Showing papers in "Communications on Pure and Applied Mathematics in 1984"

The dirichlet problem for nonlinear second‐order elliptic equations I. Monge‐ampégre equation

Abstract: On considere le probleme de Dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture Ω d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞

Stochastic differential equations with reflecting boundary conditions

Abstract: On resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de Skorokhod

Determining conductivity by boundary measurements

Abstract: A. P. Calderon pose la question: est-il possible de determiner la conductivite thermique d'un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite? On demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

Scattering and inverse scattering for first order systems

Abstract: On considere le probleme aux valeurs propres suivant: df/dx=2Jf(x)+q(x)f(x). Avec f: R→C n , J est une matrice constante (n×n) et q est une fonction a valeur matricielle. On presente des resultats sur la theorie analytique des problemes de diffusion et de diffusion inverse pour des systemes generalises AKNS

Morse-type index theory for flows and periodic solutions for Hamiltonian Equations

Abstract: An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.

The inversion problem and applications of the generalized radon transform

Abstract: We prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation. We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.

Discontinuous inverse eigenvalue problems

Abstract: On considere des problemes de Sturm-Liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur. On montre que si le potentiel est connu sur la moitie de l'intervalle et si une condition limite est donnee, alors le potentiel et l'autre condition limite sont determines de facon unique par les valeurs propres

Multiple solutions of H‐systems and Rellich's conjecture

Abstract: Soit Ω={(x,y)∈R 2 ; x 2 +y 2 <1}. On considere une fonction u:Ω→R 3 satisfaisant le systeme H: Δu=2Hu x ∧u y sur Ω avec l'une des conditions suivantes: soit Dirichlet u=γ sur δΩ soit Plateau: |U x | 2 −|U y | 2 =U x •U y =0 sur Ω, u(δΩ)=Γ et u non decroissante sur δΩ

Almost global existence to nonlinear wave equations in three space dimensions

Abstract: On considere l'existence a t grand de solutions de petite amplitude pour des equations d'onde d'ordre 2 non lineaires a 4 dimensions d'espace-temps

The inverse Sturm–Liouville problem. II

Abstract: Soit q=(α,β,q)∈(0,π) 2 ×L R 2 [0,1]. Le probleme de Sturm-Liouville −y''+q(x)y=λy, 0≤x≤1, y(0) cos α+y'(0) sin α=0; y(1) cos β+y(1) sin β=0, a un spectre discret de valeurs propres simples ν 0 (q)<ν 1 (q)<... avec des fonctions propres h 0 (x,q), h 1 (x,q),... On fixe p dans (0,π) 2 ×L R 2 [0,1] et soit M(p)={q∈(0,π) 2 ×L R 2 [0,1]: ν n (q)=ν n (p), n≥0} l'espace de tous les points q de meme spectre que p. On etudie M(p) comme une sous-variete de (0,π) 2 ×L R 2 [0,1]

Instability of difference models for hyperbolic initial boundary value problems

Abstract: A th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundstrom. In particular we investigate l2-instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.

Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I

Abstract: This paper deals with the spectral theory of the Laplace-Beltrami operator Δ acting on automorphic functions in n-dimensional hyperbolic space Hn. We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [-(1/2(n - 1))2, 0], and none less than (1/2(n -1))2. Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (-∞, -(1/2(n - 1))2). Our approach uses the non-Euclidean wave equation introduced by Faddeev and Pavlov, Energy EF is defined as (ut, ut)-(u, Lu), where the bracket is the L2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L2(R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations. utt-Lu = 0, L = Δ + (1/2(n - 1))2.

Numerical simulation of singular solutions to the two-dimensional cubic schrödinger equation

Abstract: Numerical simulations of the cubic nonlinear Schrodinger equation are presented in dimension N = 2. The emphasis is on a detailed mechanism of blow up. The numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire R2-space but also occurs with periodic boundary conditions. The structure of the solution as the singular time t* is approached has been investigated using examples with and without radial symmetry and/or periodicity. In most cases we observe that the amplitude develops something like the peaked self-similar profile predicted by Zakharov and Synakh [38] which has a (t*-t)-2/3 maximum amplitude and a (t*-t)2/3 width.

On the Crossing Rule

Abstract: : The paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue. In particular, the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues. This question arises naturally in the theory of first order hyperbolic systems of partial differential equations; the main theorem, in this context, tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic. (Author)

On the Piano Movers' Problem. V. The Case of a Rod Moving in Three- Dimensional Space Amidst Polyhedral Obstacles,

Abstract: This paper, a fifth in a series, solves some additional 3-D special cases of the „piano movers” problem, which arises in robotics. The main problem solved in this paper is that of planning the motion of a rod moving amidst polyhedral obstacles. We present polynomial-time motion-planning algorithms for this case, using the connectivity-graph technique described in the preceding papers. We also study certain more general polyhedral problems, which arise in the motion planning problem considered here but have application to other similar problems. Application of these techniques to the problem of planning the motion of a general polyhedral body moving in 3-space amidst polyhedral obstacles is also described.

On isospectral periodic potentials in Rn. II

Abstract: On considere un reseau a n dimensions L dans R n et un potentiel q, satisfaisant q(x+d)=q(x) pour d∈L. On considere spec k (−Δ+q), l'ensemble des λ avec multiplicites telles que −Δu+qu=λu a une solution satisfaisant u(x+d)=u(x)exp(2iΠk•d) ∀d∈L

Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions

Abstract: On considere le probleme aux valeurs initiales □u=0 u(x,0)=0, u t (x,0)=g(x). On donne des estimations L ∞ et L 1 pour les solutions dans le cas general a symetrie non spherique

The inverse sturn−Liouville problem III

Abstract: On considere la theorie spectrale inverse du probleme de Sturm-Liouville-y''+q(x)y+λy, 0≤x≤1 avec les conditions aux limites y(0)=0, by(1)+y'(1)=0, q∈L R 2 (0,1) et b∈R

Convergence of a large time step generalization of Godunov's method for conservation laws

Abstract: A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.

Nonlinear elliptic systems with certain unbounded coefficients

Abstract: We consider nonlinear systems of the form Under suitable assumptions, Holder continuity of weak solutions is proven in a neighborhood of the origin using weighted inequalities and a partial regularity method of M. Giaquinta and E. Giusti. These equations relate to an exterior flow problem in physics.

A particle limit for the wave equation with a variable wave speed

Abstract: On etudie les solutions courte longueur d'onde de l'equation d'onde a n dimensions U tt =(C(x)) 2 Δu

Sharkovskii's order for the appearance of superstable cycles in one‐parameter families of simple real maps: An elementary proof

Abstract: We give an elementary proof of a theorem “a la Sharkovskii” which orders, according to the period, the first occurrence of superstable cycles in some one-parameter families of endomorphisms of the real line

Asymptotic behavior of the solutions of certain parabolic equations

Abstract: Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) j≤m, where η and vj are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = uo(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § Rm the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u0(x) ≥ 0 we derive bounds d1u1 ≤ u(x, t) ≤ d2u2, Where di, i = 1, 2, depend on bounds for η, v and G, and the ui are bounds on the initial condition u0.

A construction of approximate positive parts of essentially selfadjoint pseudo-differential operators

Abstract: Soit a(x,ξ) une fonction symbole a valeur reelle appartenant a la classe symbole S 10 1 (R n ) de Hormander. On note a(x,D) la quantification de Weyl de a(x,ξ). Soit A l'extension auto-adjointe de a(x,D). On considere les proprietes spectrales de A en utilisant le comportement local de a(x,ξ)