Existence and uniqueness for some two-scale systems involving tangential operators

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. Such kind of problems have a wide range of applications in many models in which physical properties at a macroscopic level are affected by phenomena taking place at a microscopic level and which, in turn, are affected by the evolution of the macroscopic state variable. A relevant applications of these kind of problems takes place in the wellknown homogenization theory.