Asymptotic Analysis - Volume 72, issue 3-4

Authors: Miyazaki, Yoichi

Article Type: Research Article

Abstract: Let A be a 2mth-order elliptic operator in divergence form subject to the Dirichlet boundary conditions in a bounded Cρ domain Ω of Rn , whose coefficients are in Cσ with given σ>0, where ρ=1 if 0<σ≤1 and ρ=σ+1 if σ>1. Then we investigate the asymptotic formula for the partition function, namely the trace of the kernel of exp (−tA) as t→+0, and show that the well-known asymptotic formula for the C∞ case remains valid up to the order according to σ. We also give the asymptotic formula for the heat kernel in Rn , Rn + …and a bounded C1 domain. Show more

Keywords: elliptic operator, Dirichlet boundary condition, heat kernel, asymptotic formula, trace, partition function, spectral function, counting function

DOI: 10.3233/ASY-2010-1024

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 125-167, 2011

Authors: Li, Tatsien | Rao, Bopeng

Article Type: Research Article

Abstract: For 1-D linear hyperbolic systems with constant coefficients we introduce the asymptotic controllability and the asymptotic zero controllability in L2 space under the lack of boundary controls and show the duality that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4×4 system with only one control is shown to be asymptotically controllable but not exactly controllable.

Keywords: linear hyperbolic system, asymptotic (zero) controllability, exact controllability, strong (weak) observability

DOI: 10.3233/ASY-2010-1027

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 169-187, 2011

Authors: Véron, Laurent

Article Type: Research Article

Abstract: If Ω is a bounded domain in RN and f a continuous nondecreasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): ∂t u−Δu+f(u)=0 in Q∞ Ω :=Ω×(0,∞), u=∞ on the parabolic boundary ∂p Q. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in Ω.

Keywords: elliptic equations, parabolic equations, singular solutions, Wiener criterion

DOI: 10.3233/ASY-2011-1030

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 189-200, 2011

Authors: Allemand, Thibaut | Toscani, Giuseppe

Article Type: Research Article

Abstract: We discuss the grazing collision limit of certain kinetic models of Bose–Einstein particles obtained from a suitable modification of the one-dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a non-linear Fokker–Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in Phys. Rev. E 49 (1994), 5103–5110. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.

Keywords: Bose gas, Kac model, grazing collisions, Fokker–Planck

DOI: 10.3233/ASY-2011-1031

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 201-229, 2011

Authors: Gargiulo, Giuliano | Zappale, Elvira

Article Type: Research Article

Abstract: A lower semicontinuity result is proved in the space of special vector fields with bounded deformation for a fracture energetic model of the type ∫Ju Ψ([u], νu ) dℋN−1 ,  [u]·νu ≥0, ℋN−1 -a.e. on Ju . A representation of the energy density Ψ, which ensures lower semicontinuity, is also given.

Keywords: lower semicontinuity, fracture, special functions of bounded deformation

DOI: 10.3233/ASY-2011-1032

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 231-249, 2011

Article Type: Other

Citation: Asymptotic Analysis, vol. 72, no. 3-4, pp. 251-251, 2011