Weak and strong density results for the Dirichlet energy
type: Item
instance of: scholarly article
title: Weak and strong density results for the Dirichlet energy
zbMATH Open document ID: 1043.49040
DOI: 10.4171/JEMS/3
author: Mariano Giaquinta
author: Domenico Mucci
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2004-06-15T00:00:00Z
review text: Summary: Let \({\mathcal Y}\) be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in \(B^n\times{\mathcal Y}\) with equibounded Dirichlet energies, \(B^n\) being the unit ball in \(\mathbb{R}^n\). More precisely, weak limits of graphs of smooth maps \(u_k:B^n\to {\mathcal Y}\) with equibounded Dirichlet integral give rise to elements of the space \(\text{cart}^{2,1} (B^n\times{\mathcal Y})\). In this paper we prove that every element \(T\) in \(\text{cart}^{2,1} (B^n\times{\mathcal Y})\) is the weak limit of a sequence \(\{u_k\}\) of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension \(n=2\), we show that the sequence \(\{u_k\}\) can be chosen in such a way that the energy of \(u_k\) converges to the energy of \(T\).
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2075134
zbMATH Keywords: Dirichlet energy
zbMATH Keywords: energy minimizing currents
zbMATH Keywords: weak limits
MaRDI profile type: Publication
cites work: The approximation problem for Sobolev maps between two manifolds
cites work: Q3999519
cites work: Q5565773
cites work: On sequence of maps with equibounded energies
cites work: The Dirichlet integral for mappings between manifolds: Cartesian currents and homology
cites work: Q4210546
cites work: Q3261851
cites work: The problem of Plateau on a Riemannian manifold
cites work: Boundary regularity and the Dirichlet problem for harmonic maps
cites work: Q3336268
cites work: Existence of least-area mappings of N-dimensional domains
description: scientific article; zbMATH DE number 2075134