Lyapunov-type inequalities [electronic resource] : with applications to eigenvalue problems /
Pinasco, Juan Pablo.author.
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.  .
Description based upon print version of record.
Preface; Contents; Symbols and Notation; 1 Introduction; 1.1 A Few Words About Four Theorems; 1.2 Organization of the Book; 2 Lyapunov's Inequality; 2.1 The Classical Inequality; 2.1.1 The Linear Case; 2.1.1.1 Borg's Proof; 2.1.1.2 Direct Integration; 2.1.1.3 Green's Functions and Higher-Order Problems; 2.1.1.4 Hartman-Wintner Proof; 2.1.2 An Interesting Extension; 2.2 Quasilinear Problems; 2.2.1 A Simple Proof; 2.2.2 Relationship with Integral Comparison Theorems; 2.3 Some Incomplete Generalizations; 2.3.1 Higher-Order Quasilinear Problems; 2.3.2 Nonconstant Coefficients
2.3.3 Singular Coefficients2.3.4 Optimality of the Constants; 2.3.4.1 Optimality of the Power; 2.4 Eigenvalue Problems: Lower Bounds of Eigenvalues; 2.4.1 Optimality of the Bound; 2.4.2 A Different Bound; 3 Nehari-Calogero-Cohn Inequality; 3.1 The Work of Calogero and Cohn; 3.1.1 Cohn's Proof; 3.1.2 Calogero's Proof; 3.1.3 A Partial Converse; 3.2 Nehari's Proof and Generalizations; 3.2.1 Nehari's Proof for Second-Order Problems; 3.2.1.1 Notation and Preliminary Results; 3.2.1.2 A Key Lemma; 3.2.2 Nehari's Proof for Linear Higher-Order Differential Equations
3.3 The Inequality for p-Laplacian Problems3.3.1 An Extension for Different Powers; 3.4 Optimality of the Bound; 3.5 Higher Eigenvalues and Some Nonmonotonic Weights; 3.5.1 Higher Eigenvalues; 3.5.2 Nonmonotonic Weights; 4 Bargmann-Type Bounds; 4.1 Bargmann-Type Bounds; 4.1.1 Bargmann's Proof; 4.1.1.1 Proof of Inequality (1.5); 4.1.2 A Shorter Proof; 4.1.3 The Quasilinear Problem; 4.2 Proofs of Theorems D and D'; 4.3 Comparison of Inequalities; 4.4 Singular Eigenvalue Problems; 4.4.1 Introduction; 4.4.2 The Asymptotic Behavior of Eigenvalues; 4.4.2.1 Proof of Theorem 4.6
4.4.3 Proof of Theorem 4.75 Miscellaneous Topics; 5.1 Resonant Systems; 5.1.1 Lyapunov's Inequality for Resonant Systems; 5.1.2 Some Generalizations; 5.1.3 A Different Idea; 5.1.4 Other Systems of Equations; 5.1.4.1 Cycled Systems; 5.1.4.2 Vector p-Laplacian Systems; 5.1.4.3 Coupled Differential Operators of Different Orders; 5.1.5 Other Improvements; 5.1.6 A Word of Caution; 5.1.6.1 A Frequent Mistake; 5.2 Nehari-Calogero-Cohn and Resonant Systems; 5.3 Lyapunov-Type Inequalities in RN; 5.3.1 Related Inequalities; 5.3.1.1 Egorov and Kondratiev Estimates; 5.3.1.2 Anane's and Cuesta's Estimates
5.3.1.3 Case pA.1.6 Jensen's Inequality
Includes bibliographical references.
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.  .
Differential equations.
Functions of complex variables.
Difference equations.
Functional equations.
Ordinary Differential Equations.
Several Complex Variables and Analytic Spaces.
Difference and Functional Equations.