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

DOI:
https://doi.org/10.1090/S0002-9947-99-02399-5

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:
https://doi.org/10.1090/S0002-9947-99-02399-5

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