Many physical and biological phenomena take place across disordered and wild media. For instance, diffusion processes in physical membranes, the current flow across rough electrodes in electrochemistry, diffusion of sprays in the lungs are transport phenomena which occur, from the geometrical point of view, across irregular layers. Irregular structures provide appropriate frameworks to irrigation models, bronchial systems, root infiltration, tree foliage and other kinds of applications where surface effects are dominant. The role of surface roughness also in contact mechanics is relevant to processes ranging from adhesion to friction wear and lubrication. It also has a deep impact on applied sciences, including coating technology and design of microelectro-mechanical systems.
In particular, DIFFUSION PROCESSES ON IRREGULAR STRUCTURES could be applied to get a deeper insight in many physical phenomena such as particle diffusion, spin diffusion, and diffusion regimes in electrochemistry.
In this framework, fractal layers or boundaries provide new interesting setting to describe wild or irregular media and bodies in which "boundaries" are "large" while bulk is "small". Hence, in the theory of boundary value problems, a new perspective emerges to model phenomena in which the surface effects are enhanced.
The present project is concerned with the study of Partial Differential Equations arising as models of diffusion processes on irregular structures: domains with non-smooth boundaries, fractal boundaries and fractal layers.
Problems on irregular structures are particularly significant because they provide us with a deep insight into an arena where Euclidean and fractal concepts and techniques cross each other. Such an interbreeding is indeed the marking trait of the present research project.
As many phenomena in physics and in nature can be modeled by fractals, the development of mathematical theories with the construction of new tools based on fractal concepts and techniques should then help us to understand such phenomena in a better way.
For example, in the paper [RGS] it is shown experimentally that irregular morphology of the interface significantly speeds up heat transfer. This result suggests that irregular or fractal shapes can be used to increase heat removal from pulsed heat sources in electronics and microelectronic devices (see, for example, the simulations in [Fai]).
Diffusion of medical sprays and oxygen transport to and across alveolar membranes in the human lung are transport phenomena which occur across irregular layers well modeled by fractal curves and surfaces (see, for example, the numerical computations in [HU]).
Mathematical analysis of these problems are essential to confirm (or disprove) theoretical models based on experimental data (see [DFS]).
Moreover in this project we merge the fractal structures with fractional operators in time and in space.
A type of space fractional equation we consider is the one involving the fractional Laplacian, for instance. Equations involving time-space fractional derivative have been considered in order to model ground water flows and transport ([SMZZC], fractional in space and example of superdiffusion), the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells ([GC], fractional in time and example of subdiffusion). Further applications can be found in the theory of viscoelasticity ([S]) and in modeling the cardiac tissue electrode interface ([Mag]).
The expertise of the P.I. as well the participants to the group is well-documented by the papers [C1][C2][C3][C4][C5][C6][C7][CL1] [CV1][CV2][CV3][CV4][CV5][CV6][CV7][CV8][FMT][L1][L2][LV1][LV2][MOSCH] and the references in Pubblicazioni dei Componenti il gruppo di ricerca.
The P.I. and the participants to the group have given talks in the recent 8TH EUROPEAN CONGRESS OF MATHEMATICS 2021.
We think that our techniques and tools introduced up to now will be a good starting point to develop other 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.
[CV1] R. Capitanelli, M.A. Vivaldi, Trace Theorems on Scale Irregular Fractals in Classification and Application of Fractals, Nova Publishers, 2011.
[CV2] R. Capitanelli, M.A. Vivaldi, Insulating Layers and Robin Problems on Koch Mixtures, Journal of Differential Equations 251 (2011).
[CV3] R. Capitanelli, M.A. Vivaldi, On the Laplacean transfer across fractal mixtures, Asymptotic Analysis 83 (2013).
[CV4] R.Capitanelli, M.R. Lancia, M.A. Vivaldi, Insulating layers of fractal type, Differential and Integral Equations 26 (2013)
[CV5] R. Capitanelli, M.A. Vivaldi, Uniform weighted estimates on pre-fractal domains, Discrete Contin. Dyn. Syst. Ser B 19 (2014).
[CV6] R. Capitanelli, M.A. Vivaldi, Weighted estimates on fractal domains¿, Mathematika, 61 (2015)
[CV7]R. Capitanelli, M.A. Vivaldi, Quasi-filling fractal layers Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015)
[CV8] R. Capitanelli, M.A. Vivaldi, Reinforcement problems for variational inequalities on fractal sets, Calculus of Variations and Partial Differential Equations 54 (2015).
[DFS] D.S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces. Phys. Rev. E 73 (2006).
[Fai] M. S. Fairbanks and al., Fractal electronic devices: simulation and implementation, Nanotechnology 36 (2011), 365304.
[FMT] S. Filippas, L. Moschini, A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian. Arch. Ration. Mech. Anal. 208 (2013).
[GC] I. Golding, E. C. Cox, Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96 (2006), # 098102.
[HU] Hou and al., Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does Oxygen Enter the Alveolar System?, Fractals in Biology and Medicine, 2005.
[Mag] R.L. Magin, Modeling the Cardiac tissue electrode interface using frac- tional calculus. J. of Vibration and Control 14 (2008).
[MOSCH] L. Moschini, New Liouville theorems for linear second order degenerate elliptic equations in divergence form. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005).
[RGS] A. Rozanova-Pierrat, D. S. Grebenkov, B. Sapoval, Faster Diffusion across an Irregular Boundary, Phys. Rev. Lett. 108 (2012) 240602 .
[S] E. Soczkiewicz, Application of fractional calculus in the theory of vis- coelasticity. Molecular and Quantum Acoustics 23 (2002).
[SMZZC] H. Sun and al., A fractal Richards equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Advances in Water Resources 52 (2013).