Abstract
We study the boundary behavior of Q-homeomorphisms on the Finsler manifolds and formulate the conditions that are imposed on a function Q(x) and on the boundaries of domains and are such that every Q-homeomorphism admits a continuous or homeomorphic extension to the boundary.
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References
G. S. Asanov, Finsler Geometry, Relativity, and Gauge Fields, Reidel, Dordrecht, 1985.
Y. Takano, “Gravitational field in Finsler spaces,” Lett. Nuov. Cim., 10, No. 17, 747–750 (1974).
G. I. Garas’ko, S. S. Kokarev, V. N. Trishin, V. Balan, N. Brinzei, S. V. Siparov, V. M. Chernov, and V. A. Panchelyuga, Foundations of Finsler Geometry and Its Applications in Physics [in Russian], Bauman MSTU, Moscow, 2010.
R. S. Ingarden, “On physical applications of Finsler geometry,” Contemp. Math., 196, 213–223 (1996).
J. D. Clayton, “On Finsler geometry and applications in mechanics: review and new perspectives,” Adv. Math. Phys., 2015, ID 828475, (2015), 11 pp.
P. L. Antonelli, “Preface for “Applications of Finsler differential geometry to biology, engineering, and physics,” in: Finsler Geometry, Amer. Math. Soc., Providence, RI, 1996, pp. 199–202.
R. B. Gardner and G. R. Wilkens, “Preface for “Applications of Finsler geometry to control theory,” in: Finsler Geometry, Amer. Math. Soc., Providence, RI, 1996, pp. 227–229.
P. L. Antonelli, Handbook of Finsler Geometry, Kluwer Academic Publ., Dordrecht, 2003.
K. Kuratowski, Topology, Academic Press, New York, 1968, Vol. 2.
E. Cartan, Riemannian Geometry in an Orthogonal Frame, World Scientific, Singapore, 2002.
E. G. Poznyak and E. V. Shikin, Differential Geometry [in Russian], MSU, Moscow, 1990.
Yu. V. Dymchenko, “Relation between the condenser capacity and the module of a family of separating surfaces in Finsler spaces,” Zap. Nauch. Sem. POMI, 418, 74–89 (2013).
S. F. Rutz and F. M. Paiva, “Gravity in Finsler spaces,” in: Finslerian Geometries: A Meeting of Minds, Dordrecht, Kluwer, 2000, pp. 233–244.
H. Rund, The Differential Geometry of Finsler Spaces, Springer, Berlin, 1959.
X. Cheng and Z. Shen, Finsler Geometry: An Approach via Randers Spaces, Springer, Berlin, 2012.
J. Väisälä, Lectures on n−Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.
E. S. Afanas’eva, “Boundary behavior of ring Q-homeomorphisms on Riemannian manifolds,” Ukr. Mat. Zh., 63, No. 10, 1299–1313 (2011).
M. Hohmann, “Observer dependent geometries,” arXiv:1403.4005v1 [math-ph], March 17, 2014.
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.
V. I. Ryazanov and R. R. Salimov, “Weakly planar spaces and boundaries in the theory of mappings,” Ukr. Mat. Vestn., 4, No. 2, 199–234 (2007).
E. S. Smolovaya, “Boundary behavior of ring Q-homeomorphisms in metric spaces,” Ukr. Mat. Zh., 62, No. 5, 682–689 (2010).
H. Federer, Geometric Measure Theory, Springer, Berlin, 1996.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 12, No. 3, pp. 311–325, July–August, 2015.
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Afanas’eva, E.S. The boundary behavior of Q-homeomorphisms on the Finsler spaces. J Math Sci 214, 161–171 (2016). https://doi.org/10.1007/s10958-016-2766-5
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DOI: https://doi.org/10.1007/s10958-016-2766-5