On a counter-example to quantitative Jacobian bounds
[Sur un contre-exemple aux bornes quantitatives du jacobien]
Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 171-178.

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].

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].

DOI : https://doi.org/10.5802/jep.21
Classification : 35J55,  35R30,  35B27
Mots clés : Théorème de Radó-Kneser-Choquet, problèmes inverses hybrides, tomographie d’impédance, homogénéisation
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Capdeboscq, Yves. 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. http://www.numdam.org/articles/10.5802/jep.21/

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