Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincar\'e type). We show that they can be linked through new inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier, Helffer-Nier and Villani is in particular discussed in the final section

Stability, Resonance and Lyapunov Inequalities for Periodic Conservative Systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.Comment: 19 page

An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations

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 ($1\leq p\leq \infty$) 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 and by using a more accurate version of the Sturm separation theory, an explicit optimal result is obtained. Then, we establish Lyapunov inequalities for systems of equations. To this respect, a key point is the characterization of the best $L^p$ Lyapunov constant for the scalar given problem, as a minimum of some especial (constrained or unconstrained) variational problems defined in appropriate subsets of the usual Sobolev spaces. For Partial Differential Equations on a domain $\Omega \subset \real^N$, 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). This fact shows a deep difference with respect to the ordinary case. Combining the linear results with Schauder fixed point theorem, we can obtain 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 equationsComment: 36 page

Lyapunov-type Inequalities for Partial Differential Equations

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature