Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations

Abstract

This paper is devoted to the study of $L_p$ Lyapunov-typeinequalities for linear systems of equations with Neumann boundaryconditions and for any constant $p \geq 1$. We consider ordinaryand elliptic problems. The results obtained in the linear case arecombined with Schauder fixed point theorem to provide new resultsabout the existence and uniqueness of solutions for resonantnonlinear problems. The proof uses in a fundamental way thenontrivial relation between the best Lyapunov constants and theminimum value of some especial minimization problems.