Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Lipschitz property of the free boundary in the parabolic obstacle problem


Authors: D. E. Apushkinskaya, N. N. Ural'tseva and H. Shahgholian
Translated by: D. E. Apushkinskaya
Original publication: Algebra i Analiz, tom 15 (2003), nomer 3.
Journal: St. Petersburg Math. J. 15 (2004), 375-391
MSC (2000): Primary 35R35
DOI: https://doi.org/10.1090/S1061-0022-04-00813-1
Published electronically: March 25, 2004
MathSciNet review: 2052937
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A parabolic obstacle problem with zero constraint is considered. It is proved, without any additional assumptions on a free boundary, that near the fixed boundary where the homogeneous Dirichlet condition is fulfilled, the boundary of the noncoincidence set is the graph of a Lipschitz function.


References [Enhancements On Off] (What's this?)

  • [AUS1] D. E. Apushkinskaya, N. N. Ural'tseva, and H. Shahgholian, Boundary estimates for solutions to the parabolic free boundary problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 39-55. (English) MR 2002b:35205
  • [AUS2] -, On the global solutions of the parabolic obstacle problem, Algebra i Analiz 14 (2002), no. 1, 3-25; English transl., St. Petersburg Math. J. 14 (2003), no. 1, 1-17. MR 2003d:35163
  • [AtSa] I. Athanasopoulos and S. Salsa, An application of a parabolic comparison principle to free boundary problems, Indiana Univ. Math. J. 40 (1991), no. 1, 29-32. MR 92d:35306
  • [C1] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155-184. MR 56:12601
  • [C2] -, A monotonicity formula for heat functions in disjoint domains, Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 53-60. MR 95e:35096
  • [CK] L. A. Caffarelli and C. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math. 120 (1998), no. 2, 391-440. MR 99b:35081
  • [CPS] L. A. Caffarelli, A. Petrosyan, and H. Shahgholian, Regularity of a free boundary in parabolic potential theory (in preparation).
  • [W] G. S. Weiss, Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems, SIAM J. Math. Anal. 30 (1999), no. 3, 623-644. MR 2000d:35267

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 35R35

Retrieve articles in all journals with MSC (2000): 35R35


Additional Information

D. E. Apushkinskaya
Affiliation: Saarland University, Saarbrücken, Germany
Email: darya@math.uni-sb.de

N. N. Ural'tseva
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: uunur@nur.usr.pu.ru

H. Shahgholian
Affiliation: Royal Institute of Technology, Stockholm, Sweden
Email: henriks@math.kth.se

DOI: https://doi.org/10.1090/S1061-0022-04-00813-1
Keywords: Free boundary problems, regularity, parabolic variational inequality
Received by editor(s): February 18, 2003
Published electronically: March 25, 2004
Additional Notes: D. E. Apushkinskaya and N. N. Ural′tseva were partially supported by the Russian Foundation for Basic Research (grant no. 02-01-00276). H. Shahgholian was partially supported by the Swedish Natural Sciences Research Council.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society