Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems

Vancostenoble, Judith

Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 801-805.

On étudie la stabilité Lipschitzienne pour des problèmes inverses de détermination d'une source pour l'équation de la chaleur perturbée par un potentiel singulier de la forme −μ/|x|2 avec μ⩽μ⋆(N):=(N−2)2/4 où μ⋆(N) est la constante optimale de l'inégalité de Hardy. Suivant Immanuvilov et Yamamoto (1998) [9], notre preuve repose sur des inégalités de Carleman telles que celles introduites par Fursikov et Immanuvilov (1996) [8] pour l'équation de la chaleur classique. Cependant, il faut ici tenir compte de la singularité. La première étape de la preuve consiste donc en une amélioration des inegalités de Carleman spécifiquement démontrées pour des equations avec un potentiel singulier par Vancostenoble et Zuazua (2008) [15] puis Ervedoza (2008) [7]. Certaines étapes majeures reposent sur diverses formes améliorées de l'inégalité de Hardy.

We address the question of Lipschitz stability results in inverse source problems for the heat equation perturbed by a singular inverse-square potential −μ/|x|2 when μ⩽μ⋆(N):=(N−2)2/4 where μ⋆(N) is the optimal constant in the so-called Hardy inequality. Following Immanuvilov and Yamamoto (1998) [9], our proof is based on Carleman inequalities like those developed by Fursikov and Immanuvilov (1996) [8] for the classical heat equation. However, we need here to take into account the singularity. Therefore, the first step of the proof consists in some improvements of the Carleman inequalities specifically developed for equations with inverse-square potentials by Vancostenoble and Zuazua (2008) [15] and next Ervedoza (2008) [7]. Major steps rely on various improved forms of the Hardy inequality.

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