Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1994)
- Volume: 21, Issue: 4, page 595-628
- ISSN: 0391-173X
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topVelázquez, J. J. L.. "Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 21.4 (1994): 595-628. <http://eudml.org/doc/84193>.
@article{Velázquez1994,
author = {Velázquez, J. J. L.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {generalized solutions; singularities; fast and slow blow up},
language = {eng},
number = {4},
pages = {595-628},
publisher = {Scuola normale superiore},
title = {Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow},
url = {http://eudml.org/doc/84193},
volume = {21},
year = {1994},
}
TY - JOUR
AU - Velázquez, J. J. L.
TI - Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1994
PB - Scuola normale superiore
VL - 21
IS - 4
SP - 595
EP - 628
LA - eng
KW - generalized solutions; singularities; fast and slow blow up
UR - http://eudml.org/doc/84193
ER -
References
top- [AlAG] S.J. Altschuler - S.B. Angenent - Y. Giga, Motion by Mean Curvature through singularities for surfaces of rotation. Preprint (1991).
- [AG] S.J. Altschuler - M.A. Grayson, Shortening space curves and flow through singularities. IMA Preprint823 (1991). MR1158337
- [A1] S.B. Angenent, Parabolic equations for curves on surfaces. Part. I, Ann. Math. 132 (1990), 451-483. Zbl0789.58070MR1078266
- [A2] S.B. Angenent, Parabolic equations for curves on surfaces. Part. II, Ann. Math. 133 (1991), 171-215. Zbl0749.58054MR1087347
- [A3] S.B. Angenent, On the formation of singularities in the curve shortening flow. J. Differential Geom.33 (1991), 601-633. Zbl0731.53002MR1100205
- [A4] S.B. Angenent, Some recent results on Mean Curvature Flow. In: Recent Advances in PDE's. M.A. Herrero - E. Zuazua eds., Research in Applied Math., Masson & J. Wiley, 1994, 1-18. Zbl0796.35068MR1266199
- [AV1] S.B. Angenent - J.J.L. Velázquez, Nonconvex collapse at rotationally symmetric hypersurfaces evolving by mean curvature flow. In preparation.
- [AV2] S.B. Angenent - J.J.L. Velázquez, Asymptotic shape of cusp singularities in curve shortening. Duke Math. J., in press. Zbl0829.35058MR1317628
- [BDGG] E. Bombieri - E. De Giorgi - E. Giusti, Minimal cones and the Bernstein problem. Inv. Math.7 (1969), 243-268. Zbl0183.25901MR250205
- [B] K.A. Brakke, The motion of a surface by its mean curvature. Princeton University Press, Math. Notes, Princeton, New Jersey, 1978. Zbl0386.53047MR485012
- [B1] A. Bressan, On the asymptotic shape of blow-up. Indiana Univ. Math. J.39 (1990), 947-960. Zbl0705.35014MR1087180
- [B2] A. Bressan, Stable blow-up patterns. J. Differential Equations.98 (1992), 57-75. Zbl0770.35010MR1168971
- [BQ] C.J. Budd - Y.W. Qi, The existence of bounded solutions of a semilinear heat equation. J. Differential Equations82 (1989), 207-218. Zbl0709.35039MR1027967
- [CGG] Y.G. Chen - Y. Giga - S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow. J. Differential Geom.33 (1991), 749-786. Zbl0696.35087MR1100211
- [DG] E. De Giorgi, Some conjectures on flow by mean curvature. Preprint.
- [ESS] L.C. Evans - H.M. Soner - P.E. Souganidis, Phase Transitions and generalized mean curvature flow equations. To appear in Comm. Pure Appl. Math. MR1177477
- [ES] L.C. Evans - J. Spruck, Motion of level sets by mean curvature I. J. Differential Geom.33 (1991), 635-681. Zbl0726.53029MR1100206
- [F] H. Federer, Geometric measure theory. Springer Verlag, New York, 1969. Zbl0176.00801MR257325
- [FK] S. Filippas - R.V. Kohn, Refined asymptotics for the blow-up of u t-Δu=up. Comm. Pure Appl. Math.45 (1992), 821-869. Zbl0784.35010
- [FL] S. Filippas - W. Liu, On the blow-up of a multidimensional semilinear heat equation. Ann. Inst. H. Poincaré Anal. Non Linéaire10 (1993), 313-344. Zbl0815.35039MR1230711
- [GH] M. Gage - R.S. Hamilton, The heat equation shrinking plane convex curves. J. Differential Geom.23 (1986), 69-96. Zbl0621.53001MR840401
- [GP] V.A. Galaktionov - S.A. Posashkov, The equation ut=uxx +uB. Localization, asymptotic behaviour of unbounded solutions, Preprint N. 97, M. Keldish, Inst. Appl. Math. (1985), (in russian). MR832277
- [GaP] A. Galindo - P. Pascual, Quantum Mechanics. Vol. I (1990) and vol. II (1991), Springer Verlag. Zbl0824.00008
- [G] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston, 1984. Zbl0545.49018MR775682
- [GK1] Y. Giga - R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math.38 (1985), 297-319. Zbl0585.35051MR784476
- [GK2] Y. Giga - R.V. Kohn, Characterizing blow-up using similarity variables. Indiana Univ. Math. J.36 (1987), 1-40. Zbl0601.35052MR876989
- [GK3] Y. Giga - R.V. Kohn, Nondegeneracy of blow-up for semilinear heat equations. Comm. Pure Appl. Math.42 (1987), 845-884. Zbl0703.35020MR1003437
- [Gr] M. Grayson, The heat equation shrinks embedded plane curves to round points. J. Differential Geom.26 (1987), 285-314. Zbl0667.53001MR906392
- [HV1] M.A. Herrero - J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. Henri Poincaré10 (1993), 131-189. Zbl0813.35007MR1220032
- [HV2] M.A. Herrero - J.J.L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations. Differential Integral Equations5 (1992), 973-998. Zbl0767.35036MR1171974
- [HV3] M.A. Herrero - J.J.L. Velázquez, Blow-up profiles in one-dimensional semilinear parabolic problems. Comm. Partial Differential Equations17 (1992), 205-219. Zbl0772.35027MR1151261
- [HV4] M.A. Herrero - J.J.L. Velázquez, Generic behaviour of one-dimensional blow-up patterns. Ann. Scuola Norm. Sup. Pisa, Cl. Sci.19 (1992), 381-450. Zbl0798.35081MR1205406
- [HV5] M.A. Herrero - J.J.L. Velázquez, Generic behaviour near blow-up points for a N-dimensional semilinear heat equation. In preparation.
- [HV6] M.A. Herrero - J.J.L. Velázquez, A blow-up result for semilinear heat equations in the supercritical case. To appear.
- [HV7] M.A. Herrero - J.J.L. Velázquez, Explosion de solutions d'equations paraboliques semilinéaires supercritiques. C.R. Acad. Sci. Paris319 (1994), 141-145. Zbl0806.35005MR1288393
- [H] G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow. J. Differential Geom.31 (1991), 285-299. Zbl0694.53005MR1030675
- [LSU] O.A. Ladyshenskaya - N.N. Solonnikov - V.A. Uralceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical monographs, A.M.S. vol. 23, 1988. Zbl0174.15403
- [L] L.A. Lepin, Countable spectrum at eigenfunctions of a nonlinear heat conduction equation with distributed parameters. Differentsial'nye Uravneiya24, 7 (1988), 1226-1234. Zbl0675.35054MR958417
- [RS] M. Reed - B. Simon, Functional analysis, vol. II, Academic Press (1980). MR751959
- [S] J. Simons, Minimal varieties in riemannian manifolds. Ann. of Math.88 (1968), 62-106. Zbl0181.49702MR233295
- [So] H.M. Soner, Motion of a set by the curvature of its boundary. J. Differential Equations101 (1993), 313-372. Zbl0769.35070MR1204331
- [T] W.C. Troy, The existence of bounded solutions of a semilinear heat equation. SIAM J. Math. Anal. vol. 18, 2 (1987), 332-336. Zbl0655.35039MR876275
- [V1] J.J.L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions. Trans. Amer. Math. Soc.338 (1993), 441-464. Zbl0803.35015MR1134760
- [V2] J.J.L. Velázquez, Higher dimensional blow-up for semilinear parabolic equations. Comm. Partial Differential Equations17 (1992), 1567-1596. Zbl0813.35009MR1187622
- [V3] J.J.L. Velázquez, Estimates on the (N-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J.42 (1993), 446-476. Zbl0802.35073MR1237055
- [V4] J.J.L. Velázquez, Blow-up for semilinear parabolic equations. In Recent Advances in PDE's, M.A. Herrero - E. Zuazua eds., Research in Applied Math., Masson & J. Wiley, 1994, 131-145. Zbl0798.35072MR1266206
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