On a counter-example to quantitative Jacobian bounds

Yves Capdeboscq

doi:10.5802/jep.21

On a counter-example to quantitative Jacobian bounds
[Sur un contre-exemple aux bornes quantitatives du jacobien]

Yves Capdeboscq ¹

¹ Mathematical Institute, University of Oxford, Andrew Wiles Building Woodstock Road, Oxford OX2 6GG, UK

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 171-178.

Résumé
Abstract

Cette note fournit un contre-exemple à la positivité locale du déterminant jacobien des solutions de l’équation de conduction en dimension $3$ . On montre que le signe du déterminant ne peut pas être imposé par un choix a priori de données au bord dans $H^{1 / 2} (\partial Ω)$ dépendant seulement des bornes inférieure et supérieure de la conductivité, même localement. L’argument utilise une conductivité scalaire à deux phases construite par Briane, Milton & Nesi [11, 10].

This note provides a counter-example to the local positivity of the Jacobian determinant for solutions of the conductivity equation in dimension $3$ . It shows that the sign of the determinant cannot be imposed by an a priori choice of boundary data in $H^{1 / 2} (\partial Ω)$ depending only on the upper and lower bound of the conductivity, even locally. The argument uses a scalar two-phase conductivity constructed by Briane, Milton & Nesi [11, 10].

Reçu le : 2015-05-07
Accepté le : 2015-07-15

MR Zbl

DOI : 10.5802/jep.21

Classification : 35J55, 35R30, 35B27
Keywords: Radó-Kneser-Choquet Theorem, hybrid inverse problems, impedance tomography, homogenization
Mot clés : Théorème de Radó-Kneser-Choquet, problèmes inverses hybrides, tomographie d’impédance, homogénéisation

Affiliations des auteurs :

Yves Capdeboscq ¹

¹ Mathematical Institute, University of Oxford, Andrew Wiles Building Woodstock Road, Oxford OX2 6GG, UK

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Yves Capdeboscq. On a counter-example to quantitative Jacobian bounds. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 171-178. doi : 10.5802/jep.21. https://jep.centre-mersenne.org/articles/10.5802/jep.21/

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