Modeling in fractal analysis is a recent and active research field since the last twenty years. In this period, our group has acquainted an expertise on this topic. Depending on the problem, different tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean setting as well as delicate functional inequalities which are instrumental to the study of some BVPs. In view of real world applications, it is also crucial to yield a numerical approximation of the solution of the problem at hand. To achieve this goal, two steps are necessary. The former is to yield a constructive approach which allows to define an abstract operator in terms of smoother ones, the latter is to construct an efficient numerical approximation scheme. Irregular structures appear in different situations, e.g. in problems of wettability, some problems of human physiology, diffusion of sprays and gases in the lungs, tumor growth in biological systems, non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media, statistical mechanics and quantum fields on fractals. From the mathematical point of view, these problems can be modeled by either scalar or vector, linear or nonlinear, autonomous or non-autonomous problems, possibly with unusual boundary conditions.
Modeling in fractal analysis is a recent and active research field since the last twenty years. Our group has acquainted an expertise on this topic. Depending on the problem, different tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean setting, as well as delicate functional inequalities which are instrumental to the study of some BVPs. In view of real world applications, it is also crucial to yield a numerical approximation of the solution for the problems at hand. To achieve this goal, two steps are necessary. The former is to yield a constructive approach which allows to define an abstract operator in terms of smoother ones; the latter is to construct an efficient numerical approximation scheme. Irregular structures appear in different situations, e.g. in problems of wettability [Za], or some problems of human physiology, diffusion of sprays and gases in the lungs [Ku], fractal antennas [WG], tumor growth in biological systems, non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media ([D] and references therein), statistical mechanics and quantum fields on fractals [AK1,AK2,AK3]. From the mathematical point of view, these problems can be modeled by boundary value problems possibly with unusual boundary conditions (e.g. of Venttsel' type).
In all those phenomena which take place in small volumes with large surfaces, fractal surfaces can efficiently model the exchange surface. From the mathematical point of view, they are non-self-similar sets and they do not enjoy many of the properties satisfied by fractal curves.
Our group has obtained many results on some scalar BVPs in fractal domains. More recently, we focused on vector BVPs. In [4] we defined a fractal boundary integrodifferential operator and in [3] we considered the numerical approximation of a 3D Stokes flow, and we built an ad-hoc mesh allowing us to obtain an optimal rate of convergence.
Our aim is now to focus on vector BVPs (possibly nonlinear) in irregular domains, coupled possibly with non-standard boundary conditions. Moreover, we will investigate suitable numerical schemes to approximate our problems. The first examples in the literature of numerical approximation of heat-type problems in 2D fractal domains are given in [CLD] and [CLH], while that of 3D vector fractal problems by mixed methods are in [3] and [4]. These numerical schemes have a high computational cost.
We think that our techniques and tools introduced up to now will be a good starting point to develop new tools or adapt existing ones to attack this new direction of research.
The expertise of the P.I. is well-documented by the list of publications and by the references quoted therein.
The P.I. has recently given a talk during the "8th European Congress of Mathematics", which was held online from June 20 to June 26 2021. Moreover, the P.I. won a PostDoc (assegno di ricerca) for twelve months (starting 01/07/2021) at the Department of Basic and Applied Sciences for Engineering (SBAI) at Sapienza Università di Roma.
The P.I. got a grant in 2020 from the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) group of INdAM (Istituto Nazionale di Alta Matematica).
[AK1] E. Akkermans, Statistical mechanics and quantum fields on fractals, Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics, Amer. Math. Soc., Providence, RI, 2013.
[AK2] E. Akkermans, G. Dunne, A. Teplyaev, Physical consequences of complex dimensions of fractals. Europhys. Lett. 2009.
[AK3] E. Akkermans, G. Dunne, A. Teplyaev, Thermodynamics of photons on fractals. Phys. Rev. Lett. 2010.
[CLH] M. Cefalo, M.R. Lancia, H. Liang, Heat-flow problems across fractal mixtures: regularity results of the solutions and numerical approximation. Differential Integral Equations 2013.
[D] J.I. Diaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics. 106. Pitman, Boston, MA, 1985.
[WG] D.H. Werner, S. Ganguly, An overview of fractal antenna engineering research. Antennas and Propagation Magazine, IEEE, 2003.