| This dissertation is concerned with the study of free boundary problems related to degenerate diffusion. The main part of this work is devoted to the free boundary regularity for the generalized porous medium equation. We show the Cinfinity-regularity of the solutions of the Cauchy problem for the generalized porous media equation u t = Deltaphi(u) in R n x [0, infinity), u(x, 0) = u0 on Rn, where phi(u) = Smi=1ciua i with, ci > 0 and alpha i > alphai-1 > 1. The initial data u0 is considered to be nonnegative, integrable and compactly supported. It is well known that if the initial data u0 is nonnegative, integrable and compactly supported, then the Cauchy problem for the porous medium equation with phi(u) = um admits a unique solution on Rn x (0, infinity) which has constant mass. Thus the study of the free boundary Gamma = ∂( suppu ) becomes of interest. We show that when the initial data satisfy appropriate regularity assumptions, then the pressure f = phi '(u) is smooth up to the interface Gamma for all t ∈ (0, T), for some T > 0.; In addition, we show the short time existence and regularity of the Mean Curvature Flow on a 3-dimensional surface of revolution, by considering it as free boundary problem. |
