Lyapunov-type Inequalities for Differential Equations

Let us consider the linear boundary value problem

((0.1))

where and is defined by Classical Lyapunov inequality states that for any function where The constant 4/L is optimal. Let us note that Lyapunov inequality is given in terms of the usual norm in the space L¹(0, L). In this paper we review some recent results on L_p Lyapunovtype inequalities, , for ordinary and partial differential equations on a bounded and regular domain in In the last case, it is showed that the relation between the quantities p and N/2 plays a crucial role, pointing out a deep difference with respect to the ordinary case. In the proof, the best constants are obtained by using a related variational problem and Lagrange multiplier theorem. Finally, the linear results are combined with Schauder fixed point theorem in the study of resonant nonlinear problems. The authors have been supported by the Ministry of Science and Technology of Spain MTM2005- 01331 and by Junta de Andalucia (FQM116).