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

Judith Vancostenoble

doi:10.3934/dcdss.2011.4.761

Article Contents

2011, Volume 4, Issue 3: 761-790. Doi: 10.3934/dcdss.2011.4.761

This issue Previous Article Riesz systems, spectral controllability and a source identification problem for heat equations with memory Next Article

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

Judith Vancostenoble^1,

1.
Université de Toulouse; Université Paul Sabatier Toulouse III, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse

Received: March 31, 2009

Revised: November 30, 2009

Published: May 2011

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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

Keywords:

Mathematics Subject Classification: 93B05, 93C20, 93B07, 35K65.

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