First Bloch eigenvalue in high contrast media

TL;DR

This paper investigates the asymptotic behavior of the first Bloch eigenvalue in high contrast periodic media, revealing convergence properties, higher-order expansions in two dimensions, and discontinuities in three dimensions due to microstructure effects.

Contribution

It provides new insights into the asymptotic limits and expansions of the first Bloch eigenvalue in high contrast media, including dimension-specific phenomena and discontinuities.

Findings

Convergence of the first Bloch eigenvalue as contrast increases.

Higher-order expansion in two dimensions up to fourth order.

Discontinuity of the eigenvalue in three dimensions due to fiber reinforcement.

Abstract

This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast $\ep Y$-periodic conductivity. When the conductivity is bounded in $L^1$ and the constant of the Poincar\'e-Wirtinger weighted by the conductivity is very small with respect to $\ep^{-2}$, the first Bloch eigenvalue converges as $\ep\to 0$ to a limit which preserves the second-order expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourth-order under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a $L^1$-unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given low-contrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch parameter around zero.