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In this work we introduce a generalized linear model regulating the spread of population displayed in a d-dimensional spatial region Ω of R d constituted by two juxtaposed habitats having a common interface Γ. This model is described by an operator L of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in L p-spaces using the H ∞-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by L which is important for the study of the complete nonlinear generalized diffusion equation associated to it.

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The aim of this paper is to give asymptotic models for the impedance of contrasted multi-thin layers for the harmonic Maxwell's equations. We start from a transmission problem which describes the scattering of electromagnetic waves by an obstacle covered with a thin coating (superposition of different thin layers of dielectric materials).We show how to model the effect of the thin coating by an impedance boundary conditions on the boundary of the propagation domain. To this end, we use a technique of abstract differential equations.