Asymptotic Analysis - Volume 48, issue 1-2

Authors: Clément, Philippe | García-Huidobro, Marta | Guerra, Ignacio | Manásevich, Raúl

Article Type: Research Article

Abstract: We give a new region of existence of solutions to the superhomogeneous Dirichlet problem \begin{equation}\begin{array}{l}-\Delta_{p}u=v^{\delta},\quad v>0\ \mbox{in}\ B,\\-\Delta_{q}v=u^{\mu},\quad u>0\ \mbox{in}\ B,\\u=v=0\quad\mbox{on}\ \curpartial B,\end{array}\label{(S_R)}\end{equation} where B is the ball of radius R>0 centered at the origin in $\mathbb {R}^{N}$ . Here δ,μ>0 and Δm u=div(|∇u|m−2 ∇u) is the m-Laplacian operator for m>1.

Keywords: m-Laplacian, energy identities

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 1-18, 2006

Authors: Jung, Il Hyo | Kang, Yong Han

Article Type: Research Article

Abstract: We study decay estimates of the energy for the nonlinear wave equation in the whole space $\mathbb{R}^{N}$ . The dissipative term consists of the following two parts: The one part is nonlinear in a suitable ball; the other part is linear in the outside of the ball and it may be effective only at infinity. So we may call such a dissipation as a half-linear dissipation. We note that the method of proof is based on the multiplier technique and the unique continuation.

Keywords: nonlinear wave equation, energy decay, Cauchy problem

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 19-32, 2006

Authors: Pideri, C. | Seppecher, P.

Article Type: Research Article

Abstract: We study the asymptotic behavior of a non-linear elastic material lying in a thin neighborhood of a non-planar line when the diameter of the section tends to zero. We first estimate the rigidity constant in such a domain then we prove the convergence of the three-dimensional model to a one-dimensional model. This convergence is established in the framework of $\varGamma $ -convergence. The resulting model is the one classically used in mechanics. It corresponds to a non-extensional line subjected to flexion and torsion. The torsion is an internal parameter which can eventually by eliminated but this elimination leads to …a non-local energy. Indeed the non-planar geometry of the line couples the flexion and torsion terms. Show more

Keywords: beam, rod, non-linear elasticity, 3D–1D, $\varGamma $-convergence

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 33-54, 2006

Authors: Shieh, Tien-Tsan | Sternberg, Peter

Article Type: Research Article

Abstract: We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of the formal result of Richardson and Rubinstein (Proc. Roy. Soc. London A 455 (1999), 2549–2564) revealing that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 1 applied fields through the addition of a potential depending on the field.

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 55-76, 2006

Authors: Huang, Yong | Su, Ning | Zhang, Xingyou

Article Type: Research Article

Abstract: In this paper the homogenization of degenerate quasilinear parabolic equations \[\curpartial _{t}u-\mathop {\mathrm {div}}\nolimits a\Bigl(\frac{t}{\varepsilon},\frac{x}{\varepsilon},u,\nabla u\Bigr)=f(t,x)\] is studied via a weighted compensated compactness result, where a(t,y,α,λ) is periodic in (t,y).

Keywords: degenerate parabolic equations, homogenization, compensated compactness

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 77-89, 2006

Authors: Frank, Rupert L.

Article Type: Research Article

Abstract: We study the Schrödinger operator (hD−A)2 with periodic magnetic field B=curl A in an antidot lattice $\varOmega_{\infty}=\mathbb{R}^{2}\setminus\bigcup_{\alpha\in\varGamma}(U+\alpha)$ . Neumann boundary conditions lead to spectrum below hinf B. Under suitable assumptions on a “one-well problem” we prove that this spectrum is localized inside an exponentially small interval in the semi-classical limit h→0. For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.

Keywords: semi-classical analysis, tunneling effect, magnetic Schrödinger operator, periodic operator

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 91-120, 2006

Authors: Engelberg, Shlomo | Schochet, Steven

Article Type: Research Article

Abstract: Solutions of scalar viscous conservation laws whose initial data are bounded and tend at x=±∞ to values that may be connected by a shock profile are shown to converge in L∞ to a time-dependent translation of that profile. Unlike the standard theory, the initial data is not restricted to be an L1 perturbation of the shock profile, and the translation may not be linear in time. Estimates for the translation are obtained.

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 121-140, 2006

Authors: Le Dret, Hervé | Zorgati, Hamdi

Article Type: Research Article

Abstract: We consider a thin curved film made of a martensitic material. The behavior of the film is governed by a free energy composed of a bulk energy term and an interfacial energy term. We show that the minimizers of the free energy converge to the minimizers of an energy depending on a two-dimensional deformation and one Cosserat vector field when the thickness of the curved film goes to zero using $\varGamma$ -convergence arguments.

Citation: Asymptotic Analysis, vol. 48, no. 1-2, pp. 141-171, 2006