## Scalar and vector problems on extension domains

Autore:

Lancia Maria Rosaria

Descrizione:

Many engineering models are characterized by the

presence of different temporal and spatial scales and/or

by the presence of contacts among different components

through rough (fractal) interfaces. These barriers or

boundaries are often highly irregular. In quite all these

phenomena the information flows from a smaller to a

larger scale or viceversa. Fractals provide a useful tool

to describe such wild geometries. A great challenge is to

propose mathematical models which allow investigating

these phenomena with a particular regard to scale effects

and interface interactions as well as their numerical ap-

proximation. The first example in the literature in which

classical BVPs merge with the theory of fractal sets and

operators is due our group and goes back to 2002. Nu-

merical approximation of BVPs in domains with fractal

boundaries or interfaces, which model fast heat diffusion

phenomena across a Koch interface, is more recent. Nev-

ertheless, many problems are still open, but at the same

time, from the point of view of applications, rigorous

formulations and models for vector BVPs are strongly

demanding. We investigated some scalar and Vector

boundary value problems for heat flow, magnetostatics

and fluid dynamics, possibly with non-standard bound-

ary conditions.