A Variational Characterization of the Best Lyapunov Constants

Abstract

This chapter is devoted to the definition and main properties of the L p Lyapunov constant, \(1 \leq p \leq \infty,\) for scalar ordinary differential equations with different boundary conditions, in a given interval (0, L). It includes resonant problems at the first eigenvalue and nonresonant problems. A main point is the characterization of such a constant as a minimum of some especial minimization problem, defined in appropriate subsets X p of the Sobolev space H1(0, L). This variational characterization is a fundamental fact for several reasons: first, it allows to obtain an explicit expression for the L p Lyapunov constant as a function of p and L; second, it allows the extension of the results to systems of equations (Chap. 5) and to PDEs (Chap. 4). For resonant problems (Neumann or periodic boundary conditions), it is necessary to impose an additional restriction to the definition of the spaces \(X_{p}, 1 \leq p \leq \infty,\) so that we will have constrained minimization problems. This is not necessary in the case of nonresonant problems (Dirichlet or antiperiodic boundary conditions) where we will find unconstrained minimization problems. For nonlinear equations, we combine the Schauder fixed point theorem with the obtained results for linear equations.