Carlos Conca

BLOCH WAVE HOMOGENIZATION AND SPECTRAL ASYMPTOTIC ANALYSIS

We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Eloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of. the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum, which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum, which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too. 0 Elsevier, Paris

BLOCH APPROXIMATION IN HOMOGENIZATION AND APPLICATIONS

The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in this paper. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions

Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in 2

u^\varepsilon

Effective Chemical Processes in Porous Media

of boundary value problems associated with such operators when the period

Added mass and damping in fluid-structure interaction

\varepsilon>0

Existence and location of eigenvalues for fluid-solid structures

of the coefficients is small. In a previous work by C. Conca and M. Vanninathan [SIAM J. Appl. Math., 57 (1997), pp. 1639--1659], a new proof of weak convergence as

Identification of immersed obstacles via boundary measurements

\varepsilon\to 0

Bloch-wave homogenization for a spectral problem in fluid-solid structures

towards the homogenized solution was furnished using Bloch wave decomposition.Following the same approach here, we go further and introduce what we call Bloch approximation, which will provide energy norm approximation for the solution

Existence of solutions for the equations

u^\varepsilon

Motion of a rigid body in a viscous fluid

. We develop several of its main features. As a simple application of this new object, we show that it contains both the first and second order correctors. Necessarily, the Bloch approximation will have to captur...