Asymptotic Analysis

In the kinetic theory of gases, half‐space problems are important because they govern the behavior of correctors to the Hilbert expansion near the boundaries. This paper takes up the discussion in Coron et al., Comm. Pure Appl. Math. 41 (1988) by a different method, based on the index formula for Wiener–Hopf operators on the half‐line. This method leads naturally to a new result on analogues of Chandrasekhar’s $H$ function, which does not seem easy to obtain by the energy method of Coron et al. (1988).

This paper deals with the limit behaviour of solutions to certain kinds of boundary value problems with equivalued surface on a domain with thin layer. Some applications are given in resistivity well‐logging, etc.

We study a nonlinear homogenization problem in perforated domains for the Ginzburg–Landau functional with a surface energy integral term which was introduced in the theory of liquid crystals. Under certain conditions between the inclusion size and the order of the surface energy integral we compute the nonlinear homogenized problem and prove the convergence of the solutions. An additional relation between the domain size and physical parameters provides the uniqueness of the homogenized limit. Our proof is based on a variational approach and does not require strict periodicity of the perforated domains.

The problem of recovering the asymptotics of a short range perturbation of the Euclidean Laplacian on $\mathbb{R}^n$ from fixed energy scattering data is studied. It is shown that for $n \geq 3$ the asymptotics of a magnetic potential are determined, modulo Gauge invariance, by its scattering matrix at a fixed non‐zero energy. This result also holds for a wide class of scattering manifolds.

Let $u(\varepsilon)$ be the rescaled three‐dimensional displacement field solution of the linear elastic model for a clamped prismatic straight rod ${\varOmega}^\varepsilon$ having cross section with diameter of order $\varepsilon$ , and let $u^0$ be the corresponding Bernoulli–Navier displacement. In this article we establish that the error $\|u(\varepsilon)-u^0\|_{1,{\varOmega}}$ in the reference space $[H^1({\varOmega})]^3$ is of order $\varepsilon^{1/2}$ . We mainly use an auxiliar corrector function and we prove that this estimation cannot be improved using other corrector functions of the same family.