Scalar and vector problems on extension domains

Lancia Maria Rosaria
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.

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