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Mathematics - Dynamical Systems & Differential Equations | Lyapunov-type Inequalities - With Applications to Eigenvalue Problems

Lyapunov-type Inequalities

With Applications to Eigenvalue Problems

Pinasco, Juan Pablo

2013, XIII, 131 p.

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  • ​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
​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.  ​

Content Level » Research

Keywords » Lyapunov inequality - Orlicz spaces - eigenvalue bounds - integral inequalities - p-laplace operator - quasilinear operators

Related subjects » Analysis - Dynamical Systems & Differential Equations

Table of contents 

​A short history of Lyapunov inequality.- Lyapunov inequality for p-laplacian operators.- Generalizations.- Lyapunov type inequalities in RN​.

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