Skip to main content
SpringerLink
Log in

Lyapunov-type Inequalities

With Applications to Eigenvalue Problems

  • Book
  • © 2013

Overview

Authors:
  1. Juan Pablo Pinasco

    1. Departamento de Matematica, Universidad de Buenos Aires, Buenos Aires, Argentina

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Emphasizes the use of Lyapunov-type inequalities to obtain lower bounds for eigenvalues
  • Devoted to more general nonlinear equations, systems of differential equations, or partial differential equations
  • Many inequalities intertwined, including Hardy, Sobolev and Poincare inequalities

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Introduction

    • Juan Pablo Pinasco
    Pages 1-10

  3. Lyapunov’s Inequality

    • Juan Pablo Pinasco
    Pages 11-37

  4. Nehari–Calogero–Cohn Inequality

    • Juan Pablo Pinasco
    Pages 39-62

  5. Bargmann-Type Bounds

    • Juan Pablo Pinasco
    Pages 63-84

  6. Miscellaneous Topics

    • Juan Pablo Pinasco
    Pages 85-114

  7. Back Matter

    Pages 115-131
    Download chapter PDF
Back to top

Keywords

About this book

​The eigenvalue problems for quasilinear and nonlinear operators present many differences with the linear case, and a Lyapunov inequality for quasilinear resonant systems showed the existence of  eigenvalue asymptotics driven by the coupling of the equations instead of the order of the equations. For p=2, the coupling and the order of the equations are the same, so this cannot happen in linear problems.  Another striking difference between linear and quasilinear second order differential operators is the existence of Lyapunov-type inequalities in R^n when p>n. Since the linear case corresponds to p=2, for the usual Laplacian there exists a Lyapunov inequality only for one-dimensional problems. For linear higher order problems, several Lyapunov-type inequalities were found by Egorov and Kondratiev and collected in On spectral theory of elliptic operators, Birkhauser Basel 1996. However, there exists an interesting interplay between the dimension of the underlying space, the order of the differential operator, the Sobolev space where the operator is defined, and the norm of the weight appearing in the inequality which is not fully developed.   Also, the Lyapunov inequality for differential equations in Orlicz spaces can be used to develop an oscillation theory, bypassing the classical sturmian theory which is not known yet for those equations. For more general operators, like the p(x) laplacian, the possibility of existence of Lyapunov-type inequalities remains unexplored.  ​

Reviews

From the book reviews:

“This book presents in about 114 pages a good concise summary and a fairly comprehensive study of Lyapunov-type inequalities. … The presentation is clear and well prepared as a whole. The book will be useful for a graduate or even an advanced undergraduate course where Lyapunov-type inequalities are studied or needed and for mathematicians working in the field.” (Shao Zhu Chen, Mathematical Reviews, October, 2014)

“In this volume, the author presents in detail some of these aspects, from its origin to its current situation. … The book contains many explanatory remarks and many comments on the cited bibliography, which can contribute to a better understanding of the theoretical results. Finally, this volume can be useful for researchers interested in the subject and also for all those who want to start in the field.” (Antonio Cañada Villar, zbMATH, Vol. 1291, 2014)

Authors and Affiliations

  • Departamento de Matematica, Universidad de Buenos Aires, Buenos Aires, Argentina

    Juan Pablo Pinasco

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation