Asymptotic Analysis - Volume 124, issue 1-2

Authors: Engl, Dominik | Kreisbeck, Carolin

Article Type: Research Article

Abstract: We use variational convergence to derive a hierarchy of one-dimensional rod theories, starting out from three-dimensional models in nonlinear elasticity subject to local volume-preservation. The densities of the resulting Γ-limits are determined by minimization problems with a trace constraint that arises from the linearization of the determinant condition of incompressibility. While the proofs of the lower bounds rely on suitable constraint regularization, the upper bounds require a careful, explicit construction of locally volume-preserving recovery sequences. After decoupling the cross-section variables with the help of divergence-free extensions, we apply an inner perturbation argument to enforce the desired non-convex determinant constraint. To …illustrate our findings, we discuss the special case of isotropic materials. Show more

Keywords: Dimension reduction, Γ-convergence, Euler–Lagrange equations, incompressibility, rods

DOI: 10.3233/ASY-201636

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 1-28, 2021

Authors: Lin, Xiaoyan | He, Yubo | Tang, Xianhua

Article Type: Research Article

Abstract: This paper is focused on the following Schrödinger–Poisson system − Δ u + V ( x ) u + ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where V ( x ) is weakly differentiable and tends asymptotically to a constant and f ∈ C ( R , R ) …. By introducing some new tricks, we prove that the above system admits a ground state solution under some mild assumptions on V and f . Our results generalize and improve the existing ones in the literature. Show more

Keywords: Schrödinger–Poisson system, ground state solution, Pohozaev type identity

DOI: 10.3233/ASY-201639

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 29-49, 2021

Authors: Smii, Boubaker

Article Type: Research Article

Abstract: In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L ( t ) = L t , t > 0 . The transition probability density p t , t > 0 of the semigroup associated to the solution u t , t ⩾ 0 of the SDE is given by a power series expansion. The series expansion of p t can be re-expressed in …terms of Feynman graphs and rules. We will also prove that p t , t > 0 has an asymptotic expansion in power of a parameter β > 0 , and it can be given by a convergent integral. A remark on some applications will be given in this work. Show more

Keywords: Stochastic differential equations, transition probability densities, Lévy processes, Feynman graphs and rules, Borel summability, neural networks

DOI: 10.3233/ASY-201640

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 51-68, 2021

Authors: Gittins, K. | Helffer, B.

Article Type: Research Article

Abstract: We consider the cases where there is equality in Courant’s nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter h > 0 is large, small respectively. In this paper we investigate the case where h < 0 .

Keywords: Courant-sharp, Robin eigenvalues, negative parameter, squar

DOI: 10.3233/ASY-201642

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 69-107, 2021

Authors: Yamazaki, Taeko

Article Type: Research Article

Abstract: We show diffusion phenomenon for linear abstract dissipative wave equations with time dependant coefficients of propagation speed and dissipation. Coefficients are decaying in time but not assumed to be monotone.

Keywords: Diffusion phenomenon, abstract linear dissipative wave equation, time dependent propagation speed, time dependent dissipation

DOI: 10.3233/ASY-201665

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 109-161, 2021

Authors: Zhao, Liang | Xi, Shuai

Article Type: Research Article

Abstract: It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100 (5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more …generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper. Show more

Keywords: Convergence rate, first order balance laws, partial dissipation, isotropic parabolic system, stream function

DOI: 10.3233/ASY-211687

Citation: Asymptotic Analysis, vol. 124, no. 1-2, pp. 163-198, 2021