Vector Boundary value problems on Fractafolds
Fractal analysis is a recent research field (15-20 years old). The rigorous mathematical formulation of problems coming from mathematical physics, well-established in the classical Euclidean setting, requires new tools and techniques in the fractal case, according to the problem at hand.
We have an expertise in the study of scalar BVPs in fractal domains. Although some problems are still open and the research is still ongoing, the study of vector BVPs is at the very beginning. Aim of this project is to focus mainly on vector BVPs in 3D domains with fractal boundary (fractafolds) as well as to study linear and quasilinear scalar problems describing diffusion processes possibly anomalous in/within fractal domains.
Our goal is also to test our numerical results through laboratory experiments by producing a fractal prototype useful for industrial applications.
Our research will be divided in 3 main topics:
1) GAFFNEY INEQUALITY: A KEY TOOL FOR VECTOR ANALYSIS ON FRACTAFOLDS
i) Maxwell equations in fractal domains
ii) Navier-Stokes equations with non-standard boundary conditions
2) NONLOCAL DIFFUSION WITH DYNAMICAL BOUNDARY CONDITIONS IN FRACTAL DOMAINS
i) Linear case
ii) Space p-fractional diffusion
iii) Time fractional equations
3) QUASILINEAR PROBLEMS
i) Regularity results for the solution of obstacle problems for the p-Laplacian in polygonal domains
ii) Obstacle problems for the p-Laplacian in non-smooth domains