The Harnack inequality for quasilinear elliptic equations under minimal assumptions

In this note we prove the Harnack inequality for non negative solutions of the quasilinear equation

On the singular Bochner-Martinelli integral

The Bochner-Martinelli (B.-M.) kernel inherits, forn2, only some of properties of the Cauchy kernel in . For instance it is known that the singular B.-M. operatorM _n is not an involution forn2. M. Shapiro and N. Vasilevski found a formula forM ₂ ² using methods of quaternionic analysis which are essentially complex-twodimensional. The aim of this article is to present a formula forM _n ² for anyn2. We use now Clifford Analysis but forn=2 our formula coincides, of course, with the above-mentioned one.     

Orlicz-Pettis Polynomials on Banach Spaces

 We introduce the class of Orlicz-Pettis polynomials between Banach spaces, defined by their action on weakly unconditionally Cauchy series. We give a number of equivalent definitions, examples and counterexamples which highlight the differences between these polynomials and the corresponding linear operators. (Received 17 May 1999; in revised form 6 October 1999)     

O’Neil Inequality for Multilinear Convolutions and Some Applications

In this paper we prove the O’Neil inequality for the k-linear convolution f ⊗ g. By using the O’Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the k-linear convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator M _{Ω, α} and k-linear fractional integral operator I _{Ω, α} with rough kernels from the spaces V.S. Guliyev partially supported by the grant of INTAS (project 05-1000008-8157).     

On the Riccati equations

In this article we give, in terms of so-called Berezin symbols, some necessary conditions for the solvability of the Riccati equation on the set of all Toeplitz operators on the Hardy space . Author’s address: Department of Technical Programs, Isparta (MYO) Vocational School, Suleyman Demirel University, 32260 Isparta, Turkey     

Sharp Lorentz space estimates for rough operators

We demonstrate the or mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. Received December 10, 1999 / Published online April 12, 2001     

The asymmetric random cluster model and comparison of Ising and Potts models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β _c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model. Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001     

Hessian measures of semi-convex functions and applications to support measures of convex bodies

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. Received: 15 July 1999     

Finite determinacy relative to closed¶and finitely generated ideals

In this paper we propose to develop a theory of finite determinacy and unfolding in the spirit of Thom–Mather theory for C ^∞ function germs in a finitely generated and closed ideal with three additionnal properties. It may be used as the beginning of a more systematic study of non isolated real singularities. Received: 26 January 2000 / Revised version: 30 June 2000     

Large-Time Behaviour of Solutions of the Exterior-Domain Cauchy–Dirichlet Problem for the Porous Media Equation with Homogeneous Boundary Data

The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries. This behaviour is decidedly different from when the boundary data are fixed and not homogeneous.     

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P4

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P ⁴ such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronesé variety V ²⊂P ⁵ and the theory of associativity equations of two-dimensional topological field theory. Received: 15 August 2001     

An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if is an open subset of the plane with smooth boundary, satisfiesLu=0 on , has tempered growth at the boundary, and its weak boundary value is a measure , then is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of . Received March 10, 2000 / Published online April 12, 2001     

The Anosov Relation for Nielsen Numbers of Maps of Infra-Nilmanifolds

The relative ranks of the monoid of endomorphisms of a strong independence algebra of infinite rank modulo two distinguished subsets are calculated. These subsets are the group of automorphisms and the endomorphisms α satisfying α² = α. The extra generators are characterised in each case.     

Compact-like manifolds associated to JB*-triples

Finite dimensional JB ^\ast;-triples have a uniquely associated simply connected compact symmetric manifold. In infinite dimensions, different constructions have been used to yield three possibly different Banach manifolds from a JB ^\ast; all of which display compact like behaviour. We clarify the relationship between these three manifolds and characterise a case when all three of these manifolds coincide as in the finite dimensional case. These results have been announced in [11]. Received: 29 May 2001     

The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices. Authors’ addresses: M. A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain; A. Schürmann, Institut für Algebra und Geometrie, Otto-von-Guericke Universit?t Magdeburg, 39106 Magdeburg, Germany; F. Vallentin, Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data

We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem -div(a(x,Du))=μ in Ω, u=0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω. Received: December 27, 2000 Published online: December 19, 2001     

Hypersurfaces and Codazzi tensors

In this paper we deal with the following problem. Let (M ⁿ ,〈,〉) be an n-dimensional Riemannian manifold and an isometric immersion. Find all Riemannian metrics on M ⁿ that can be realized isometrically as immersed hypersurfaces in the Euclidean space . More precisely, given another Riemannian metric on M ⁿ , find necessary and sufficient conditions such that the Riemannian manifold admits an isometric immersion into the Euclidean space . If such an isometric immersion exists, how can one describe in terms of f? Author’s address: Thomas Hasanis and Theodoros Vlachos, Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece