*Published Paper*

**Inserted:** 16 mar 2006

**Last Updated:** 11 jan 2009

**Journal:** J. Analyse Math.

**Year:** 2006

**Abstract:**

We consider two--phase metrics of the form $\phi(x,xi):= \alpha
1_{B_\alpha}(x)\,

xi

+ \beta 1_{B_\alpha}(x)\,

xi

$, 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

xi

<= \phi (x,xi) <= \beta

xi

\quad\mbox{ on } R^{N\times} R^{N.
}
$$
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

**Download:**