Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

Abstract

We consider the following class of degenerate/singular parabolic operators:$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters$\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities