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Harmonic maps from manifolds of L ∞-Riemannian metrics

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For a bounded domain Ω ⊂ R n endowed with L ∞-metric g, and a C 5-Riemannian manifold (N, h) ⊂ R k without boundary, let u ∈ W 1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C α (Ω, N) for n  =  2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H n-2(Σ) = 0, such that \({u \in C^\alpha(\Omega{\setminus}\Sigma, N)}\) for some α ∈ (0, 1).

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Authors and Affiliations

  1. Providence College, Providence, RI, 02918, USA

    W. Ishizuka

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    C. Y. Wang

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  1. W. Ishizuka
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  2. C. Y. Wang
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Correspondence to W. Ishizuka.

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C. Y. Wang Partially supported by NSF.

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Ishizuka, W., Wang, C.Y. Harmonic maps from manifolds of L ∞-Riemannian metrics. Calc. Var. 32, 387–405 (2008). https://doi.org/10.1007/s00526-007-0149-y

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