Our group is working on fractal analysis and irregular structures since the end of nineties. Fractal analysis is a rather recent research field (15/20 years). According to the problem, new tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean setting. When dealing with such an irregular framework, a constructive approach is crucial. On one side it allows to define abstract operators as limit of smoother ones on the other side it is important in view of numerical approximations.
Our expertise in the study of scalar BVPs in fractal domains is well established, some problems are still open. More recent is the study of vector BVPs in irregular domains, [CL], [C2]. Aim of this project is to focus on vector BVPs in 3D domains with fractal boundary (fractafolds) as well as to study linear, quasilinear, possibly nonlocal and non autonomous problems in/within irregular domains.
Numerical simulations and laboratory tests will be crucial to produce a fractal prototype useful for industrial applications.
Our group is working on some problems of fractal analysis since the end of nineties. Fractal analysis is a rather recent research field (15/20 years). According to the problem, new tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean setting. When dealing with such an irregular framework, a constructive approach is crucial. On one side it allows to define abstract operators as limit of smoother ones on the other side it is important in view of numerical approximations.
Irregular structures occur in many situations. For instance when considering 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]. All the above problems can be modeled by either scalar or vector BVPs linear or nonlinear, autonomous or non autonomous.
Our expertise in the study of scalar BVPs in fractal domains is well established, some problems are still open (see 2,4,5). On the other side the more recent study of vector BVPs is at the very beginning [CL]. We have been focusing mainly on some vector BVPs in 3D domains with fractal boundary (fractafolds).
From the point of view of applications, fractal surfaces are useful tools in all those physical phenomena that take place in small volumes with large surfaces.
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 [C2].
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]. The P.I. has organized a mini symposium at ICIAM19, July 2019, Valencia, on "Fractals in engineering: models and theoretical results". 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.
[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).
[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).
[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.