Showing papers in "Duke Mathematical Journal in 1990"

Geometric bounds on the density of resonances for semiclassical problems

Abstract: In this paper, we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real, axis, for semiclassical operators like −h 2 Δ+V(x), when h is small

Horocycle flow on geometrically finite surfaces

Abstract: acts on T S. It is our main goal to determine all N-invariant Radon measures on T1S. Our first remark is that if C is the cone of positive N-invariant Radon measures in the space ’(Tx S) of all Radon measures with the vague topology, then C is the closed convex hull of the union of its extremal generators [B, II No. 2]; moreover it is easily seen that a measure is on an extremal generator of C if and only if it is ergodic. This reduces the problem to the classification of all ergodic measures. To proceed further we consider the following decomposition of T S: Let S be the ideal boundary of D and A c S be the limit set of F. Using the visual map:

Wavefront propagation for reaction-diffusion systems of PDE

Abstract: M. I. Freidlin has introduced probabilistic techniques to study propagation for systems of reaction-diffusion PDE. The motivating idea is that should a reaction-diffusion system possess only a single unstable and a single stable equilibrium, then the solution u of the system will presumably tend to «switch» for large times from near the former to near the latter state. This paper brings to bear purely PDE techniques to this problem, especially the theory of viscosity solutions on Hamilton-Jacobi equations, due to Crandall-Lions

Betti numbers of hypersurfaces and defects of linear systems

Abstract: 0. Introduction. Let w (Wo,..., wn) be a set of integer positive weights and denote by S the polynomial ring C[Xo, xn] graded by the conditions deg(x) wt for 0, n. For any graded object M, let Mk denote the homogeneous component of degree k. Let f SN be a weighted homogeneous polynomial of degree N with respect to w. Let V be the hypersurface defined by f 0 in the weighted projective space P(w) Proj S Cn+l\\{0}/C* where the C*-action on C+1 is defined by t.x (tWxo,..., W\"xn) for C*, x C+. Assume that the singular locus E(f) of f is 1-dimensional, namely

Global existence of small analytic solutions to nonlinear Schrödinger equations

Abstract: In this paper we consider the following nonlinear Schrodinger equation in R n (n≥2): i∂u+1/2Δu=F(u, △u, ū − , △u), (t,x)eR×R n , u(0,x)=φ(x), xeR n

Sur le théorème local des cycles invariants

Abstract: Si on considere une famille de varietes projectives complexes non singulieres, c’est un fait aujourd’hui bien connu que les possibles varietes singulieres vers lesquelles peut degenerer cette famille doivent verifier certaines contraintes, parmi lesquelles une importante relation entre la cohomologie de la fibre singuliere, la cohomologie de la fibre generique et la monodromie de la famille, qui est precisee par le theoreme local des cycles invariants prouve par Clemens, Deligne et Steenbrink ([1], [4], [13]) : tous les cocycles de la fibre generique qui sont invariants par la monodromie autour d’une fibre singuliere proviennent par specialisation de la cohomologie de cette fibre singuliere. Le but de ce travail est de donner une preuve de ce theoreme en suivant de pres l’argumentation de [13] qui se base sur l’utilisation des structures de Hodge mixtes qui y sont presentes. Concretement, nous prouverons que la filtration par le poids de la structure de Hodge mixte construite par Steenbrink sur la cohomologie de la fibre limite coincide avec la filtration definie par la monodromie, d’ou on deduit aisement le theoreme des cycles invariants par un argument de Deligne. Or, dans notre demonstration, pour prouver la coincidence de ces deux filtrations, nous utiliserons aussi un resultat recent de Deligne-Saito sur les modules de Hodge-Lefschetz polarises ([5], [11]). Ceci nous permettra, comme dans [11], de completer la preuve de Steenbrink qui etait insuffisante, comme l’avait remarque El Zein ([6]). Les principaux ingredients de la demonstration que nous presentons ici du theoreme des cycles invariants sont, comme nous venons de dire, dus a Deligne, Steenbrink et Saito, et notre seule contribution dans cet article est de prouver que le terme E1 de la suite spectrale de Steenbrink est un module de Hodge-Lefschetz polarise au sens de Deligne, ce qui rend possible l’application du resultat de [5]. Notre dernier objectif a

Bott-Chern currents and complex immersions

Abstract: In the whole paper, we will say that i:M'→M is an immersion of smooth manifolds if M' is a submanifold of M, and if i is the corresponding injection map. In particular the topology of M' is the topology induced by the topology of M. In differential geometry, such maps i:M'→M are also called embeddings

On long-range scattering

Abstract: Even if special cases are not well understood it pays sometimes to consider a problem at hand in the most general framework. Thus addressing the existence of wave operators for the most general potentials Hormander [Ho1] has discovered a natural asymptotic evolution for the long-range scattering. It is determined by a solution, S(k, t), of a certain Hamilton-Jacobi equation

On the convexity of a solution of Liouville's equation

Abstract: The main purpose of the present note is to give a simple, complex variable proof of the fact [3], [10], [11] that h(z) is convex whenever Ω is convex. The major step in our proof can be formulated as a coefficient inequality for convex univalent functions