Presence Of Fractal Boundaries Research Articles

A Cauchy transform for polyanalytic functions on fractal domains

A polyanalytic Cauchy transform over fractal curves is introduced and discussed, which allows us to obtain, in particular, integral representation formulas for smooth and polyanalytic functions in the presence of fractal boundaries.

Scaling of the density of state of the weighted Laplacian in the presence of fractal boundaries

Spectral properties of the weighted Laplace operator in the presence of fractal boundaries are numerically investigated for both Neumann and Dirichlet boundary conditions. This corresponds to the characterization of heat and mass transport in microchannels with irregular and rough surfaces induced by the microfabrication process. The axial velocity field with no-slip boundary conditions, representing the weighting function of the Laplace operator, influences the localization properties of the eigenfunctions and the scaling of the integrated density of state (IDOS) N(lambda). The results indicate that N(lambda) deviates from the form given by the modified Weyl-Berry-Lapidus conjecture as it shows a correction of DeltaN(lambda) approximately lambda(D(f)/4) to the leading-order Weil term. Numerical results are presented for Koch and Koch snowflake fractal boundaries. The role of slip or no-slip boundary conditions of the velocity field on the IDOS is also investigated.

Anomalous Warburg impedance and universal surface magnetic exponent for Gaussian models in the presence of fractal boundaries.

Recent results on the anomalous frequency dependence of the diffusion impedance in electrolytic cells with rough electrodes are shown to be relevant for the problem of surface critical exponents of Gaussian models in the presence of fractal boundaries. On the basis of this, surface magnetic exponents are predicted, which turn out to be independent of the surface fractal dimension. Generalizations of the above results to the cases of anomalous diffusion and of cells with fractal bulk properties are also presented.