Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations
type: Item
instance of: scholarly article
title: Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations
zbMATH Open document ID: 1202.35336
author: L. Robbiano
author: Jérôme Le Rousseau
published in: Archive for Rational Mechanics and Analysis
publication date: 2010-07-06T00:00:00Z
review text: The authors consider a second-order elliptic operator \(A = - \partial_{x_0}^{2} - \nabla_{x}(c(x)\nabla_{x})\) in a bounded domain of \(\mathbb R^{n+1}\), \(n \geq 2\), where the coefficient \(c(x)\) is piecewise smooth yet discontinuous across a smooth interface \(S\). They prove a local Carleman estimate for \(A\) in the neighbourhood of any point of \(S\). The Calderón projector technique is used to obtain the estimate. Further, using the method of \textit{G. Lebeau} and \textit{L. Robbiano} [Commun. Partial Differ. Equations 20, No.~1--2, 335--356 (1995; Zbl 0819.35071)] they prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator \(\partial_{t} - \nabla_{x}(c(x)\nabla_{x})\). The results of this paper opens perspective for future research towards the null controllability of semi-linear parabolic equations in space of dimension \(n \geq 2\) and towards more complicated situations, for instance, in the case of coefficients with singularities that do not lie on a smooth interface.
reviewed by: Wiesław Kotarski
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 5731005
zbMATH Keywords: elliptic operators
zbMATH Keywords: Carleman estimate
zbMATH Keywords: null-controllability of linear parabolic equations
MaRDI profile type: Publication
full work available at URL: s00205-009-0242-9
OpenAlex ID: W2027369553
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DOI: 10.1007/S00205-009-0242-9
description: scientific article; zbMATH DE number 5731005