Showing papers in "Indiana University Mathematics Journal in 1999"

The Toeplitz algebra of a Hilbert bimodule

Abstract: Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.

Global Regularity of 3D Rotating Navier-Stokes Equations for Resonant Domains

Abstract: We prove existence on infinite time intervals of regular solutions to the 3D rotating Navier-Stokes equations in the limit of strong rotation (large Coriolis parameter Ω). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear “ 21 2 - dimensional ” limit equations; smoothness assumptions are the same as for local existence theorems. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D rotating Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit equations and convergence theorems.

Geometry of self-affine tiles ii

Abstract: We continue the study in part I of geometric properties of self similar and self a ne tiles We give some experimental results from implementing the algorithm in part I for computing the dimension of the boundary of a self similar tile and we describe some conjectures that result We prove that the dimension of the boundary may assume values arbitrarily close to the dimension of the tile We give a formula for the area of the convex hull of a planar self a ne tile We prove that the extreme points of the convex hull form a set of dimension zero and we describe a natural gauge function for this set Introduction to Part II This paper is a continuation of SW which we refer to as part I and the sections are numbered accordingly In Section of part I we obtained an algorithm for computing the dimension box and Hausdor dimensions are equal of the boundary in the case of a self similar tile satisfying

A commutant lifting theorem on the polydisc

Abstract: 0. Introduction Interpolation problems for bounded analytic functions in the unit disk have been studied for at least one century. The simplest ones are the Nevanlinna-Pick case, in which the constraints on the functions are the values in a finite number of points, and the Caratheodory-Fejer, where the first finite number of Taylor coefficients of the development of a function are prescribed. In all these cases, one imposes a size constraint on the function: say, its supremum norm should be smaller than 1. Starting with the paper of Sarason ([S]), it has been realized that there exists a natural operatorial frame which unifies all function theoretic problems. The setting is the following: the algebra of bounded analytic functions is identified with the algebra of analytic multiplication (Toeplitz) operators acting on the Hardy Hilbert space H(D), while the interpolation conditions are translated in the existence of a subspace of H, semiinvariant with respect to Toeplitz operators, and the compression of the multiplication operator to this subspace. The most general result in this direction is the intertwining lifting theorem of Sz-Nagy and Foias ([SNF]), which has found subsequently many applications, including applied areas like system theory. The generalization of these interpolation theorems to several variables (that is, to bounded analytic functions on the polydisc) is a relatively new subject. To paraphrase the one-dimensional result, it would imply the consideration of a subspace M ⊂ H(D), semiinvariant to the d operators of multiplication by the variables, and of a contraction X on M commuting with the compression of these multiplications. The problem would be to lift X to a contractive Toeplitz operator on the whole H(D). The most simple case, the Nevanlinna-Pick problem, has been solved only recently ([Ag], [AgMC], [BT]). It also points out that a direct analogue of the onedimensional problem is not possible, and that we have either to restrict the hypothesis or to relax the conclusion. The second alternative has been achieved, most notably by M. Cotlar and C. Sadosky in [CS1], where they obtain two-variable commutant lifting theorems in the more general context of abstract scattering systems. When specialized to the intertwining lifting problem, their result produces two “partial” interpolating Toeplitz

Hopf-Lax Formulas for Semicontinuous Data

Abstract: The equations u t + H (Du) = 0 and u t + H (u; Du) = 0, with initial condition u(0; x) = g(x) have an explicit solution when the hamiltonian is convex in the gradient variable (Lax formula) or the initial data is convex, or quasiconvex (Hopf formula). This paper extends these formulas to initial functions g which are only lower semicontinuous (lsc), and possibly innnite. It is proved that the Lax formulas give a lsc viscosity solution, and the Hopf formulas result in the minimal superso-lution. A level set approach is used to give the most general results.

Mean value inequalities

Abstract: In this article, we will discuss various issues concerning when a complete Riemannian manifold possesses a global mean value inequality for positive subsolutions of either the Laplace equation or the heat equation. This study is motivated by the recent result of the first author [L1]. In that paper, he proved estimates on the dimensions of spaces of harmonic functions of at most polynomial growth of degree d on manifolds satisfying the weak volume growth condition and the mean value inequality . Let us first recall the weak volume growth condition.

Boundary conditions for hyperbolic systems with stiff source terms

Abstract: This work is concerned with boundary conditions for multi-dimensional first-order hyperbolic systems with stiff source terms (also called relaxation). It is observed that usual relaxation stability conditions and the uniform Kreiss condition are not enough for the existence of the zero relaxation limit. To remedy this, we propose a so-called generalized Kreiss condition for initial-boundary value problems (henceforth, IBVPs) of the relaxation systems. By assuming that the relaxation system admits the quasi-stability condition and the prescribed boundary condition satisfies the generalized Kreiss condition, we derive a reduced boundary condition, for the corresponding equilibrium system, satisfying the uniform Kreiss condition and show the existence of boundary-layers. Moreover, if the relaxation system admits a more restrictive relaxation stability condition, then Friedrichs’ strictly dissipative boundary conditions, which induce certain uniform stability estimates, are shown to satisfy the generalized Kreiss condition. The present results are expected to be used as theoretical criteria to construct relaxation approximations for IBVPs of conservation laws, which are of practical interest.

Static shells for the Vlasov-Poisson and Vlasov-Einstein systems

Abstract: We prove the existence of static, spherically symmetric solutions of the stellar dynamic Vlasov-Poisson and Vlasov-Einstein systems, which have the property that their spatial support is a finite, spherically symmetric shell with a vacuum region at the center.

The weak-type (1,1) of L \log L homogeneous convolution operators

Abstract: We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the result of Seeger. The proof is in a similar spirit to that of Christ and Rubio de Francia.

Existence and stability of Camm type steady states in galactic dynamics

Abstract: We prove the existence and nonlinear stability of Camm type steady states of the Vlasov-Poisson system in the gravitational case. The paper demonstrates the effectiveness of an approach to the existence and stability problem for steady states, which was used in previous work by the authors: The steady states are obtained as minimizers of an energy-Casimir functional, and from this fact their dynamical stability is deduced.

P-cross-section bodies

Abstract: If K is a convex body in E, its cross-section body CK has a radial function in any direction u ∈ Sn−1 equal to the maximal volume of hyperplane sections of K orthogonal to u. A generalization called the p-cross-section body CpK of K, where p > −1, is introduced. The radial function of CpK in any direction u ∈ Sn−1 is the pth mean of the volumes of hyperplane sections of K orthogonal to u through points in K. It is shown that C1K is convex but CpK is generally not convex when p > 1. An inclusion of the form an,qCqK ⊆ an,pCpK, where −1 < p < q and the constant an,p is the best possible, is established. This is applied to disprove a conjecture of Makai and Martini.