Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

Covering spheres of Banach spaces by balls

If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.     

On some variational properties of metric spaces

We use the basic formulation of Ekeland’s variational principle to establish characterizations of complete path metric spaces which, being described in terms of the strong slope, are called coherent as in [3]. We also provide some basic nonlinear error bound and metric regularity results, in the context of coherent spaces.     

Every monotone open 2-homogeneous metric continuum is locally connected

It is observed that all results of Whittington's note [17] can be extended in two directions. The assumption of homogeneity of the considered spaces can be weakened to monotone open homogeneity, and homeomorphisms used in the results can be replaced by monotone open and closed mappings. In particular, the statement in the title is obtained.     

Onm-compact spaces

We give characterizations ofm-compact andm-closed spaces by means of nets and filterbases. We study the images ofm-compact andm-closed sets under various modifications ofM-continuous functions. We introduce the notion ofmc-compact space and investigate some relations betweenmc-compact spaces andM-continuous functions.     

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains. Received: 12 September 2007     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Growth of conjugacy classes of Schottky groups in higher rank symmetric spaces

Let X be a globally symmetric space of noncompact type and rank greater that one, and a Schottky group of axial isometries. Then is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homotopy with period less than t.     

Weak and strong solvability of parabolic variational inequalities in Banach spaces

We consider parabolic variational inequalities having the strong formulation

Bicompleting weightable quasi-metric spaces and partial metric spaces

We show that the bicompletion of a weightable quasi-metric space is a weightable quasi-metric space. From this result we deduce that any partial metric space has an (up to isometry) unique partial metric bicompletion. Some other consequences are derived. In particular, applications to two interesting examples of partial metric spaces which appear in Computer Science, as the domain of words and the complexity space, are given. The second and third listed authors are partially supported by grants from Spanish Ministry of Science and Technology, BFM 2000-III, and Polytechnical University of Valencia.     

Strong density results in trace spaces of maps between manifolds

We deal with strong density results of smooth maps between two manifolds and in the fractional spaces given by the traces of Sobolev maps in W ^1,p .     

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre. N. Garofalo is supported in part by NSF grant DMS-0701001. A. Petrosyan is supported in part by NSF grant DMS-0701015.     

Regularity theory in Orlicz spaces for the parabolic polyharmonic equations

In this paper we generalize classical L ^p estimates to Orlicz spaces for the parabolic polyharmonic equations. Our argument is based on the iteration-covering procedure. Received: 10 September 2007     

Adjacency in generalized projective Veronese spaces

A bijection of strong subspaces of a generalized Veronese space preserving the adjacency need not to be determined by an automorphism of the underlying space. We give conditions which assure that the adjacency preserving bijection of points (or lines) of a generalized Veronese space is determined by an automorphism of this space. The results are applied to Veronese spaces associated with projective structures.     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds

We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E _α ) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.     

Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L¹([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of . Related general results are shown for subspaces X of or of reflexive K?the function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis. Received: 25 September 2008     

The mixed problem in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> for some two-dimensional Lipschitz domains

We consider the mixed problem,