Asymptotic Analysis - Volume 41, issue 3-4

Authors: Dolbeault, Jean | Escobedo, Miguel

Article Type: Research Article

Abstract: In this paper, using entropy techniques, we study the rate of convergence of nonnegative solutions of a simple scalar conservation law to their asymptotic states in a weighted L1 norm. After an appropriate rescaling and for a well chosen weight, we obtain an exponential rate of convergence. Written in the original coordinates, this provides intermediate asymptotics estimates in L1 , with an algebraic rate. We also prove a uniform convergence result on the support of the solutions, provided the initial data is compactly supported and has an appropriate behaviour on a neighborhood of the lower end of its support.

Keywords: scalar conservation laws, asymptotics, entropy, shocks, weighted $L^{1}$ norm, self‐similar solutions, N‐waves, time‐dependent rescaling, Rankine–Hugoniot condition, uniform convergence, graph convergence

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 189-213, 2005

Authors: Bonnaillie, Virginie

Article Type: Research Article

Abstract: The superconducting properties of a sample submitted to an external magnetic field are mathematically described by the minimizers of the Ginzburg–Landau's functional. The analysis of the Hessian of the functional leads to estimate the fundamental state for the Schrödinger operator with intense magnetic field for which the superconductivity appears. So we are interested in the asymptotic behavior of the energy for the Schrödinger operator with a magnetic field. A lot of papers have been devoted to this problem, we can quote the works of Bernoff–Sternberg, Lu–Pan, Helffer–Mohamed. These papers deal with estimates of the energy in a regular domain and …our goal is to establish similar results in a domain with corners. Although this problem is often mentioned in the physical literature, there are very few mathematical papers. We only know the contributions by Pan and Jadallah which deal with very particular domains like a square or a quarter plane. The physicists Brosens, Devreese, Fomin, Moshchalkov, Schweigert and Peeters propose a non optimal upper bound for the energy. Here, we present a more rigorous analysis and give an asymptotics of the smallest eigenvalue of the operator in a sector Ωα of angle α when α is closed to 0, an estimate for the eigenfunctions and we use these results to study the fundamental state in the semi‐classical case. Show more

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 215-258, 2005

Authors: Camar‐Eddine, M. | Milton, G.W.

Article Type: Research Article

Abstract: It is a very well‐known fact that the effective properties of a heterogeneous electrical medium may contain a non‐local term. The same phenomenon can occur within a heterogeneous thermally conducting medium. We are interested in the set of all non‐local interactions which may arise from the homogenization of a thermoelectric medium where there are couplings between the temperature and the electric field. We consider a bounded open subset Ω⊂$\mathbb{R}^{3}$ and we show that any non‐local energy of the kind F(u,v)=∫Ω×Ω $\pmatrix{u(x)-u(y)\cr v(x)-v(y)\cr}$ ·μ(dx,dy) $\pmatrix{u(x)-u(y)\cr v(x)-v(y)\cr}$ , belongs to the closure of the …set of thermoelectric functionals, provided that the positive definite symmetric matrix‐valued measure μ(dx,dy) makes this energy functional continuous in the strong topology of L2 (Ω,$\mathbb{R}^{2}$ ). Show more

Keywords: homogenization, gamma‐convergence, Mosco‐convergence, composite materials, thermoelectricity, non‐local phenomena

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 259-276, 2005

Authors: Matei, Basarab

Article Type: Research Article

Abstract: The aim of this paper is to characterize the smoothness of a function belonging to a Hölder or to a Besov space, by the rate of decay of its nonlinear multiscale coefficients. The study is carried out in the general framework of nonlinear multiscale representations of data due to A. Harten. We also study the stability of these representations, a key issue in their applications to design adaptive algorithms.

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 277-309, 2005

Authors: Bensoussan, Alain | Touzi, Nizar | Menaldi, José Luis

Article Type: Research Article

Abstract: In this paper, we consider the problem of super‐replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super‐replication value is the smallest function which lies above the Black–Scholes price function and which is stable for the so‐called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.

Keywords: hedging under portfolio constraints, penalization, viscosity solutions

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 311-330, 2005

Authors: Raguž, Andrija

Article Type: Research Article

Abstract: In this paper we study asymptotic behavior as ε→0 of Ginzburg–Landau functional Iε (v):=∫Ω (ε2 v″2 (s)+W(v′(s))+a(s)(v(s)+g(s))2 ) ds for v∈Hper 2 (Ω), where Ω⊆R is a bounded open interval, W is a non‐negative continuous function vanishing at ±1, a∈L1 (Ω), and g is 1‐Lipschitz. Our consideration follows the approach introduced in the original paper by G. Alberti and S. Müller (Comm. Pure Appl. Math. 54 (2001), 761–825), where the case g=0 was studied. We show that their program can be modified in the case of functional Iε : we define suitable relaxation of Iε and prove …a Γ‐convergence result in the topology of the so‐called Young measures on micropatterns. Moreover, we identify a unique minimizing measure for the functional in the limit, which is the unique translation‐invariant measure supported on the orbit of a particular periodic sawtooth function having minimal period and slope dependent on a derivative of g. Show more

Keywords: Young measures, relaxation, Ginzburg–Landau functional, Gamma convergence

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 331-361, 2005

Article Type: Other

Citation: Asymptotic Analysis, vol. 41, no. 3-4, pp. 363-364, 2005