Geodesic
On the affine heat equation for non-convex curves
Angenent, Sigurd1 ; Sapiro, Guillermo2 ; Tannenbaum, Allen
1 Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
2 Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 601-634

Voir la notice de l'article provenant de la source American Mathematical Society

In this paper, we extend to the non-convex case the affine invariant geometric heat equation studied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-known Euclidean geometric heat equation.
DOI : 10.1090/S0894-0347-98-00262-8

Angenent, Sigurd 1 ; Sapiro, Guillermo 2 ; Tannenbaum, Allen 

1 Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
2 Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455 
@article{10_1090_S0894_0347_98_00262_8,
     author = {Angenent, Sigurd and Sapiro, Guillermo and Tannenbaum, Allen},
     title = {On the affine heat equation for non-convex curves},
     journal = {Journal of the American Mathematical Society},
     pages = {601--634},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {1998},
     doi = {10.1090/S0894-0347-98-00262-8},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-98-00262-8/}
}
TY  - JOUR
AU  - Angenent, Sigurd
AU  - Sapiro, Guillermo
AU  - Tannenbaum, Allen
TI  - On the affine heat equation for non-convex curves
JO  - Journal of the American Mathematical Society
PY  - 1998
SP  - 601
EP  - 634
VL  - 11
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-98-00262-8/
DO  - 10.1090/S0894-0347-98-00262-8
ID  - 10_1090_S0894_0347_98_00262_8
ER  - 
%0 Journal Article
%A Angenent, Sigurd
%A Sapiro, Guillermo
%A Tannenbaum, Allen
%T On the affine heat equation for non-convex curves
%J Journal of the American Mathematical Society
%D 1998
%P 601-634
%V 11
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-98-00262-8/
%R 10.1090/S0894-0347-98-00262-8
%F 10_1090_S0894_0347_98_00262_8
Angenent, Sigurd; Sapiro, Guillermo; Tannenbaum, Allen. On the affine heat equation for non-convex curves. Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 601-634. doi: 10.1090/S0894-0347-98-00262-8

[1] ÁLvarez, Luis, Guichard, Frã©Dã©Ric, Lions, Pierre-Louis, Morel, Jean-Michel Axiomes et équations fondamentales du traitement d’images (analyse multiéchelle et EDP) C. R. Acad. Sci. Paris Sér. I Math. 1992 135 138

[2] Alvarez, Luis, Guichard, Frã©Dã©Ric, Lions, Pierre-Louis, Morel, Jean-Michel Axiomatisation et nouveaux opérateurs de la morphologie mathématique C. R. Acad. Sci. Paris Sér. I Math. 1992 265 268

[3] Andrews, Ben Contraction of convex hypersurfaces by their affine normal J. Differential Geom. 1996 207 230

[4] Angenent, Sigurd Parabolic equations for curves on surfaces. I. Curves with 𝑝-integrable curvature Ann. of Math. (2) 1990 451 483

[5] Angenent, Sigurd Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions Ann. of Math. (2) 1991 171 215

[6] Angenent, Sigurd On the formation of singularities in the curve shortening flow J. Differential Geom. 1991 601 633

[7] Angenent, Sigurd The zero set of a solution of a parabolic equation J. Reine Angew. Math. 1988 79 96

[8] Su, Bu Chin Affine differential geometry 1983

[9] Dieudonnã©, Jean A., Carrell, James B. Invariant theory, old and new 1971

[10] Chen, Xu-Yan, Matano, Hiroshi Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations J. Differential Equations 1989 160 190

[11] Epstein, C. L., Gage, Michael The curve shortening flow 1987 15 59

[12] Gage, Michael E. An isoperimetric inequality with applications to curve shortening Duke Math. J. 1983 1225 1229

[13] Gage, M. E. Curve shortening makes convex curves circular Invent. Math. 1984 357 364

[14] Gage, M., Hamilton, R. S. The heat equation shrinking convex plane curves J. Differential Geom. 1986 69 96

[15] Grayson, Matthew A. The heat equation shrinks embedded plane curves to round points J. Differential Geom. 1987 285 314

[16] Grayson, Matthew A. Shortening embedded curves Ann. of Math. (2) 1989 71 111

[17] Guggenheimer, Heinrich W. Differential geometry 1963

[18] Kimia, Benjamin B., Tannenbaum, Allen, Zucker, Steven W. On the evolution of curves via a function of curvature. I. The classical case J. Math. Anal. Appl. 1992 438 458

[19] Matano, Hiroshi Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1982 401 441

[20] Olver, Peter J. Applications of Lie groups to differential equations 1993

[21] Olver, Peter J., Sapiro, Guillermo, Tannenbaum, Allen Classification and uniqueness of invariant geometric flows C. R. Acad. Sci. Paris Sér. I Math. 1994 339 344

[22] Olver, Peter J., Sapiro, Guillermo, Tannenbaum, Allen Invariant geometric evolutions of surfaces and volumetric smoothing SIAM J. Appl. Math. 1997 176 194

[23] Osher, Stanley, Sethian, James A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 1988 12 49

[24] Protter, Murray H., Weinberger, Hans F. Maximum principles in differential equations 1967

[25] Sapiro, Guillermo, Tannenbaum, Allen On affine plane curve evolution J. Funct. Anal. 1994 79 120

[26] Sapiro, Guillermo, Tannenbaum, Allen On invariant curve evolution and image analysis Indiana Univ. Math. J. 1993 985 1009

[27] Spivak, Michael A comprehensive introduction to differential geometry. Vol. I 1979

[28] White, Brian Some recent developments in differential geometry Math. Intelligencer 1989 41 47

Cité par Sources :