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We characterize the principal eigenvalues and eigenfunctions in R N , and present comparison results, for higher dimensional p-Laplacian. Our main tool is Picone's identity. In this way we extend several recent results on spectral and comparison properties for differential equations. Academic Press

The strong oscillation of a second order symmetric elliptic operator is shown to be equivalent to the oscillation of all solutions of the associated homogeneous equation. Extensions of a nonoscillation theorem and of an existence theorem are obtained as applications.

In this paper we present and analyze a nutrient-oxygen-phytoplankton-zooplankton mathematical model simulating lagoon ecological interactions. We obtain sufficient conditions, based on principal eigenvalue criteria -- for the existence of periodic solutions. A decoupled model which arises in the high nutrient regime is then considered in further detail for gathering some explicit conditions on parameters and averages of exogenous inputs needed for coexistence. An application to Italian coastal lagoons is finally obtained by parameter estimation and comparison with real data. A biological interpretation of the mathematical results is also presented.

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Abstract. In this paper we discuss the multiscale analysis of Maxwell's equations in composite materials with a periodic microstructure. The new contributions in this paper are the determination of higher-order correctors and the explicit convergence rate for the approximate solutions (see Theorem 2.3). Consequently, we present the multiscale finite element method and derive the convergence result (see Theorem 4.1). The numerical results demonstrate that higher-order correctors are essential for solving Maxwell's equations in composite materials.

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