TITLE

Flat convergence for integral currents in metric spaces

AUTHOR(S)
Wenger, Stefan
PUB. DATE
February 2007
SOURCE
Calculus of Variations & Partial Differential Equations;Feb2007, Vol. 28 Issue 2, p139
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In compact local Lipschitz neighborhood retracts in $$\mathbb{R}^n$$ weak convergence for integral currents is equivalent to convergence with respect to the flat distance. This comes as a consequence of the deformation theorem for currents in Euclidean space. Working in the setting of metric integral currents (the theory of which was developed by Ambrosio and Kirchheim) we prove that the equivalence of weak and flat convergence remains true in the more general context of metric spaces admitting local cone type inequalities. These include in particular all Banach spaces and all CAT(Îº)-spaces. As an application we obtain the existence of a minimal element in a fixed homology class and show that the weak limit of a sequence of minimizers is itself a minimizer.
ACCESSION #
23218190

Related Articles

• Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces. Ozdemir, Murat; Akbulut, Sezgin; Kiziltunc, Hukmi // Discrete Dynamics in Nature & Society;2010, Special section p1

We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the...

• Existence of zero points for pseudomonotone operators in Banach spaces. Matsushita, Shin-ya; Takahashi, Wataru // Journal of Global Optimization;Dec2008, Vol. 42 Issue 4, p549

The purpose of this paper is to study the existence of zero points for set-valued pseudomonotone operators in a Banach space by using a new condition which was recently proposed by the authors (Matsushita and Takahashi, Set-Valued Analysis 15:251ï¿½264, 2007).

• Nonlinear Full Invariant of Compact Banach-Space Maps. Wu Junde; Tang Zhifeng; Cui Chengri // International Journal of Theoretical Physics;Mar2005, Vol. 44 Issue 3, p277

We characterize a nonlinear full invariant of compact Banach-space maps: Let ( X, ?.?) and ( Y, ?.?) be two Banach spaces and P C( X, Y) be all compact maps which map ( X, ?.?) to ( Y, ?.?). Then each weak operator-topology subseries-convergent series ? i P i in P c( X, Y) is also...

• Towards variational analysis in metric spaces: metric regularity and fixed points. Ioffe, A. D. // Mathematical Programming;May2010, Vol. 123 Issue 1, p241

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a â€œpartial compositionâ€ of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutinâ€™s theorem about additive perturbation of a mapping...

• Fixed points for some non-obviously contractive operators defined in a space of continuous functions. Avramescu, Cezar; Vladimirescu, Christian // Electronic Journal of Qualitative Theory of Differential Equatio;Mar2004, p1

Let X be an arbitrary (real or complex) Banach space, endowed with the norm |.|. Consider the space of the continuous functions C ([0; T] ; X) (T > 0), endowed with the usual topology, and let M be a closed subset of it. One proves that each operator A : M â†’ M fullling for all x, y...

• Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Paolo Cubiotti; Jen-chih Yao // Journal of Global Optimization;Feb2007, Vol. 37 Issue 2, p263

Abstract??We consider the following implicit quasi-variational inequality problem: given two topological vector spacesEandF, two nonempty setsX$$\sqsubseteq$$EandC$$\sqsubseteq$$F, two multifunctions ? :?X? 2Xand ? :X? 2C, and a single-valued map?:$$x\times c\times x\to ir$$, find a pair(\hat...

• On the generalized 2-D stochastic Ginzburg-Landau equation. Yang, De Sheng // Acta Mathematica Sinica;Aug2010, Vol. 26 Issue 8, p1601

This paper proves the existence and uniqueness of solutions in a Banach space for the generalized stochastic Ginzburg-Landau equation with a multiplicative noise in two spatial dimensions. The noise is white in time and correlated in spatial variables. The condition on the parameters is the same...

• Isometries on the quasi-Banach spaces L p (0 < p < 1). Lei Li; Wei Ren // Acta Mathematica Sinica;Aug2010, Vol. 26 Issue 8, p1519

We study the extension of isometries between the unit spheres of quasi-Banach spaces L p for 0 < p < 1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of L p( ï¿½) into that of another L p( ?) can be extended to be a linear isometry defined on the...

• CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES. Yongfu Su; Xiaolong Qin; Meijuan Shang // Acta Mathematica Universitatis Comenianae;2008, Vol. 77 Issue 1, p31

Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T : K ? E be an asymptotically nonexpansive P mapping with {kn} ? [1,8) such that ?n=18(kn -1) < 8 and let F(T) be nonempty, where F(T) denotes the fixed points...

Share