Laplacian transport towards irregular surfaces: the mathematics

Denis Grebenkov
Seconda Università di Napoli
Laboratoire de Physique de la Matiere Condensee

Laplacian transport is ubiquitous and present in such various fields as physiology, electrochemistry, catalysis, etc. In a simple macroscopic model, stationary diffusive motion in the bulk is represented by the Laplace operator, while the microscopic "interaction" on the surface is incorporated via Fourier or mixed boundary condition. The influence of a geometrical irregularity of the boundary is known to be crucial for these phenomena.

In this lecture, we present a theoretical approach to study the Laplacian transport towards irregular surfaces in a systematic rigorous way. In this frame, the influence of a geometrical irregularity can be fully taken into account using a mathematical operator called Dirichlet-to-Neumann operator. Its spectral properties completely determine the linear response of the system in question. For simple geometrical shapes like disk or sphere, one can derive a number of analytical results. A numerical analysis is however required to get quantitative description for more irregular interfaces. We focus on the Laplacian transport towards deterministic and stochastic Von Koch boundaries. In this case, it has been established that the proportion of the Dirichlet-to-Neumann operator eigenmodes contributing to the impedance of the interface is very low. It has been shown that its eigenvalues can be interpreted as the inverses of characteristic lengths of the boundary. We propose an analytical model of the impedance of self-similar fractals which is based on a hierarchy of characteristic scales of the boundary. At last, the experimental study of Von Koch electrodes confirmed a possibility for taking the geometrical irregularity into account without specific knowledge of the microscopic transport mechanism. Several open problems and further perspectives will be discussed.

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