relationship with the spectral theory developed for the homogeneous case. Both stability and constant sign solutions of the equation will be considered in the treatment. Reviews of present classical and recent results made by the authors and by other key authors will be included. Evaluates classical topics in the Hill's equation that are crucial for understanding to modern physical models and non-linear applications Describes explicit and effective conditions on maximum and anti-maximum principles Collates information from disparate sources in one self-contained volume, with extensive referencing throughout

About The Author

Department of Mathematical Analysis, Faculty of Mathematics, Universidade de Santiago de Compostela, Santiago de Compostela, Spain.Author or co-author of more than 130 journal articles.Department of Mathematics, Universidade de Vigo, Campus de Ourense , Spain.Author or co-author of more than 40 journal articles.Department of Mathematic...
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Details & Specs

Title:Maximum Principles For The Hill's EquationFormat:PaperbackDimensions:160 pages, 8.75 × 6.35 × 0.68 inPublished:October 1, 2017Publisher:Academic PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:012804117X

ISBN - 13:9780128041178

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Table of Contents

1. Introduction

1.1. Hill's equation

1.2. Stability in the sense of Lyapunov

2. Homogeneous equation

2.1. Introduction

2.2. Sturm comparison theory.

2.3. Spectral properties of the Dirichlet Problem.

2.4. Spectral properties of the Periodic Problem: intervals of stability and instability

3. Non homogeneous equation

3.1. Introduction

3.2. The Green's function

3.3. Some spectral characterizations for the Maximum and Anti-Maximum principles

3.4. Explicit conditions for the Maximum and Anti-Maximum principles

4. Nonlinear equations

4.1. Lower and upper solutions

4.2. Monotone iterative techniques

4.3. Fixed points for positive operators

4.4. Problems with singularities

4.5. Problems with parametric dependence

5. Appendix

5.1. Floquet theory

5.2. Sobolev Inequalities