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(¿). The results indicate that N(¿) deviates from the form given by the modified Weyl-Berry-Lapidus conjecture as it shows a correction of ¿N(¿)~¿Df/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.