Asymptotic Analysis - Volume 119, issue 3-4

Authors: Chen, Jingrun | Lin, Ling | Zhang, Zhiwen | Zhou, Xiang

Article Type: Research Article

Abstract: We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unperturbed reference domains, which significantly reduces computations in practice, especially for random perturbations. Our setting is quite general and allows two types of elliptic problems: the perturbation of the domain boundary where the Dirchlet condition is imposed and the perturbation of the interface where the transmission condition is imposed. As the …perturbation size and the ratio of the conductivities tends to zero, the two-parameter asymptotic expansions on the reference domain are derived to any order after the single parameter expansions are solved beforehand. The results suggest the emergence of the Neumann or Robin boundary condition, depending on the relation of the two asymptotic parameters. Our method is the classic asymptotic analysis techniques but in a new unified approach to both problems. Show more

Keywords: Asymptotic analysis, interface problem, high-contrast ratio, two-parameter expansion

DOI: 10.3233/ASY-191571

Citation: Asymptotic Analysis, vol. 119, no. 3-4, pp. 153-198, 2020

Authors: Aghajani, Asadollah | Mosleh Tehrani, Alireza

Article Type: Research Article

Abstract: We consider the nonlinear eigenvalue problem L u = λ f ( u ) , posed in a smooth bounded domain Ω ⊆ R N with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, λ > 0 and f : [ 0 , a f ) → R + (0 < a f ⩽ ∞ ) is a smooth, increasing and convex nonlinearity such that f ( 0 ) …> 0 and which blows up at a f . First we present some upper and lower bounds for the extremal parameter λ ∗ and the extremal solution u ∗ . Then we apply the results to the operator L A = − Δ + A c ( x ) with A > 0 and c ( x ) is a divergence-free flow in Ω. We show that, if ψ A , Ω is the maximum of the solution ψ A ( x ) of the equation L A u = 1 in Ω with Dirichlet boundary condition, then for any incompressible flow c ( x ) we have, ψ A , Ω ⟶ 0 as A ⟶ ∞ if and only if c ( x ) has no non-zero first integrals in H 0 1 ( Ω ) . Also, taking c ( x ) = − x ρ ( | x | ) where ρ is a smooth real function on [ 0 , 1 ] then c ( x ) is never divergence-free in unit ball B ⊂ R N , but our results completely determine the behaviour of the extremal parameter λ A ∗ as A ⟶ ∞ . Show more

Keywords: Semilinear elliptic problem, nonlinear eigenvalue problem, extremal solution

DOI: 10.3233/ASY-191572

Citation: Asymptotic Analysis, vol. 119, no. 3-4, pp. 199-219, 2020

Authors: Akil, Mohammad | Chitour, Yacine | Ghader, Mouhammad | Wehbe, Ali

Article Type: Research Article

Abstract: In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain …the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet–Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small. Show more

Keywords: Timoshenko system, boundary damping, strong stability, exponential stability, polynomial stability, observability inequality, exact controllability

DOI: 10.3233/ASY-191574

Citation: Asymptotic Analysis, vol. 119, no. 3-4, pp. 221-280, 2020

Authors: Nualart, David | Zheng, Guangqu

Article Type: Research Article

Abstract: In this paper, we present an oscillatory version of the celebrated Breuer–Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures.

Keywords: Oscillatory integral, Breuer–Major theorem, Malliavin calculus, random homogenization

DOI: 10.3233/ASY-191575

Citation: Asymptotic Analysis, vol. 119, no. 3-4, pp. 281-300, 2020