Abstract 

Physical or chemical processes governed by the Laplace equation with mixed boundary
condition are generally called Laplacian transport phenomena. Its examples 
are found in physiology (oxygen diffusion towards and through alveolar membranes), 
in electrochemistry (electric transport through metallic electrodes), in heterogeneous 
catalysis (diffusion of reactive molecules towards catalytic surfaces).
The influence of an irregular geometry, crucial for these phenomena,
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.
An extensive numerical study of different aspects of the Laplacian transport towards
irregular interfaces, modelled by deterministic or stochastic Koch boundaries, has
led us to numerous results. Most importantly, it has been established that
the proportion of eigenmodes of the Dirichlet-to-Neumann operator 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. 
An analytical model of the impedance of self-similar fractals was developed. In particular, 
this model is based on a hierarchy of characteristic scales of the boundary
and allows to study finite generations (prefractals) of arbitrarily high order. 
The experimental study of Koch electrodes showed that this theoretical 
approach allows to take into account the geometrical irregularity without specific
knowledge of the microscopic transport mechanism. This result opens a new branch of 
possible applications, in electrochemistry or in other fields.