Asymptotic Analysis - Volume 52, issue 3-4

Authors: Maz'ya, V.G. | Movchan, A.B. | Nieves, M.J.

Article Type: Research Article

Abstract: We present uniform asymptotic approximations of Green's kernels for boundary value problems of elasticity in singularly perturbed domains containing a small hole. We consider the cases of two and three dimensions, for an isotropic Lamé operator and the Dirichlet boundary conditions. The main feature of the asymptotic approximations mentioned is their uniformity with respect to the independent spatial variables. The formal asymptotic formulae are supplied with rigorous derivations and the remainder estimates.

Keywords: Green's tensors, uniform asymptotics, singularly perturbed domains, compound asymptotic expansions

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 173-206, 2007

Authors: Han, Jongmin | Kim, Namkwon

Article Type: Research Article

Abstract: In this paper we are concerned with the asymptotic behavior of the relativistic Maxwell–Chern–Simons (MCS) vortices on the unit disc. We establish the existence of a radial solution of the static MCS equations which is a minimizer of the MCS functional in the class of radial functions, and derive the asymptotic limit of the radial solution. As a consequence, we obtain a set of elliptic linear equations which is a generalization of the London equation for the Ginzburg–Landau model.

Keywords: Maxwell–Chern–Simons Higgs model, asymptotics, London equation

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 207-225, 2007

Authors: Menaldi, Jose-Luis | Tarzia, Domingo Alberto

Article Type: Research Article

Abstract: We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions. The parameter α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion $\varGamma _{1}$ of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control $\hat{g}_{\alpha}$ with optimal state $u_{\hat{g}_{\alpha}\alpha}$ and optimal adjoint state $p_{\hat{g}_{\alpha}\alpha}$ are convergent as α→∞ (in norm of a …suitable Sobolev parabolic space) to $\hat{g},$ $u_{\hat{g}}$ and $p_{\hat{g}},$ respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on $\varGamma _{1}.$ The main techniques used are derived from the parabolic variational inequality theory. Show more

Keywords: parabolic variational inequalities, distributed evolution optimal control, mixed boundary conditions, adjoint state, optimality condition, asymptotic

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 227-241, 2007

Authors: Guerrero, S. | Mercado, A. | Osses, A.

Article Type: Research Article

Abstract: In this paper we prove an inverse inequality for the parabolic equation \[u_{t}-\varepsilon\Delta u+M\cdot\nabla u=g\mathbh{1}_{\omega}\] in a bounded domain Ω⊂$\mathbb{R}^{n}$ with Dirichlet boundary conditions. With the motivation of finding an estimate of g in terms on the trace of the solution in 𝒪×(0,T) for ε small, our approach consists in studying the convergence of the solutions of this equation to the solutions of some transport equation when ε→0, and then recover some inverse inequality from the properties of the last one. Under some conditions on the open sets ω, 𝒪 and the time T, we …are able to prove that, in the particular case when g∈H0 1 (ω) and it does not depend on time, we have: |g|L2 (ω) ≤C(|u|H1 (0,T;L2 (𝒪)) +ε1/2 |g|H1 (ω) ). On the other hand, we prove that this estimate implies a regional controllability result for the same equation but with a control acting in 𝒪×(0,T) through the right-hand side: for any fixed f∈L2 (ω) , the L2 -norm of the control needed to have |u(T)|ω −f|H−1 (ω) ≤γ remains bounded with respect to γ if ε≤Cγ2 . Show more

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 243-257, 2007

Authors: Díaz, J.I. | Sanchez-Palencia, E.

Article Type: Research Article

Abstract: We study the rigidification phenomenon for several thin slender bodies or shells, with a small curvature in the transversal direction to the main length, for which the propagation of singularities through the characteristics is of parabolic type. The asymptotic behavior is obtained starting with the two-dimensional Love–Kirchoff theory of plates. We consider, in a progressive study, a starting basic geometry, we pass then to consider the “V-shaped” structure formed by two slender plates pasted together along two long edges forming a small angle between their planes and, finally, we analyze the periodic extension to a infinite slab. We introduce a …scalar potential φ and prove that the equation and constrains satisfied by the limit displacements are equivalent to a parabolic higher-order equation for φ. We get some global informations on φ, some on them easely associated to the different momenta and others of a different nature. Finally, we study the associate obstacle problem and obtain a global comparison result between the third component of the displacements with and without obstacle. Show more

Keywords: thin shells, V-shaped structures, asymptotic behavior, scalar potential, parabolic higher-order equations, one-side problems

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 259-297, 2007

Article Type: Other

Citation: Asymptotic Analysis, vol. 52, no. 3-4, pp. 299-299, 2007