Elliptic estimates in composite media with smooth inclusions: an integral equation approach

Ammari, Habib; Bonnetier, Eric; Triki, Faouzi; Vogelius, Michael

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 2, pp. 453-495.

Nous étudions des milieux composites constitués d'inclusions homogènes de forme ${𝒞}^{1,{\alpha }_0}$, immergées dans une phase matrice constante. Lorsque les inclusions ne se touchent pas, la solution de l'équation de diffusion peut être représentée à l'aide de potentiels de surface, solutions d'un système d'équations intégrales. Nous étudions ce système lorsque la distance inter-inclusion tend vers 0. Nous montrons que les potentiels de surface convergent dans ${𝒞}^{0,\alpha }$, $0<\alpha <{\alpha }_0$, vers des potentiels limites, qui permettent d'obtenir une représentation intégrale du problème limite. Nous en déduisons des estimations sur les solutions dans ${𝒞}^{1,\alpha }$, uniformes par rapport à la distance inter-inclusions.

We consider a scalar elliptic equation for a composite medium consisting of homogeneous ${𝒞}^{1,{\alpha }_0}$ inclusions, $0<{\alpha }_0\le 1$, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functions converge in ${𝒞}^{0,\alpha }$, $0<\alpha <{\alpha }_0$ to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform ${𝒞}^{1,\alpha }$ bounds on the solution, which are independent of the inter–inclusion distances.

Classification : 45F15, 35J15.
Keywords: Composite media, uniform Hölder estimates.
Mot clés : Milieux composites, estimations hölderiennes uniformes.

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