Remarks on Lyapunov-type inequalities for (p,q)-Laplace 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 ≤ 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 ⊂ ℝN, 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.