Abstract
The symmetry group of a centro-affine invariant flow is presented and a corresponding optimal system is found. Group invariant solutions associated to the optimal system are obtained and classified.
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References
B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), 207–230.
S. Angenent, Parabolic equations for curves on surfaces, Part II. Intersections, Ann. of Math. (2) 133 (1991), 45–52.
S. Angenent, E. Pichon, and A. Tannenbaum, Mathematical methods in medical image processing, Bull. Amer. Math. Soc. 43 (2006), 365–396.
S. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991), 491–514.
S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), 601–633.
E. Calabi, P. Olver, and A. Tannenbaum, Affine geometry, curve flows, and invariant numerical approximations, Adv. Math. 124 (1996), 154–196.
F Cao, Geometric Curve Evolution and Image Processing, Springer-Verlag, Berlin, 2003.
K.S. Chou and K.S. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001.
K.S. Chou and G.X. Li, Optimal systems and invariant solutions for the curve shortening problem, Comm. Anal. Geom. 10 (2002), 241–274.
M. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993.
M. Ivaki, Centro-affine curvature flows on centrally symmetric convex curves, Trans. Amer. Math. Soc. 366 (2014), 5671–5692.
M. Ivaki, Centro-affine normal flows on curves: Harnack estimates and ancient solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 1189–1197.
B. Kimia, Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space, Int. J. Comput. Vision. 15 (1995), 189–224.
W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900–904.
P. Olver, G. Sapiro, and A. Tannenbaum, Affine invariant detection: edge maps, anisotropic diffusion, and active contours, Acta Appl. Math. 59 (1999), 45–77.
P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1998.
P. Olver, G. Sapiro, and A. Tannenbaum, Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach, In: Geometry-Driven Diffusion in Computer Vision Computational Imaging and Vision, Volume 1, Springer, Dordrecht, 1994, 255–306.
L.V. Ovisannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
G. Sapiro, Geometric partial differential equations in image analysis: past, present, and future, Cambridge University Press, New York, 2001.
G. Sapiro and A. Tannenbaum, Affine invariant scale-space. Int. J. Comput. Vision 11 (1993), 25–44.
A. Stancu, Centro-affine invariants for smooth convex bodies, Int. Math. Res. Not. 114 (2012), 2289–2320.
W.F. Wo, C.Z. Qu, and X.L. Wang, The centro-affine invariant geometric heat flow, preprint, 2016.
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Wo, W., Yang, S. & Wang, X. Group invariant solutions to a centro-affine invariant flow. Arch. Math. 108, 495–505 (2017). https://doi.org/10.1007/s00013-016-1010-3
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DOI: https://doi.org/10.1007/s00013-016-1010-3