On the one-dimensional parabolic obstacle problem with variable   coefficients
  This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results. 
Adrien Blanchet (CEREMADE, Cermics), Jean Dolbeault (CEREMADE), Regis   Monneau (CERMICS)