Harmonic maps from manifolds of \(L^{\infty}\)-Riemannian metrics
type: Item
instance of: scholarly article
title: Harmonic maps from manifolds of \(L^{\infty}\)-Riemannian metrics
zbMATH Open document ID: 1154.58005
author: W. Ishizuka
author: Changyou Wang
published in: Calculus of Variations and Partial Differential Equations
publication date: 2008-06-04T00:00:00Z
review text: Regularity of harmonic maps from manifolds with \(C^\infty\)-Riemannian metrics has been studied by \textit{R. Schoen} and \textit{K. Uhlenbeck} [J. Differ. Geom. 17, 307--335 (1982); correction ibid. 19, 329 (1984; Zbl 0521.58021)], \textit{M. Giaquinta} and \textit{E. Giusti} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 45--55 (1984; Zbl 0543.49018)], \textit{F. Helein} [C. R. Acad. Sci., Paris, Sér. I 312, No. 8, 591--596 (1991; Zbl 0728.35015)], \textit{L. C. Evans} [Arch. Ration. Mech. Anal. 116, No. 2, 101--113 (1991; Zbl 0754.58007)], and \textit{F. Bethuel} [Manuscr. Math. 78, No. 4, 417--443 (1993; Zbl 0792.53039)]. From the point of view in mathematical homogenization, it is important to consider domains endowed with rough metrics. In this paper, the authors consider the regularity issue of weakly harmonic maps from \((\Omega,g)\), a bounded domain in \(\mathbb R^n\) endowed with an \(L^\infty\) measurable metric \(g\), to \((N,h) \in \mathbb R^k\), a \(C^5\)-Riemannian manifold without boundary. Assume \(u \in W^{1,2}(\Omega,N)\) is a weakly harmonic map. The first result here shows that for \(n=2\), \(u \in C^\alpha (\Omega, N)\) for some \(\alpha\in (0,1)\). The second one shows that for \(n \geq 3\), if in addition \(g \in \text{VMO} (\Omega)\) and \(u\) satisfies a certain quasi-monotonicity inequality, then there exists a closed set \(\Sigma \in \Omega\), with \(H^{n-2}(\Sigma) = 0\), such that \(u \in C^\alpha(\Omega-\Sigma,N)\) for some \(\alpha \in(0,1)\).
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35J55
zbMATH DE Number: 5284782
zbMATH Keywords: harmonic maps
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: quasi-monocity inequality
Wikidata QID: Q115387661
MaRDI profile type: Publication
full work available at URL: s00526-007-0149-y
OpenAlex ID: W2130372369
cites work: On the singular set of stationary harmonic maps
cites work: Elliptic equations with BMO coefficients in Reifenberg domains
cites work: Compensated compactness and Hardy spaces
cites work: Q4389910
cites work: Q5689283
cites work: Partial regularity for stationary harmonic maps into spheres
cites work: \(H^p\) spaces of several variables
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Q3331814
cites work: Q3834387
cites work: Elliptic partial differential equations of second order
cites work: Q3351776
cites work: Q4537523
cites work: Q3998311
cites work: Gradient estimates and blow-up analysis for stationary harmonic maps
cites work: Q3782476
cites work: Q5334555
cites work: A monotonicity formula for Yang-Mills fields
cites work: Everywhere discontinuous harmonic maps into spheres
cites work: A regularity theory for harmonic maps
cites work: A partial regularity result of harmonic maps from manifolds with bounded measurable Riemannian metrics
DOI: 10.1007/S00526-007-0149-Y
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description: scientific article; zbMATH DE number 5284782