The Hubbard Model at Half-Filling, Part III: the Lower Bound on the Self-Energy

We complete the proof that the two-dimensional Hubbard model at half- filling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first nontrivial self-energy graph.submitted 20/12/04, accepted 10/12/05     

Bernstein-type techniques for 2D free boundary graphs

We prove an a priori bound for the Lipschitz constant of a smooth one-phase free boundary graph F(u) in two dimensions. The function u satisfies an elliptic equation in its positive side, and on F(u).     

2-d stability of the Néel wall

We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.We prove stability of the 1-d transition layer under 2-d perturbations. This amounts to the investigation of the following singularly perturbed energy functional:

The Eigenvalue Problem of Singular Ergodic Control

We consider the problem of finding a real number λ and a function u satisfying the PDE Here f is a convex, superlinear function. We prove that there is a unique λ* such that the above PDE has a viscosity solution u satisfying $\lim_{|x|\rightarrow \infty}u(x)/|x|=1$ . Moreover, we show that associated to λ* is a convex solution u* with D²u^*∈ $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L^{\infty}(\R^N)$ and give two min‐max formulae for λ*. λ* has a probabilistic interpretation as being the least, long‐time averaged (ergodic) cost for a singular control problem involving f. © 2011 Wiley Periodicals, Inc.     

Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations

We use the concept of the complex WKB–Maslov method to construct semiclassically concentrated solutions for Hartree-type equations. Formal solutions of the Cauchy problem for this equation that are asymptotic (with respect to a small parameter , 0) are constructed with the power-law accuracy O(^N/2), where N3 is a positive integer. The system of Hamilton–Ehrenfest equations (for averaged and centered moments) derived in this paper plays a significant role in constructing semiclassically concentrated solutions. In the class of semiclassically concentrated solutions of Hartree-type equations, we construct an approximate Green's function and state a nonlinear superposition principle.     

Cauchy-Schwarz and means inequalities for elementary operators into norm ideals

The Cauchy-Schwarz norm inequality for normal elementary operators

implies a means inequality for generalized normal derivations

for all , as well as an inequality for normal contractions and

for all in and for all unitarily invariant norms

     

Infinite lifetime for the starlike dynamics in Hele-Shaw cells

One of the ``folklore" questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain , the Hele-Shaw chain of subordinating domains , , exists for an infinite time under injection at the point of starlikeness.

     

A weak-type inequality of subharmonic functions

We prove the weak-type inequality , , between a non-negative subharmonic function and an -valued smooth function , defined on an open set containing the closure of a bounded domain in a Euclidean space , satisfying , and , where is a constant. Here is the harmonic measure on with respect to 0. This inequality extends Burkholder's inequality in which and , a Euclidean space.

     

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of sites, where ω _d is the volume of the unit ball in , we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(r ^α ) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr ² particles, we show that the inradius is at least , and the outradius is at most . This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig. Yuval Peres is partially supported by NSF grant DMS-0605166.     

Uncountable categoricity for gross models

A model is said to be gross if all infinite definable sets in have the same cardinality (as ). We prove that if for some uncountable , has a unique gross model of cardinality , then for any uncountable , has a unique gross model of cardinality .

     

Soliton-Type Asymptotics for the Coupled Maxwell-Lorentz Equations

We consider a Maxwell field translation invariantly coupled to a single charge. This Hamiltonian system admits soliton-type solutions, where the charge and the co-moving field travel with constant velocity. We prove that a solution of finite energy converges, in suitable local energy seminorms, to a certain soliton-type solution in the long time limit t ±.Communicated by Viencent Rivasseausubmitted 01/12/03, accepted 26/05/04     

On criteria for extremality of Teichmüller mappings

Let be a Teichmüller self-mapping of the unit disk corresponding to a holomorphic quadratic differential . If satisfies the growth condition (as ), for any given 0$">, then is extremal, and for any given , there exists a subsequence of such that

is a Hamilton sequence. In addition, it is shown that there exists with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.

     

Mean curvature of extrinsic spheres in submanifolds of real space forms

Given a submanifold P^m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P^m in terms of the mean curvature of the geodesic spheres in with same radius, and the mean curvature of P^m.Received: 4 April 2003     

Bel-Robinson Energy and Constant Mean Curvature Foliations

An energy estimate is proved for the Bel-Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an bound on the second fundamental form, and a bound on a spacetime version of Bel-Robinson energy. Communicated by Sergiu KlainermanSubmitted 25/07/03, accepted 27/01/04     

The modulus of Rudin-Carleson extensions

We prove the followingTheorem. LetF be a closed subset of the unit circleT which has Lebesgue measure zero. Suppose thatp is a smooth positive function onT. GivenfC(F) which satisfies|f(s)|p(s) (sF) and a neighbourhoodU ofF there is an extension off in the disc algebra such that and .     

Depth and cohomological connectivity in modular invariant theory

Let be a finite group acting linearly on a finite-dimensional vector space over a field of characteristic . Assume that  divides the order of so that is a modular representation and let be a Sylow -subgroup for . Define the cohomological connectivity of the symmetric algebra to be the smallest positive integer such that . We show that is a lower bound for the depth of . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if is -nilpotent and is cyclic, then, for any modular representation, the depth of is .

     

Devaney's chaos implies existence of -scrambled sets

Let be a complete metric space without isolated points, and let be a continuous map. In this paper we prove that if is transitive and has a periodic point of period , then has a scrambled set consisting of transitive points such that each is a synchronously proximal Cantor set, and is dense in . Furthermore, if is sensitive (for example, if is chaotic in the sense of Devaney), with being a sensitivity constant, then this is an -scrambled set.

     

Decay and growth for a nonlinear parabolic difference equation

We prove a difference equation analogue of the decay-of-mass result for the nonlinear parabolic equation when and a new growth result when 0$">.

     

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function which is a frame wavelet and is defined by a Lebesgue measurable set such that . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the -norm. This result also holds for s-elementary -dilation frame wavelets in in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.