Calculus of Variations and Geometric Measure Theory

A. Davini - M. Ponsiglione

Homogenization of two--phase metrics and applications

created by ponsiglio on 16 Mar 2006
modified by davini on 11 Jan 2009


Published Paper

Inserted: 16 mar 2006
Last Updated: 11 jan 2009

Journal: J. Analyse Math.
Year: 2006


We consider two--phase metrics of the form $\phi(x,xi):= \alpha 1_{B_\alpha}(x)\,
+ \beta 1_{B_\alpha}(x)\,
$, where $\alpha$,$\beta$ are fixed positive constants, and $B_\alpha$, $B_\beta$ are disjoint Borel sets whose union is $ R^N$, and we prove that they are dense in the class of symmetric Finsler metrics $\phi$ satisfying $$ \alpha
<= \phi (x,xi) <= \beta
\quad\mbox{ on } RN\times RN. $$ Then we study the closure $Cl(M_t^{\alpha,\beta})$ of the class $M_t^{\alpha,\beta}$ of two--phase periodic metrics with prescribed volume fraction $t$ of the phase $\alpha$. We have not a complete answer to this problem at the moment: we give upper and lower bounds for the class $Cl(M_t^{\alpha,\beta})$, and we localize the problem, generalizing the bounds to the non--periodic setting. Finally, we apply our results to study the closure, in terms of $\Gamma$--convergence, of two--phase gradient-constraints in composites of the type $f(x, D u) <= C(x)$, with $C(x)$ is in $\{\alpha, \, \beta\}$ for almost every $x$.

Keywords: Homogenization, composites, Finsler metrics, $\Gamma$--convergence