Asymptotic Analysis - Volume 35, issue 3-4

Authors: Fontelos, Marco A. | Friedman, Avner

Article Type: Research Article

Abstract: In this paper we consider free boundary problems for systems of partial differential equations. The system has solutions which are spherically symmetric with free boundary r=R, for any value of a parameter γ. We establish the existence of symmetry‐breaking bifurcation branches of solutions with free boundary of the form r=R+εF.

Keywords: free boundary problem, steady states, bifurcation, symmetry‐breaking, tumor growth

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 187-206, 2003

Authors: Wigniolle, Jérôme

Article Type: Research Article

Abstract: Let Ω be a smooth bounded domain in RN , N≥2; let a, f, h be smooth functions on $\overline{\varOmega }$ , f being positive on $\overline{\varOmega }$ and a satisfying the following condition: ∫Ω |∇u|+∫Ω a|u|≥C∫Ω |u|, ∀u∈W1,1 0 (Ω), where C is some positive constant. We look for some u∈BV(Ω), u not identically 0, which satisfies: \[\left\{\begin{array}{l}-\mathop{\mathrm{div}}\sigma+a(x)\mathop{\mathrm{sign}}(u)=f(x)|u|^{1^{*}-2}u+h(x)|u|^{q-2}u\quad \mbox{in}\ \varOmega ,\\\sigma\in L^{\infty}(\varOmega ,\mathbf{R}^{N}),\quad \sigma\cdot\nabla u=|\nabla u|\quad \mbox{in}\ \varOmega ,\\-\sigma\cdot\vec{n}u=|u|\quad\mbox{on}\ \curpartial \varOmega ,\end{array}\right.\] where 1* =N/(N−1) denotes the critical Sobolev exponent for the embedding of W1,1 (Ω) and BV(Ω) into Lk (Ω), …q is a real in ]1,1* [ and sign(u) is some L∞ function such that sign(u) u=|u|. Show more

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 207-234, 2003

Authors: Stoyanov, Luchezar

Article Type: Research Article

Abstract: Billiard trajectories in the exterior of two strictly convex domains in the plane are considered incoming from a given source S and arriving at a given target P. An asymptotic with an exponentially small error term for the sequence of travelling times of these trajectories is obtained involving the distance between the two domains and the curvatures at the ends of the shortest segment connecting the domains.

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 235-255, 2003

Authors: Bellassoued, Mourad

Article Type: Research Article

Abstract: The purpose of this paper is to study the resonances associated to the transparent obstacles. We prove the existence of an exponentially small neighborhood of the real axis free from the poles of the resolvent. As an application of these results, we get information about decay‐speed of the local energy. We prove that for regular data, the energy decays at least as fast as the inverse of the logarithm of the time.

Keywords: transparent obstacle, resonances, energy decay, stabilization

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 257-279, 2003

Authors: Elias, Uri | Gingold, Harry

Article Type: Research Article

Abstract: A new method for asymptotic integration of linear systems of ordinary differential equations is proposed and studied. It is based on the introduction of a certain integral equation that pinpoints sufficient conditions for asymptotic integration. These conditions serve as a framework from which new and old theorems are derived. In particular the fundamental theorems of Levinson and Hartman–Wintner are shown to follow from one and the same scheme. The new theorems in asymptotic integration are shown to be best possible in a certain sense. Examples are given that are not amenable to other techniques.

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 281-300, 2003

Authors: Nédélec, L.

Article Type: Research Article

Abstract: We study the resonances of matrix Schrödinger operators, in the semi‐classical limit, motivated by the Born–Oppenheimer approximation. Precisely, we are interested in a lower bound on the number of resonances near a point. We obtain the lower bound Ch−1 |ln h|−3/2 for n=3.

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 301-324, 2003

Authors: Bartolucci, D.

Article Type: Research Article

Abstract: We establish compactness and existence results for the following class of nonlinear elliptic problems: −Δu=K(x) eu in Ω, with ∫Ω K(x) eu =λ, where Ω is a smooth two‐dimensional bounded domain with nontrivial topology and K(x)=$\prod_{j=1}^{m}$ |x−xj |2αj  eσ(x) is a nonnegative function which admits a zero at the point xj ∈Ω with multiplicity αj ∈(0,1), for j=1,…,m. We consider the case where λ∈(8π,min j=1,…,m 4π(3−αj ))\{8π(1+αj )}j=1,…,m and extend the work of Struwe and Tarantello [34] and Ding et al. [12].

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 325-347, 2003

Authors: Nicoleau, François

Article Type: Research Article

Abstract: We consider the Stark Hamiltonian H=½p2 −x1 +V(x) which describes the scattering of a quantum mechanical particle in $\mathbb{R}^{n}$ by a short‐range potential in the presence of a constant electric field. We show that the electric potential V is uniquely determined by the high energy limit of the scattering operator, if the dimension n≥3. We prove our results using the Enss–Weder time‐dependent method.

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 349-359, 2003

Article Type: Other

Citation: Asymptotic Analysis, vol. 35, no. 3-4, pp. 361-362, 2003