On a counter-example to quantitative Jacobian bounds
Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 171-178.

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 (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].

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 (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].

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
Capdeboscq, Yves 1

1 Mathematical Institute, University of Oxford, Andrew Wiles Building Woodstock Road, Oxford OX2 6GG, UK
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Capdeboscq, Yves. On a counter-example to quantitative Jacobian bounds. Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 171-178. doi : 10.5802/jep.21. http://www.numdam.org/articles/10.5802/jep.21/

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