On Positive Solutions of Elliptic Equations with Periodic Coefficients in [Figure presented]^N, Spectral Results and Extensions to Elliptic Operators on Riemannian Manifolds

This chapter discusses the properties of positive solutions and related spectral results for elliptic operators with periodic coefficients on ℝⁿ. It also discusses the extensions of the results to a larger class of elliptic operators on certain noncompact Riemannian manifolds. The chapter focuses on periodic case. Whenever the elliptic equation Pu = 0 admits a positive solution on ℝⁿ, it also admits a positive exponential type solution. The chapter describes the family of positive solutions of Pu = 0 that are exponentials. It presents the various properties of this family. Some results in spectral theory of elliptic operators with periodic coefficients that are based on the knowledge of the family of positive exponential solutions are discussed in the chapter. The chapter also describes the extensions of the results to a more general class of elliptic operators defined on certain noncompact Riemannian manifolds. There are close connections between the spectral properties of second order elliptic operators and the properties of positive solutions of elliptic equations.