Ricerc@Sapienza

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We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials.

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through
interfaces, which is mediated by a suitable Laplace-Beltrami type equation.

We will present some existence and uniqueness theorems for two different two-scale problems ([1]). In this framework, a two-scale problem is a system of PDEs involving two unknowns (u; u_1), the first one just depending on a set of space variables denoted by x (usually called the macroscopic or slow variables) and on the time t, the second one depending on a second set of spatial variables y, beside the old ones (i.e. u_1 depends on (x; y; t)). The second set of space variables y are usually called microscopic or fast variables.

In this paper we prove existence and uniqueness for a parabolic and a parabolic/elliptic two-scale system of partial differential equations typically arising in homogenization theory or in macroscale-microscale applied models.