Regularity of a free boundary in parabolic potential theory

Authors:
Luis Caffarelli, Arshak Petrosyan and Henrik Shahgholian

Journal:
J. Amer. Math. Soc. **17** (2004), 827-869

MSC (2000):
Primary 35R35

Published electronically:
August 27, 2004

MathSciNet review:
2083469

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the regularity of the free boundary in a Stefan-type problem

with no sign assumptions on and the time derivative .

**[ACF84]**Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,*Variational problems with two phases and their free boundaries*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 431–461. MR**732100**, 10.1090/S0002-9947-1984-0732100-6**[ACS96]**I. Athanasopoulos, L. Caffarelli, and S. Salsa,*Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems*, Ann. of Math. (2)**143**(1996), no. 3, 413–434. MR**1394964**, 10.2307/2118531**[Caf93]**Luis A. Caffarelli,*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**1260438****[Caf98]**L. A. Caffarelli,*The obstacle problem revisited*, J. Fourier Anal. Appl.**4**(1998), no. 4-5, 383–402. MR**1658612**, 10.1007/BF02498216**[CK98]**Luis A. Caffarelli and Carlos E. Kenig,*Gradient estimates for variable coefficient parabolic equations and singular perturbation problems*, Amer. J. Math.**120**(1998), no. 2, 391–439. MR**1613650****[CKS00]**Luis A. Caffarelli, Lavi Karp, and Henrik Shahgholian,*Regularity of a free boundary with application to the Pompeiu problem*, Ann. of Math. (2)**151**(2000), no. 1, 269–292. MR**1745013**, 10.2307/121117**[Fri64]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[Fri88]**Avner Friedman,*Variational principles and free-boundary problems*, 2nd ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988. MR**1009785****[KN77]**D. Kinderlehrer and L. Nirenberg,*Regularity in free boundary problems*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**4**(1977), no. 2, 373–391. MR**0440187****[KN78]**David Kinderlehrer and Louis Nirenberg,*The smoothness of the free boundary in the one phase Stefan problem*, Comm. Pure Appl. Math.**31**(1978), no. 3, 257–282. MR**480348**, 10.1002/cpa.3160310302**[Lie96]**Gary M. Lieberman,*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184****[Wei99]**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 (electronic). MR**1677947**, 10.1137/S0036141097327409

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
35R35

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

Additional Information

**Luis Caffarelli**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Email:
caffarel@math.utexas.edu

**Arshak Petrosyan**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Address at time of publication:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
arshak@math.utexas.edu, arshak@math.purdue.edu

**Henrik Shahgholian**

Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

Email:
henriksh@math.kth.se

DOI:
http://dx.doi.org/10.1090/S0894-0347-04-00466-7

Keywords:
Free boundary problems,
Stefan problem,
regularity,
global solutions,
monotonicity formulas.

Received by editor(s):
December 20, 2002

Published electronically:
August 27, 2004

Additional Notes:
The first author was supported in part by the NSF

The second author thanks the Göran Gustafsson Foundation and the Department of Mathematics, Royal Institute of Technology, for the visiting appointment

The third author was supported in part by the Swedish Research Council

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.