Asymptotic Analysis - Volume 55, issue 1-2

Authors: Cassanas, Roch

Article Type: Research Article

Abstract: Under conditions of clean flow we compute the leading term in the STF when the set of periods of the energy surface is discrete. Comparing to the case of non-degenerate periodic orbits, we obtain a supplementary term which is given in terms of the linearized flow. As particular cases, we give a STF for quadratic Hamiltonians and we obtain the Berry–Tabor formula for integrable systems. For conservative systems (i.e. systems with several first integrals), we give practical conditions to get a clean flow and interpret the leading term of the STF for a compact symmetry. We give several examples to …illustrate our computation. Show more

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 1-32, 2007

Authors: Bildhauer, M. | Fuchs, M.

Article Type: Research Article

Abstract: We consider local minimizers u: Rn ⊃Ω→RN of anisotropic variational integrals of (p, q)-growth with exponents 2≤p≤q≤min {2+p, p n /(n−2) }. If the integrand is of splitting-type, then partial C1 -regularity of u is established.

Keywords: anisotropic energies, vector-valued problems, local minimizers, splitting functionals, partial regularity

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 33-47, 2007

Authors: Moulin, Simon | Vodev, Georgi

Article Type: Research Article

Abstract: For a large class of real-valued potentials, V(x), x∈Rn , n≥4, we prove dispersive estimates for the low-frequency part of eit(−Δ+V) Pac , provided the zero is neither an eigenvalue nor a resonance of −Δ+V, where Pac is the spectral projection onto the absolutely continuous spectrum of −Δ+V. This class includes potentials V∈L∞ (Rn ) satisfying V(x)=O(〈x〉−(n+2)/2−ε ), ε>0. As a consequence, we extend the results in Commun. Pure Appl. Math. 44 (1991) to a larger class of potentials.

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 49-71, 2007

Authors: Gladiali, Francesca | Grossi, Massimo

Article Type: Research Article

Abstract: In this paper we study the radial solutions of the problem \[\cases{-\Delta u=\lambda\mathrm{e}^{u}& \mbox{in} \varOmega,\cr u=0& \mbox{on} \partial\varOmega,}\] where Ω is an annulus of RN , N≥2 and λ is close to zero. Among the other results we show the existence of a singular limit and some qualitative properties of the solution.

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 73-83, 2007

Authors: Gruais, Isabelle | Poliševski, Dan

Article Type: Research Article

Abstract: We study the homogenization of a diffusion process which takes place in a binary structure formed by an ambient connected phase surrounding a suspension of very small spheres distributed in an ε-periodic network. We consider the critical radius case with finite diffusivities in both phases. The asymptotic distribution of the concentration is determined, as ε→0, assuming that the suspension has mass of unity order and vanishing volume. It appears that the ambient macroscopic concentration is satisfying a Volterra integro-differential equation and it is defining straightly the macroscopic concentration associated to the suspension.

Keywords: diffusion, homogenization, fine-scale substructure, Volterra integro-differential equation

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 85-101, 2007

Authors: Jollivet, A.

Article Type: Research Article

Abstract: We consider the multidimensional Newton–Einstein equation in static electromagnetic field p· =F(x, x· ), F(x, x· )=−∇V(x)+(1/c)B(x)x· , p=x· /\sqrt(1−|x· |2 /c2 )), p· =dp/dt, x· =dx/dt, x∈C1 (R, Rd ),  (*) where V∈C2 (Rd , R), B(x) is the d×d real antisymmetric matrix with elements Bi,k (x)=∂/∂xi  Ak (x)−∂/∂xk  Ai (x), and |∂x j Ai (x)|+|∂x j V(x)|≤β|j| (1+|x|)−(α+|j|) for x∈Rd , |j|≤2, i=1, …, d, and some α>1. We give estimates and asymptotics for scattering solutions and scattering data for the equation (*) for the case of small angle scattering. We show that at high energies the velocity …valued component of the scattering operator uniquely determines the X-ray transforms P∇V and PBi,k for i, k=1, …, d, i≠k. Applying results on inversion of the X-ray transform P we obtain that for d≥2 the velocity valued component of the scattering operator at high energies uniquely determines (V, B). In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely V when d≥2 and B when d=2 but that it uniquely determines B when d≥3. Show more

Citation: Asymptotic Analysis, vol. 55, no. 1-2, pp. 103-123, 2007