Asymptotic Analysis

This paper is concerned the transient Landau Lifschitz equations for the magnetic moment without exchange interaction coupled with Maxwell's equations as well as the equations for a nonlinear dielectric polarization. The main goals is the thin film limit for very flat domains. In this limit one obtains ordinary differential equations for the magnetic moment and the dielectric polarization respectively.

In this paper, solutions of the Sturm–Liouville problem associated with the one‐parameter family of non‐Hermitian Hamiltonians H_α=p²+i(q³+αq) are investigated. We also present some initial results on the distribution of zeros of eigenfunctions. This distribution seems to exhibit a complex version of the classical results on real oscillators.

In this paper we study the global existence and the relaxation limit for a 3×3 hyperbolic system of conservation laws with globally Lipschitz relaxation term. In particular, the convergence for solutions in Sobolev spaces toward the solutions to the equilibrium system, which is a 2×2 incompletely parabolic system, is proved. This is the first example of a semilinear relaxation approximation for an incompletely parabolic system. Thanks to the scaling we use, such convergence can also be viewed as the passage from the viscosity of the memory type to the viscosity of the rate type in the study of viscoelastic materials. Finally, a result of convergence for travelling waves is also provided.

The paper deals with homogenization of initial value problems for the standard parabolic equation in a periodically or randomly perforated domain. The boundary of the domain is assumed to be semi‐permeable, and this implies Signorini boundary conditions for the sought variable. Convergence of solutions to the solution of the homogenized problem is proved under the minimal number of assumptions with respect to regularity of initial data and geometry of the boundaries.

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding. We give the upper bound of the distance between the unfolded gradient of a function belonging to H¹(Ω) and the space ∇_xH¹(Ω)⌖∇_yL²(Ω;H¹_per(Y)). These distances are obtained thanks to a technical result presented in Theorem 2.3: the periodic defect of a harmonic function belonging to H¹(Y) is written with the help of the norms H^1/2 of its traces differences on the opposite faces of the cell Y. The error estimate is obtained without any supplementary hypothesis of regularity on correctors.

In this paper we obtain the exact controllability for wave equations with an equivalued boundary on a “hole” in a bounded domain, and prove that when the “hole” shrinks to a point, the HUM solutions converge in a suitable sense.