Asymptotic Analysis - Volume 110, issue 1-2

Authors: Scilla, Giovanni | Solombrino, Francesco

Article Type: Research Article

Abstract: In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Euclidean space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time t ∗ of the limit. For our class of functionals, t ∗ coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time t …c where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of t ∗ , rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini [Discrete Contin. Dyn. Syst. Ser. A 18 (2007 ), 657–675], where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times. Show more

Keywords: Gradient flow, heteroclinic solutions, singular perturbations, dynamical systems, variational methods

DOI: 10.3233/ASY-181475

Citation: Asymptotic Analysis, vol. 110, no. 1-2, pp. 1-19, 2018

Authors: Esposito, Teresa

Article Type: Research Article

Abstract: Motivated by applications to image denoising, we propose an approximation of functionals of the form F ( u ) = ∫ Ω | ∇ u | d x + ∫ S u g ( | u + − u − | ) d H n − 1 + | D c u | ( Ω ) , u ∈ BV ( Ω ) , …with g : [ 0 , + ∞ ) → [ 0 , + ∞ ) increasing and bounded. The approximating functionals are of Ambrosio–Tortorelli type and depend on the Hessian or on the Laplacian of the edge variable v which thus belongs to W 2 , 2 ( Ω ) . When the space dimension is equal to two and three v is then continuous and this improved regularity leads to a sequence of approximating functionals which are ready to be used for numerical simulations. Show more

Keywords: Ambrosio–Tortorelli approximation, free-discontinuity problems, Γ-convergence

DOI: 10.3233/ASY-181476

Citation: Asymptotic Analysis, vol. 110, no. 1-2, pp. 21-52, 2018

Authors: Bao, Weizhu | Ruan, Xinran

Article Type: Research Article

Abstract: We study asymptotically and numerically the fundamental gaps (i.e. the difference between the first excited state and the ground state) in energy and chemical potential of the Gross–Pitaevskii equation (GPE) – nonlinear Schrödinger equation with cubic nonlinearity – with repulsive interaction under different trapping potentials including box potential and harmonic potential. Based on our asymptotic and numerical results, we formulate a gap conjecture on the fundamental gaps in energy and chemical potential of the GPE on bounded domains with the homogeneous Dirichlet boundary condition, and in the whole space with a convex trapping potential growing at least quadratically in the …far field. We then extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition. Show more

Keywords: Gross–Pitaevskii equation, fundamental gap, ground state, first excited state, energy asymptotics, repulsive interaction

DOI: 10.3233/ASY-181477

Citation: Asymptotic Analysis, vol. 110, no. 1-2, pp. 53-82, 2018

Authors: Holzmann, Markus | Lotoreichik, Vladimir

Article Type: Research Article

Abstract: In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists …a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in L 2 ( R 2 ) . The proofs rely on perturbation theory of linear operators, Floquet–Bloch analysis, and properties of Schrödinger operators with point interactions. Show more

Keywords: Photonic crystals, spectral gaps, inverse problem, thin rods, electromagnetic waves, TE- and TM-modes, periodic differential operators, perturbation theory, Floquet–Bloch analysis, point interactions

DOI: 10.3233/ASY-181478

Citation: Asymptotic Analysis, vol. 110, no. 1-2, pp. 83-112, 2018