A diffusion model for patch formation on cellular surfaces

A number of phenomena of interest in the biology of the cellular membrane can be modeled by a diffusion process which is undergone by certain objects on the cellular surface.We consider one such model and are led to the task of estimating the principal eigenvalue of an elliptic operator with homogeneous Dirichlet boundary conditions.The requested estimate is carried out by two different methods: one makes use of a Lyapunov-type technique and the other applies the Cameron—Martin—Girsanov formula. These methods apply to much more general situations than the specific one considered here.This work was carried out within the frame of a research program partially supported by C.N.R.-GNAFA (G.D.G.) and C.N.R.-GNFM (A.G., F.M.).