Quantitative Jacobian Bounds For The Conductivity Equation In High Contrast Composite Media
Résumé
We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes both positive and negatives values. In this work, we construct a class of inclusions $ Q $ and boundary conditions $ \phi $ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.
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