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Partial regularity of $ p(x)$-harmonic maps


Authors: Maria Alessandra Ragusa, Atsushi Tachikawa and Hiroshi Takabayashi
Journal: Trans. Amer. Math. Soc. 365 (2013), 3329-3353
MSC (2010): Primary 35J20, 35J47, 35J60, 49N60, 58E20
DOI: https://doi.org/10.1090/S0002-9947-2012-05780-1
Published electronically: October 4, 2012
MathSciNet review: 3034468
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Abstract: Let $ (g^{\alpha \beta }(x))$ and $ (h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $ h_{ij}(u)$ and $ p(x) \, ( \, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional

$\displaystyle {\mathcal F}(u) = \int _\Omega (g^{\alpha \beta }(x) h_{ij}(u) D_\alpha u^iD_\beta u^j)^{ p(x)/2} dx, $

under the nonstandard growth conditions of $ p(x)$-type. If $ g^{\alpha \beta }(x)$ are in the class $ VMO$, we have partial Hölder regularity. Moreover, if $ g^{\alpha \beta }$ are Hölder continuous, we can show partial $ C^{1,\alpha }$-regularity.

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Additional Information

Maria Alessandra Ragusa
Affiliation: Dipartimento di Matematica e Informatica, Universitá di Catania, Viale Andrea Doria, 6-95128 Catania, Italy
Email: maragusa@dmi.unict.it

Atsushi Tachikawa
Affiliation: Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan
Email: tachikawa$\textunderscore$atsushi@ma.noda.tus.ac.jp

Hiroshi Takabayashi
Affiliation: Kasa Ai 103, 1-20 Mukaihara-cho, Kashiwa, Chiba 277-0851, Japan
Email: h.takaba119@hotmail.co.jp

DOI: https://doi.org/10.1090/S0002-9947-2012-05780-1
Received by editor(s): March 31, 2011
Received by editor(s) in revised form: July 19, 2011, October 3, 2011, and December 14, 2011
Published electronically: October 4, 2012
Additional Notes: This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 22540207, 2010
Dedicated: In memory of the Japanese victims of the earthquake and tsunami that occurred on 11 March 2011
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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