Partial regularity of $p(x)$-harmonic maps
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- by Maria Alessandra Ragusa, Atsushi Tachikawa and Hiroshi Takabayashi PDF
- Trans. Amer. Math. Soc. 365 (2013), 3329-3353 Request permission
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 \[ {\mathcal F}(u) = \int _\Omega (g^{\alpha \beta }(x) h_{ij}(u) D_\alpha u^iD_\beta u^j)^{ p(x)/2} dx, \] 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.References
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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$_$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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3034468
Dedicated: In memory of the Japanese victims of the earthquake and tsunami that occurred on 11 March 2011