Ricerc@Sapienza

We study a Stokes flow in a cylindrical-type fractal domain with homogeneousDirichlet boundary conditions. We consider its numerical approximation bymixed methods: finite elements in space and finite differences in time. We intro-duce a suitably refined mesh `a la Grisvard, which in turn will allow us to obtainan optimal a priori error estimate

Fractal structures or geometries have nowadays a key role in all those

> models for natural and industrial processes which exhibit the

> formation of rough surfaces and interfaces. Computer simulations,

> analytical theories and experiments have led to significant advances in

> modeling these phenomena across wild media. Many problems coming from

> engineering, physics or biology are characterized by both the presence

> of different temporal and spatial scales and the presence of contacts

We study a Stokes problem in a three dimensional fractal domain
of Koch type and in the corresponding prefractal approximating domains. We
prove that the prefractal solutions do converge to the limit fractal one in a
suitable sense. Namely the approximating velocity vector elds as well as the
approximating associated pressures converge to the limit fractal ones respec-
tively.

In this paper, we study the periodic homogenization of the stationary heat equation in
a domain with two connected components, separated by an oscillating interface defined
on prefractal Koch type curves. The problem depends both on the parameter n, which
is the index of the prefractal iteration, and ε, that defines the periodic structure of the
composite material. First, we study the limit as n goes to infinity, giving rise to a limit
problem defined on a domain with fractal interface. Then, we compute the limit as ε

We study obstacle problems involving p-Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.

We consider planar skew Brownian motion (BM) across pre-fractal Koch interfaces $\partial \Omega_n$ and moving on $\Omega^\epsilon_n = \Sigma_n \cup \Omega_n$. We study the asymptotic behaviour of the corresponding multiplicative additive functionals when thickness of $\Sigma_n$ and skewness coefficients vanish with different rates.