Harmonic maps from manifolds of \(L^{\infty}\)-Riemannian metrics
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 \(\alpha \in(0,1)\).
scientific article; zbMATH DE number 5284782
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