Asymptotic Analysis - Volume 92, issue 1-2

Authors: Chourabi, Imen | Donato, Patrizia

Article Type: Research Article

Abstract: In this paper, we study the homogenization and the correctors for a class of linear elliptic problems in a periodically perforated domain when the oscillating matrix field also depend on a weakly converging sequence. We prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. Using the periodic unfolding method, we first derive the homogenized problem, then we study the convergence of the energy of the solution and the related corrector, which are the main results of this paper. As a particular case, we obtain a corrector result for the …Laplacian with a linear nonhomogeneous Neumann condition on the hole, in the case of a nonzero Neumann data with a zero average. This remained an open problem since the corresponding homogenized results given in [Asymptotic Anal. 1 (1988), 115–138]. Show more

Keywords: homogenization, perforated domains, elliptic equations, nonlinear nonhomogeneous Robin boundary conditions

DOI: 10.3233/ASY-151288

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 1-43, 2015

Authors: Mei, Hongwei | Yin, George

Article Type: Research Article

Abstract: This work is concerned with switching diffusions in which the switching depends on the diffusion process. It has been shown in the literature that if such a process is positive recurrent then it is ergodic. This paper further develops a law of iterated logarithms for the estimates on the aforementioned convergence rate. The law of iterated logarithms gives us a sharp bound on the estimation error sequence.

Keywords: switching diffusion, stationary distribution, law of iterated logarithm

DOI: 10.3233/ASY-141265

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 45-64, 2015

Authors: Yoshino, Masafumi

Article Type: Research Article

Abstract: We study the Borel summability and the analytic behavior of the Borel sum of a formal solution of first-order semilinear system with a singular perturbative parameter. By virtue of the representation formula of the Borel sum of a formal series solution expanded in terms of a parameter, we show that the analytic continuation of the Borel sum with respect to the parameter to a regular point in a singular direction coincides with the solution of the initial value problem expanded in the space variable. We also show that a similar phenomenon occurs outside the origin of the independent variables.

Keywords: Borel sum, singular equation, singular direction

DOI: 10.3233/ASY-141270

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 65-84, 2015

Authors: Duong, Manh Hong

Article Type: Research Article

Abstract: In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional Kh , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, Kh is asymptotically equivalent, in the sense of Γ-convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an …important role. Show more

Keywords: porous medium equation, Gamma-convergence, Wasserstein gradient flow, variational methods

DOI: 10.3233/ASY-141272

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 85-106, 2015

Authors: Gomilko, Alexander | Rzepnicki, Łukasz

Article Type: Research Article

Abstract: This paper is concerned with the equation of a nonhomogeneous string of length one with one end fixed and the other one damped with a parameter h∈C. This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator iAh acting on an appropriate energy Hilbert space H. Under assumptions that the density function of the string ρ∈W2 1 [0,1] is strictly positive and has ρ(1)≠h2 (if h∈R), we prove that the set of root vectors of Ah form a basis with parentheses in H. We show that with the additional condition …∫0 1 ω1 2 (ρ′,τ)/τ2  dτ<∞, where ω1 is the integral modulus of continuity, the root vectors of the operator Ah form a Riesz basis in H. Show more

Keywords: nonhomogeneous damped string, Hilbert space, Riesz basis, modulus of continuity, basis with parentheses

DOI: 10.3233/ASY-141273

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 107-140, 2015

Authors: Anahtarci, Berkay | Djakov, Plamen

Article Type: Research Article

Abstract: The one-dimensional Dirac operator \[L=\mathrm{i}\pmatrix{1&0\cr0&-1}\frac{\mathrm{d}}{\mathrm{d}x}+\pmatrix{0&P(x)\crQ(x)&0},\quad P,Q\inL^{2}([0,\uppi ]),\] considered on $[0,\uppi ]$ with periodic or antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n∈Z, there are two (counted with multiplicity) eigenvalues λn − , λn + (periodic if n is even, or antiperiodic if n is odd) such that |λn ± −n|<1/2. We study the asymptotics of spectral gaps γn =λn + −λn − in the case P(x)=ae−2ix +Ae2ix ,  Q(x)=be−2ix +Be2ix , where a, A, b, B are any complex numbers. We show, for large enough m, that γ±2m =0 and …\begin{eqnarray*}\lefteqn{\gamma_{2m+1}=\pm2\frac{\sqrt{(Ab)^{m}(aB)^{m+1}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr],}\\\lefteqn{\gamma_{-(2m+1)}=\pm2\frac{\sqrt{(Ab)^{m+1}(aB)^{m}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr].}\end{eqnarray*} Show more

Keywords: 1D Dirac operator, spectral gap asymptotics

DOI: 10.3233/ASY-141274

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 141-160, 2015

Authors: Benmansour, Khadidja | Pousin, Jérôme

Article Type: Research Article

Abstract: The result presented in this article is an adaptation of the gradient flow technique applied to a periodic problem combined with a singular perturbation method. The motivation for considering such a question originates from problems of cardiac images segmentation and of tracking cardiac images in medical image analysis. The ‘elastic deformable template’ model introduced previously for image segmentation is improved and adapted to the image tracking problem. A result of convergence is proved, for a periodic linear elastic model, the elastic tensor of which vanishes, when the parameter associated to the quasi-static technique goes to infinity.

Keywords: images tracking, linear elasticity, singular perturbations techniques

DOI: 10.3233/ASY-141267

Citation: Asymptotic Analysis, vol. 92, no. 1-2, pp. 161-185, 2015