On a counter-example to quantitative Jacobian bounds

Capdeboscq, Yves

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 $H^{1/2}\left(\partial \Omega \right)$ 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 $H^{1/2}\left(\partial \Omega \right)$ 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].

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