Vancostenoble, Judith Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. (English) Zbl 1213.93018 Discrete Contin. Dyn. Syst., Ser. S 4, No. 3, 761-790 (2011). Summary: We consider the following class of degenerate/singular parabolic operators: \[ Pu = u_t-(x^\alpha u_x)_x - \frac {\lambda}{x^\beta} u, \quad x\in (0,1) \]associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters \(\alpha \geq 0\), \(\beta,\lambda \in \mathbb R\), 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. Cited in 30 Documents MSC: 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 93B07 Observability 35K65 Degenerate parabolic equations Keywords:parabolic equation; degenerate diffusion; singular potential; null controllability; Carleman estimates; observability; Hardy-Poincaré’s inequalities PDFBibTeX XMLCite \textit{J. Vancostenoble}, Discrete Contin. Dyn. Syst., Ser. S 4, No. 3, 761--790 (2011; Zbl 1213.93018) Full Text: DOI