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An applied mathematical excursion through Lyapunov inequalities, classical analysis and Differential Equations

Content Provider	Springer Nature Link
Author	Villegas, Salvador Cañada, Antonio
Copyright Year	2012
Abstract	Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of L $_{p}$ Lyapunov inequalities (1 ≤ p ≤ Ω) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues. Then, we establish Lyapunov inequalities for systems of equations. For Partial Differential Equations on a domain Ω ⊂ ℝ$^{N}$, we consider the Laplace equation with Neumann or Dirichlet boundary conditions. It is proved that the relation between the quantities p and N/2 plays a crucial role in order to obtain nontrivial L $_{p}$ Lyapunov type inequalities (which are called Sobolev inequalities by many authors). One of the main applications of Lyapunov inequalities is its use in the study of nonlinear resonant problems. To this respect, combining the linear results with Schauder fixed point theorem, we show some new results about the existence and uniqueness of solutions for resonant nonlinear problems for ODE or PDE, both in the scalar case and in the case of systems of equations.
Starting Page	69
Ending Page	106
Page Count	38
File Format	PDF
ISSN	15759822
Journal	SeMA Journal
Volume Number	57
Issue Number	1
e-ISSN	22543902
Language	English
Publisher	Springer-Verlag
Publisher Date	2013-04-19
Publisher Institution	Sociedad Española de Matemática Aplicada
Publisher Place	Milan
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Boundary value problems for second-order elliptic equations Variational methods for second-order elliptic equations Optimal control problems involving ordinary differential equations Mathematics resonance ordinary differential equations Linear boundary value problems Lyapunov inequalities boundary value problems Nonlinear boundary value problems partial differential equations Oscillation theory, zeros, disconjugacy and comparison theory Optimal control problems involving partial differential equations Applications of Mathematics stability
Content Type	Text
Resource Type	Article

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