Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities.

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations.

Minimization methods for the one-particle dirac equation.

Taking into account relativistic effects in quantum chemistry is crucial for accurate computations involving heavy atoms. Standard numerical methods can deal with the problem of variational collapse and the appearance of spurious roots only in special cases. The goal of this Letter is to provide a general and robust method to compute particle bound states of the Dirac equation.