Using Aleksandrov reflection to estimate the location of the center of expansion

Lin, Yu-Chu; Tsai, Dong-Ho

doi:10.1090/S0002-9939-09-10155-7

Using Aleksandrov reflection to estimate the location of the center of expansion
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by Yu-Chu Lin and Dong-Ho Tsai PDF

Proc. Amer. Math. Soc. 138 (2010), 557-565 Request permission

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 $\gamma _{0}\subset \mathbb {R}^{2}$ lies on a certain convex plane region interior to $\gamma _{0}.$

References

Ben Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315–371. MR 1660843, DOI 10.1007/s005260050111
Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
Bennett Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom. 5 (1997), no. 2, 389–409. MR 1483984, DOI 10.4310/CAG.1997.v5.n2.a5
Bennett Chow and Robert Gulliver, Aleksandrov reflection and nonlinear evolution equations. I. The $n$-sphere and $n$-ball, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 249–264. MR 1386736, DOI 10.1007/BF01254346
Richard Courant and Fritz John, Introduction to calculus and analysis. Vol. II, Springer-Verlag, New York, 1989. With the assistance of Albert A. Blank and Alan Solomon; Reprint of the 1974 edition. MR 1016380
Bennett Chow, Lii-Perng Liou, and Dong-Ho Tsai, Expansion of embedded curves with turning angle greater than $-\pi$, Invent. Math. 123 (1996), no. 3, 415–429. MR 1383955, DOI 10.1007/s002220050034
Bennett Chow and Dong-Ho Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), no. 2, 312–330. MR 1425578
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401–441. MR 672070
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Dong-Ho Tsai, Geometric expansion of starshaped plane curves, Comm. Anal. Geom. 4 (1996), no. 3, 459–480. MR 1415752, DOI 10.4310/CAG.1996.v4.n3.a5
Dong-Ho Tsai, Asymptotic closeness to limiting shapes for expanding embedded plane curves, Invent. Math. 162 (2005), no. 3, 473–492. MR 2198219, DOI 10.1007/s00222-005-0449-9
Dong-Ho Tsai, Behavior of the gradient for solutions of parabolic equations on the circle, Calc. Var. Partial Differential Equations 23 (2005), no. 3, 251–270. MR 2142063, DOI 10.1007/s00526-004-0298-1
John I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), no. 1, 91–125. MR 1085136
Hiroki Yagisita, Asymptotic behaviors of star-shaped curves expanding by $V=1-K$, Differential Integral Equations 18 (2005), no. 2, 225–232. MR 2106103