Abstract
Let \(f : U\subseteq \mathbb {R}^m\rightarrow \fancyscript{Q}_Q(\ell _2)\) be of Sobolev class \(W^{1,p}\), \(1 < p < \infty \). If \(f\) almost minimizes its \(p\) Dirichlet energy then \(f\) is Hölder continuous. If \(p=2\) and \(f\) is squeeze and squash stationary then \(f\) is in VMO.
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References
Almgren, F.: Almgren’s big regularity paper. World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Con. Inc., River Edge, NJ (2000)
De Lellis, C., Spadaro, E.N.: Q-valued functions revisited. Mem. Am. Math. Soc. 211, 991 (2011)
De Lellis, C.: Errata to Q-valued functions revisited. http://user.math.uzh.ch/delellis
Bouafia, P., De Pauw, T., Goblet, J.: Existence of \(p\)-harmonic multiple valued maps into a separable Hilbert space. Ann. Inst. Fourier (To appear)
De Lellis, C., Grisanti, C.R., Tilli, P.: Regular selections for multiple-valued functions. Ann. Mat. Pura Appl. 183(4) no. 1, 79–95 (2004)
Dellis, C., Focardi, M., Spadaro, E.: Lower semicontinuous functionals for Almgren’s multiple valued functions. Ann. Acad. Sci. Fenn. Math. 36(2), 393–410 (2011)
Golbert, J.: Lipschitz extension of multiple Banach-valued functions in the sense of Almgren. Houst. J. Math. 35(1), 223–231 (2009)
Goblet, J., Zhu, W.: Regularity of Dirichlet nearly minimizing multiple-valued functions. J. Geom. Anal. 18(3), 765–794 (2008)
Hirsch, J.: Partial Hölder continuity for Q-valued energy minimizing maps. Preprint arXiv:1402.2651
Hirsch, J.: Boundary regularity of Dirichlet minimizing Q-valued functions. Preprint arXiv:1402.2559
Lin, C.C.: Interior continuity of two-dimensional weakly stationary-harmonic multiple-valued functions. J. Geom. Anal. 24(3), 1547–1582 (2014)
Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J. 37(2), 349–367 (1988)
Mattila, P.: Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions. Trans. Am. Math. Soc. 280(2), 589–610 (1983)
Morrey, C.B.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130, pp. ix+506. Springer-Verlag New York Inc, New York (1966)
Moser, R.: Partial regularity for harmonic maps and related problems. World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ (2005)
Zhu, W.: A Theorem on Frequency Function for Multiple-Valued Dirichlet Minimizing Functions. Preprint arXiv:math/0607576
Zhu, W.: A regularity theory for multiple-valued Dirichlet minimizing maps. Preprint arXiv:math/0608178
Zhu, W.: An Energy Reducing Flow for Multiple-Valued Functions. Preprint arXiv:math/0606478
Acknowledgments
The second author was supported in part by the Project ANR-12-BS01-0014-01 Geometry. The third author is partially supported by NSF grant DMS 1522869, NSFC grant 11128102. The authors would like to thank the referees for their constructive comments.
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Communicated by F. H. Lin.
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Bouafia, P., De Pauw, T. & Wang, C. Multiple valued maps into a separable Hilbert space that almost minimize their \(p\) Dirichlet energy or are squeeze and squash stationary. Calc. Var. 54, 2167–2196 (2015). https://doi.org/10.1007/s00526-015-0861-y
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DOI: https://doi.org/10.1007/s00526-015-0861-y