Lyapunov-type inequalities for differential equations

Let us consider the linear boundary value problem u"(x) + a(x)u(x) = 0, x is an element of (0, L), u'(0) = u(L) = 0, where a is an element of Lambda(0) and Lambda(0) is defined Lambda(0) = {a is an element of L-infinity(0, L) \ {0} : integral(L)(0) a(x) dx >= 0, (0.1) has nontrivial solutions}. Classical Lyapunov inequality states that integral(L)(0) a(+) (x) dx > 4/L for any function a is an element of Lambda(0), where a(+) (x) = max{a(x), 0}. The constant 4/L is optimal. Let us note that Lyapunov inequality is given in terms of parallel to a(+)parallel to(1), the usual norm in the space L-1 (0, L). In this paper we review some recent results on L-p Lyapunov-type inequalities, 1 < p <= +infinity, for ordinary and partial differential equations on a bounded and regular domain in R-N. 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.