[hal-00828333, v1] First Bloch eigenvalue in high contrast media
نویسندگان
چکیده
This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast εY -periodic conductivity. When the conductivity is bounded in L1 and the constant of the Poincaré-Wirtinger weighted by the conductivity is very small with respect to ε−2, the first Bloch eigenvalue converges as ε → 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 L1-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.
منابع مشابه
Spectral convergence for high contrast media with a defect via homogenization
We consider an eigenvalue problem for a divergence form elliptic operator Aε with locally perturbed high contrast periodic coefficients. Periodicity size ε is a small parameter. Local perturbation of coefficients for such operator could result in emergence of localized waves eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for a doubl...
متن کاملBloch Wave Homogenization of Linear Elasticity System
In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a re...
متن کاملA comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures
In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is wellknown that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogeniz...
متن کاملA Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrödinger equations in periodic media. For the underlying equation subject to a highly oscillatory initial data, a hybrid of the WKB approximation and homogenization leads to the Bloch eigenvalue problem and an associated HamiltonJacobi system for the phase in each Bloch band, with the Bloch eigenva...
متن کامل