Sobolev Inequalities and Ellipticity of Planar Linear Hamiltonian Systems

Abstract

Abstract In this paper we will establish two different classes of ellipticity criteria, called the Lp criteria and the Lp-Lq criteria respectively, for planar linear Hamiltonian systems with periodic coefficients. The criteria are explicitly expressed using the Lp and Lq norms of coefficients and some known Sobolev constants. These results can be considered as the extensions of the famous Lyapunov stability criterion for Hill’s equations.