Global Positivity Estimates and Harnack Inequalities for the Fast Diffusion Equation

We investigate local and global properties of positive solutions to the fast diffusion equation ut = ∆u m in the range (d− 2)+/d < m < 1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space R we prove sharp Local Positivity Estimates (Weak Harnack Inequalities) and Elliptic Harnack inequalities; we use them to derive sharp Global Positivity Estimates and a Global Harnack Principle. For the mixed initial and boundary value problem posed in a bounded domain of R with homogeneous Dirichlet condition, we prove Weak and Elliptic Harnack Inequalities.