Several physical and natural phenomena are characterized on one hand by the presence of different temporal and spatial scales, on the other by the presence of contacts among different components through rough (fractal) interfaces; in all these phenomena the information flows from a smaller to a larger scale or viceversa. Different fields of application are biology, human respiration, engineering disciplines. Our aim is to propose mathematical models to investigate these phenomena as well as their numerical approximation. The lack of regularity of the underlying fractal structures requires new tools and techniques for PDEs . Taking into account our expertise in the study of scalar BVPs in irregular domains we will focus on vector BVPs in fractal domains which involve in the boundary conditions integrodifferential operators.
Our research will be divided in 4 main topics:
1) HEAT DIFFUSION WITH NONLOCAL DYNAMICAL BOUNDARY CONDITIONS IN FRACTAL BOUNDARIES
2) QUASILINEAR PROBLEMS
3) VECTOR ANALYSIS ON FRACTAFOLDS: NAVIER-STOKES EQUATIONS and MAXWELL EQUATIONS IN FRACTAL DOMAINS: THEORY AND APPROXIMATIONS
4) FRACTIONAL DIFFUSION ACROSS FRACTAL INTERFACES
Many natural and industrial processes lead to 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. Fractals provide a good tool to describe such wild geometries. Many irregular geometries can be modeled by fractal curves as well as fractal surfaces. While the former ones are hugely studied, fractal surfaces are not so common because they are typically non self-similar sets and many of the results which hold for fractal curves cannot be shifted to the latter ones. 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 because, from the point of view of applications, fractal surfaces turn out to be useful tools in all those physical phenomena which take place in small volumes with large surfaces. Among the possible applications, 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 (see [DIA] and references therein), statistical mechanics and quantum fields on fractals [AK1,AK2,AK3]. For an educational paper see [Vu].
These problems can be modeled by both scalar and vector BVPs (linear or nonlinear) possibly with unusual boundary conditions. The expertise of the P.I. as well as of most of the participants to the group is well substantiated by the papers [L1],[L2],[LVe1],[LVe2],[LV3],[LV1],[LV2],[LMV],[MV1],[MV2],[MV3]. We think that the techniques and tools developed 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.
[DIA] J. I. Diaz: Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics. 106. Pitman, Boston, MA, 1985.
[KU] V.Kulish, Human respiration, WIT press, Boston 2006.
[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).
[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).
[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).
[LMV] M.R. Lancia, U. Mosco, M.A. Vivaldi: Homogenization for conductive thin layers of pre-fractal type. J. Math. Anal. Appl. 347 (2008).
[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).
[Vu] A. Vulpiani, Funzioni irregolari e oggetti frattali: da Weierstrass a Mandelbrot, in Lettera matematica Prinstem 97, Springer ,2016.
[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) 36-59.