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
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 (see 2 and 3), 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).
Many industrial and physical phenomena can be modeled by both scalar and vector BVPs (linear or nonlinear) possibly with unusual boundary conditions (e.g. Venttsel' boundary conditions). We mention: problems of wettability [ZaLiGeCle], or some problems of human physiology, diffusion of sprays and gases in the lungs [ZaLiGeCle], [Ku], fractal antennas [WG], [PU], tumor growth in biological systems, non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media ([DIA] and references therein), statistical mechanics and quantum fields on fractals [AK1,AK2,AK3].
From the point of view of applications, fractal surfaces turn out to be useful tools in all those physical phenomena that take place in small volumes with large surfaces. Fractal surfaces are not so diffused because they are not self-similar sets and they do not enjoy many of the properties satisfied by fractal curves.
In the last decades, there was a growing interest in studying the properties of fractal surfaces both from a dynamical point of view as well as from a static one. More recent is the attempt to define integro-differential operators on such irregular sets [CS2].
The expertise of the P.I. as well as of most of the participants to the group is well-documented by the papers [L1],[L2],[LVe1],[LVe2],[LV3],[LV1],[LV2],[LMV],[MV1],[MV2],[MV3],[CLD].
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 collaboration with physicists and engineers will be crucial in order to construct meaningful models of interest for the applications and to interpret the numerical results.
[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, Contemp. Math.601, Amer. Math. Soc., Providence, RI, 2013, 1-21.
[AK2] E. Akkermans, G. Dunne, A. Teplyaev, Physical Consequences of Complex Dimensions of Fractals,Europhys. Lett. 88, 40007 (2009).
[AK3] E. Akkermans, G. Dunne, A. Teplyaev, Thermodynamics of photons on fractals. Phys. Rev. Lett.105(23):230407, 2010.
[CLD] M. Cefalo, G. Dell'Acqua, M.R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers. Appl. Math. Comput. 218 (2012).
[DIA] J. I. Diaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics. 106. Pitman, Boston, MA, 1985.
[L1] M.R.Lancia, A transmission problem with a fractal interface. Z. Anal. Anwendungen 21 (2002).
[L2] M.R.Lancia, Second order Trasmission Problems across a fractal surface. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 27 (2003).
[LMV] M.R. Lancia, U. Mosco, M.A. Vivaldi, Homogenization for conductive thin layers of pre-fractal type. J. Math. Anal. Appl. 347 (2008).
[LVe1] M.R.Lancia, P.Vernole, Convergence results for parabolic transmission problems across highly conductive layers with small capacity. Advances in Mathematical Sciences and Applications, 16, (2006).
[LVe2] M.R.Lancia, P.Vernole, Irregular heat flow problems. SIAM J. Math. Anal. 42 (4) (2010).
[LV1] M.R. Lancia, M.A. Vivaldi, Lipschitz spaces and Besov traces on self-similar fractals. Rend. Acc. Naz. Sci. XL Mem. Mat. Appl. (5), 23, (1999).
[LV2] M.R.Lancia, M.A.Vivaldi, On the regularity of the solutions for transmission problems. Adv. Math. Sci.Appl. 12 (2002).
[LV3] M.R.Lancia, M.A.Vivaldi, Asymptotic convergence of transmission energy form. Adv. Math. Sci. Appl. 13 (2003).
[MV1] U.Mosco, M.A.Vivaldi, An example of fractal singular homogenization. Georgian Math. J. 14 (2007).
[MV2] U.Mosco, M.A.Vivaldi, Fractal reinforcement of elastic membranes. Arch. Ration. Mech. Anal. 194 (2009).
[MV3] U. Mosco, M. A. Vivaldi, Vanishing viscosity for fractal sets. Discrete Cont. Dyn. Syst. Ser. B. 28 (2010).
[WG] D.H. Werner, S. Ganguly, An overview of fractal antenna engineering research. Antennas and Propagation Magazine, IEEE, 45 (1) (2003).
[ZaLiGeCle] D. Zang F. Li, X. Geng, K. Lin, P. Clegg, Tuning the wettability of an alluminium surface via chemically deposited fractal dendrite structure. Eur. Phys. J. E (2013).