On sequences of maps with finite energies in trace spaces between manifolds
type: Item
instance of: scholarly article
title: On sequences of maps with finite energies in trace spaces between manifolds
zbMATH Open document ID: 1260.49083
author: Domenico Mucci
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 6031438
zbMATH Keywords: relaxed energies
zbMATH Keywords: trace spaces
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: currents
MaRDI profile type: Publication
full work available at URL: acv.2011.012
OpenAlex ID: W2333947330
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cites work: Topology of Sobolev mappings III
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cites work: On the minimizers of the relaxed energy functional of mappings from higher dimensional balls into \(S^{2}\)
DOI: 10.1515/ACV.2011.012
published in: Advances in Calculus of Variations
review text: The author studies some properties of mappings between Riemanniann manifolds that belong to trace spaces of Sobolev functions, equipped with a natural energy equivalent to the fractional norm. Denote by \(\mathcal{X, Y}\) two smooth, connected, compact, oriented Riemannian manifolds embedded in \(\mathbb{R}^l\) and \(\mathbb{R}^N\) respectively. Denote by \(W^{1/p}= W^{1-1/p,p}= W^{1/p}(\mathcal{X}), \;p>1\) the fractional Sobolev space of real valued functions \(u\) in \(L^p(\mathcal{X})\) with finite \(W^{1/p}\)-seminorm. The author recalls some definitions and results concerning the trace spaces, the \(g_{1/p}\)-energy, and relaxed energy. First, he considers the case where the manifold \(\mathcal{Y}\) is the sphere \(S^{\mathfrak{p}-1}\), where \(\mathfrak{p}= [p]\) is the integer part of \(p\), and obtains a certain strong density property for \(W^{1/p}\)-maps \(u_k:\Omega \to S^{\mathfrak{p}-1}\) and the corresponding energies \(\mathcal{E}_{1/\mathfrak{p}}\). After recalling the definition of current \(G_u\) carried by the graph of a function \(u\in W^{1/p}(\mathcal{X},\mathcal{Y})\) he discusses the homological singularities of \(u\). Then, he introduces the class of Cartesian currents in \(\mathcal{X}\times \mathcal{Y}\) with finite \(W^{1/p}\)-energy, enumerating their main properties. Using the density property of graphs of smooth maps in \(cart^{1/\mathfrak{p}}(\mathcal{X}\times \mathcal{Y})\), the author considers the relaxed \(\mathcal{E}_{1/p}\)-energy of smooth maps in \(W^{1/p}(\mathcal{X},\mathcal{Y})\) for general target manifolds. Then, he studies the weak approximation of spherical cycles, approximates spherical \((p-1)\)-cycles and obtains a proof of the density theorem 5.16, by first considering several special cases, then presenting the proof in the general case.
reviewed by: Vasile Oproiu
description: scientific article; zbMATH DE number 6031438