Global higher integrability for parabolic quasiminimizers in nonsmooth domains

We study the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. We show that if the lateral boundary satisfies a capacity density condition and if boundary and initial values are smooth enough, then quasiminimizers globally belong to a higher Sobolev space than assumed a priori. We derive estimates near the lateral and the initial boundaries.     

Global integrability of p-superharmonic functions on metric spaces

We consider a metric space equipped with a doubling measure and a length metric. We prove that p-superharmonic functions are integrable with a small exponent on Hölder domains of the space.     

Stability of quasiminimizers of the p-Dirichlet integral with varying p on metric spaces

We prove a stability result, with respect to the varying exponentp, for a family of quasiminimizers of the p-Dirichlet energyfunctional on a doubling metric measure space. In addition,we prove global higher integrability for upper gradients ofquasiminimizers with fixed boundary data, provided that theboundary data belong to a slightly better Newtonian space.     

Quasiconformality, homeomorphisms between metric measure spaces preserving quasiminimizers, and uniform density property

We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.     

Sobolev-type functions with variable integrability exponent on metric measure spaces

We consider various embedding theorems for Sobolev-type function classes with variable integrability exponent on a metric space.     

On the property of higher integrability for parabolic systems of variable order of nonlinearity

We study a parabolic system of the form ? _t u = div _x A(x, t, ? _x u) in a bounded cylinder Q _T = Ω × (0, T) ? ? _x,t ⁿ⁺¹ . Here the matrix function A(x, t, ζ) is subject to the conditions of power growth in the variable ζ and coercitivity with variable exponent p(x, t). It is assumed that p(x, t) has a logarithmic modulus of continuity and satisfies the estimate $$ \frac{{2n}} {{n + 2}} < \alpha \leqslant p(x,t) \leqslant \beta < \infty . $$ For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinderQ _T . The method of a solution is based on a localization of a special kind and a local variant (adapted for parabolic problems) of Gehring’s lemma with variable exponent of integrability proved in the paper.     

A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

LetF be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that ifF admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.The second author is partially supported, by a Seed Grant from The Ohio State University.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Ultrametricity and metric betweenness in tangent spaces to metric spaces

The paper deals with pretangent spaces to general metric spaces. An ultrametricity criterion for pretangent spaces is found and it is closely related to the metric betweenness in the pretangent spaces.     

Implicit mapping theorem for extended metric regularity in metric spaces

In this paper we prove an extension of the contraction mapping principle for set-valued mappings stated by A.L. Dontchev and W.W. Hager dealing with more general assumptions containing modulus instead of pseudo-contractive multifunctions. Using this result we show that the update Graves theorem, in company with the stability of metric regularity under perturbations can be extended to a much broader framework of set-valued mappings acting in abstract spaces.     

Chirality in metric spaces

A definition of chirality based on group theory is presented. It is shown to be equivalent to the usual one in the case of Euclidean spaces, and it permits to define chirality in metric spaces which are not Euclidean.     

Capacities in metric spaces

We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the axiomatic framework of [17].46E35, 31C15, 31C45     

Plasticity in metric spaces

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.     

Fuzzy metric spaces

In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

Weakly metric spaces

The paper contains some initial results of the theory of weakly metric spaces. The weak triangle axiom: for any ε > 0 there exists δ > 0 such that for any points x, y, and z with d(y, z) ≤ δ, the inequality d(x, z) ≤ d(x, y) + ε holds. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 94–105.