A Vanishing Theorem for the Heat Content of a Riemannian Manifold

The paper considers the asymptotic behaviour of the heat contentof a riemannian manifold M, when Dirichlet conditions are imposedon a small compact subset of M. 2000 Mathematics Subject Classification58J35, 35K05.     

Two-Sided Estimates of the Heat Kernel of the Schrodinger Operator

Two-sided bounds for the heat kernel of the Schrödingeroperator with a singular potential are obtained with explicitconstants. 1991 Mathematics Subject Classification 35B05, 35J10.     

On the Existence of a Solution in Weighted Sobolev Space to the Riemann-Hilbert Problem for an Elliptic System with Piecewise Continuous Boundary Data

Given a first order elliptic partial differential equation weconstruct a solution which solves a given Riemann–Hilbertboundary value problem whose coefficients have singularitiesof the first kind at a finite number of some prescribed isolatedpoints and are Holder–continuous outside those pointswhile the free term has a finite number of integrable powersingularities at some prescribed points. It is shown that thesolution belongs to some weighted Sobolev space W_1,p(D;), wherethe weight function = (z;D) is the distance of the variablepoint z from the boundary D raised to a certain power. The problemis solved by first reducing it to an analogous problem for holomorphicfunctions. The latter is then solved. 2000 Mathematics SubjectClassification 35F15, 35F30, 35J55, 35J65, 36J67.     

Heat Content and a Hardy Inequality for Complete Riemannian Manifolds

Upper bounds are obtained for the heat content and the torsionalrigidity of a complete Riemannian manifold M, assuming a Hardyinequality for the Dirichlet Laplacian on M. 2000 MathematicsSubject Classification 35J05, 35K05, 58J35.     

Another Proof for the Presentation of the Quantum Cohomology of the Moduli of Bundles over a Riemann Surface

The presentation of the quantum cohomology of the moduli spaceof stable vector bundles of rank two and odd degree with fixeddeterminant over a Riemann surface of genus g > 2 is obtained.The argument avoids the use of gauge theory, providing an alternativeproof to that given by the author in Duke Math. J. 98 (1999)525–540. 2000 Mathematics Subject Classification 14N35(primary); 14H60, 53D45 (secondary).     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

On the Existence of Extremals for the Sobolev Trace Embedding Theorem with Critical Exponent

In this paper, the existence problem is studied for extremalsof the Sobolev trace inequality W^1,p()L^p*(), where is a boundedsmooth domain in R^N, p*=p(N–1)/(N–p), is the criticalSobolev exponent, and 1 < p < N. 2000 Mathematics SubjectClassification 35J65 (primary), 35B33 (secondary).     

Minimal Solutions for a Class of Elliptic Systems

There exists a set U in the plane, such that elements of U correspondto minimal stable solutions of a two-parameter elliptic system.For points on the boundary of U, there exist weak solutionsto the elliptic system. The paper studies the properties ofthe curve . 2000 Mathematics Subject Classification 35J55, 35B05,35K50, 35K55.     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

A Short Proof of the Harris-Kesten Theorem

We give a short proof of the fundamental result that the criticalprobability for bond percolation in the planar square latticeZ² is equal to 1/2. The lower bound was proved by Harris, whoshowed in 1960 that percolation does not occur at p = 1/2. Theother, more difficult, bound was proved by Kesten, who showedin 1980 that percolation does occur for any p > 1/2. 2000Mathematics Subject Classification 60K35, 82B43.     

Uniqueness and Nonexistence of Positive Solutions to Semipositone Problems

We consider the uniqueness of the positive solution to a semilinearelliptic equation with Dirichlet boundary condition and thenonlinearity satisfying f(0)<0 and having asymptotic sublineargrowth rate. A similar idea is also applied to the nonexistenceof a positive solution to a superlinear problem. 2000 MathematicsSubject Classification 35J65 (primary), 35B40 (secondary).     

Some Sharp Bounds for the Cone Multiplier of Negative Order in R3

This paper considers the cone multiplier operator which is definedby where and . For –3/2 < µ < –3/14, sharp L^p – L^q estimatesand endpoint estimates for S^µ are obtained. 2000 MathematicsSubject Classification 42B15 (primary).     

Characteristic Zero Loop Space Homology of Two-Cones

Given a principal ideal domain R of characteristic zero, containing1/2, and a two-cone X of appropriate connectedness and dimension,we present sufficient algebraic conditions for the Hopf algebraFH(X; R) to be isomorphic with the universal enveloping algebraof an R-free graded Lie algebra; here, F stands for free part(that is, quotient by the R-torsion), H for homology, and forthe Moore loop space functor. 2000 Mathematics Subject Classification55P35, 55P62, 57T05, 17B70, 17B35.     

On The Curvature of the Real Milnor Fiber

Let C be a germ at O R² of a real analytic plane curve, andC^C its complexification; let C_t B be a fiber of a real smoothdeformation of C in the ball B = B(O,). The following inequalityis proved between the integrals of real curvature k of C_t andthose of Gaussian curvature K of : The sharpness of this inequality is proved in the case whereC is a real irreducible germ. Similar results are proved foran affine algebraic curve C R² of degree d. 2000 MathematicsSubject Classification 14H20, 14H50, 53A04.     

Strict Positivity of a Relativistic Hamiltonian Due to Brown and Ravenhall

We prove that the spectrum of a suitably projected Dirac operator—sometimescalled the no-pair operator—introduced by Brown and Ravenhallis bounded from below by some positive constant when the nuclearcharge is below some critical charge. This bound improves aresult obtained recently by Evans et al., and disproves a conjecturemade by Hardekopf and Sucher. 1991 Mathematics Subject Classification35P15, 35Q40, 81Q10.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

Regularity of the Fractional Maximal Function

The purpose of this work is to show that the fractional maximaloperator has somewhat unexpected regularity properties. Themain result shows that the fractional maximal operator mapsL^p-spaces boundedly into certain first-order Sobolev spaces.It is also proved that the fractional maximal operator preservesfirst-order Sobolev spaces. This extends known results for theHardy-Littlewood maximal operator. 2000 Mathematics SubjectClassification 42B25, 46E35.     

Some Singularly Perturbed Problems on Annuli and a Counterexample to a Problem of Gidas, Ni and Nirenberg

1991 Mathematics Subject Classification 35B05, 35J60.     

On the Approximation Numbers for the Finite Sections of Block Toeplitz Matrices

In this paper we discuss the asymptotic distribution of theapproximation numbers of the finite sections for a Toeplitzoperator and µ R, witha continuous matrix-valued generating function a. We prove thatthe approximation numbers of the finite sections T_n(a) = P_nT(a)P_nhave the k-splitting property, provided T(a) is a Fredholm operatoron . 2000 Mathematics SubjectClassification 47B35 (primary), 15A18, 47A58, 47A75, 65F20 (secondary).     

Integral Estimates for Transport Densities

The integration-by-parts methods introduced in this paper improveupon the L^p estimates on transport densities given in the recentpaper by L. De Pascale and A. Pratelli (Calc. Var. Partial DifferentialEquations 14 (2002) 249–274). 2000 Mathematics SubjectClassification 35Q99, 35B99.