Using Aleksandrov reflection to estimate the location of the center of expansion
Authors:
YuChu Lin and DongHo Tsai
Journal:
Proc. Amer. Math. Soc. 138 (2010), 557565
MSC (2000):
Primary 35K15, 35K55
Published electronically:
September 30, 2009
MathSciNet review:
2557173
Fulltext PDF
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Abstract: We use the Aleksandrov reflection result of Chow and Gulliver to show that the center of expansion in expanding a given convex embedded closed curve lies on a certain convex plane region interior to
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 B. Andrews, Evolving convex curves, Cal. of Var. & PDEs, 7 (1998), no. 4, 315371. MR 1660843 (99k:58038)
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 S. Angenent, The zero set of a solution of a parabolic equation, J. für die reine and angewandte Mathematik, 390 (1988), 7996. MR 953678 (89j:35015)
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 B. Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. & Geom., 5 (1997), no. 2, 389409. MR 1483984 (98k:53045)
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 B. Chow; R. Gulliver, Aleksandrov reflection and nonlinear evolution equations, I: The nsphere and nball, Cal. of Var. & PDEs, 4 (1994), 249264. MR 1386736 (97f:53064)
 [CJ]
 R. Courant; F. John, Introduction to Calculus and Analysis, Vol. II, John Wiley & Sons, 1974; reprint of the 1974 edition, SpringerVerlag, 1989. MR 1016380 (90j:00002b)
 [CLT]
 B. Chow; L.P. Liou; D.H. Tsai, Expansion of embedded curves with turning angle greater than , Invent. Math., 123 (1996), 415429. MR 1383955 (97c:58025)
 [CT]
 B. Chow; D.H. Tsai, Geometric expansion of convex plane curves, J. of Diff. Geom., 44 (1996), 312330. MR 1425578 (97m:58041)
 [GH]
 M. Gage; R. Hamilton, The heat equation shrinking convex plane curves, J. of Diff. Geom., 23 (1986), 6996. MR 840401 (87m:53003)
 [M]
 H. Matano, Nonincrease of the lapnumber of a solution for a onedimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA Math., 29 (1982), 401441. MR 672070 (84m:35060)
 [S]
 R. Schneider, Convex Bodies: The BrunnMinkowski Theory, Cambridge University Press, 1993. MR 1216521 (94d:52007)
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 D.H. Tsai, Geometric expansion of starshaped plane curves, Comm. Anal. & Geom., 4 (1996), no. 3, 459480. MR 1415752 (97k:58042)
 [T2]
 D.H. Tsai, Asymptotic closeness to limiting shapes for expanding embedded plane curves, Invent. Math., 162 (2005), 473492. MR 2198219 (2006j:53099)
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 D.H. Tsai, Behavior of the gradient for solutions of parabolic equations on the circle, Cal. of Var. & PDEs, 23 (2005), 251270. MR 2142063 (2006d:35116)
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 J. Urbas, An expansion of convex hypersurfaces, J. of Diff. Geom., 33 (1991), 91125. Correction, ibid., 35 (1992), 763765. MR 1085136 (91j:58155); MR 1163459 (93b:58142)
 [Y]
 H. Yagisita, Asymptotic behaviors of starshaped curves expanding by , Diff. & Integ. Eqs., 18 (2005), no. 2, 225232. MR 2106103 (2005m:53128)
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Additional Information
YuChu Lin
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
Email:
yclin@math.nthu.edu.tw
DongHo Tsai
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
Email:
dhtsai@math.nthu.edu.tw
DOI:
http://dx.doi.org/10.1090/S0002993909101557
Received by editor(s):
August 4, 2008
Published electronically:
September 30, 2009
Additional Notes:
The research of the second author was supported by NSC (grant number 952115M007009) and the research center NCTS of Taiwan.
Communicated by:
ChuuLian Terng
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
