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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Thermal capacity estimates on
the Allen-Cahn equation


Authors: Richard B. Sowers and Jang-Mei Wu
Journal: Trans. Amer. Math. Soc. 351 (1999), 2553-2567
MSC (1991): Primary 31B35, 35K57, 60J45
Published electronically: February 9, 1999
MathSciNet review: 1624210
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Allen-Cahn equation in a well-known scaling regime which gives motion by mean curvature. A well-known transformation of this PDE, using its standing wave, yields a PDE the solution of which is approximately the distance function to an interface moving by mean curvature. We give bounds on this last fact in terms of thermal capacity. Our techniques hinge upon the analysis of a certain semimartingale associated with a certain PDE (the PDE for the approximate distance function) and an analogue of some results by Bañuelos and Øksendal relating lifetimes of diffusions to exterior capacities.


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Additional Information

Richard B. Sowers
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2917
Email: r-sowers@math.uiuc.edu

Jang-Mei Wu
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2917
Email: wu@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02399-5
PII: S 0002-9947(99)02399-5
Keywords: Allen-Cahn equation, mean curvature, thermal capacity
Received by editor(s): October 20, 1997
Received by editor(s) in revised form: April 28, 1998
Published electronically: February 9, 1999
Additional Notes: The work of R. S. was supported by NSF Grants DMS 96-26398 and DMS 96-15877 and the Research Board of the University of Illinois at Urbana-Champaign.
The work of J.-M. W. was supported by NSF Grant DMS 97-05227.
Article copyright: © Copyright 1999 American Mathematical Society