Asymptotic Analysis

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 Δ_mu=div(|∇u|^m−2∇u) is the m-Laplacian operator for m>1.

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.

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.

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.

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).

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.

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.

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.