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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 $Ω\subset \real^N$, 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 equations