Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-99-02399-5
Published electronically: February 9, 1999
MathSciNet review: 1624210
Full-text PDF

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.


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

  • [BanOks] R. Bañuelos and B. Øksendal, Exit times for elliptic diffusions and BMO, Proceedings of the Edinburgh Mathematical Society 30 (1987), 273-287. MR 88m:58199
  • [BSS] G. Barles, H.M. Soner, and P.E. Souganidis, Front propagation and phase field theory, SIAM J. Control and Optimization 31 (1993), 439-469. MR 94c:35005
  • [CGG] Y.-G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci 65 (1989), 207-210. MR 91b:35049
  • [Chung] K. L. Chung, Lectures from Markov Processes to Brownian Motion, Springer-Verlag, New York, 1982. MR 84c:60091
  • [ESI] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geometry 33 (1991), 635-681. MR 92h:35097
  • [ESII] L. C. Evans and J. Spruck, Motion of level sets by mean curvature II, Trans. Amer. Math Soc. 330 (1992), 321-332. MR 92f:58050
  • [ESIII] L. C. Evans and J. Spruck, Motion of level sets by mean curvature III, Journal of Geometric Analysis 2 (1992), 121-150. MR 93d:58044
  • [ESIV] L. C. Evans and J. Spruck, Motion of level sets by mean curvature IV, Journal of Geometric Analysis 5 (1995), 77-113. MR 96a:35077
  • [ESS] L.C. Evans, H.M. Soner, and P.E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math 45 (1992), 1097-1123. MR 93g:35064
  • [HayPomm] W. K. Hayman and Ch. Pommerenke, On analytic functions of bounded mean oscillation, Bulletin of the London Mathematical Society 10 (1978), 219-224. MR 81g:30044
  • [Hunt] G. A. Hunt, Markoff processes and potentials. III, Illinois J. Math. 2 (1958), 151-213. MR 21:5824
  • [Illmanen] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom. 38 (1993), 417-461. MR 94h:58051
  • [KS] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus (2nd ed.), Springer, New York, 1991. MR 92h:60127
  • [Mueller] C. Mueller, A characterization of BMO and BMO$_{\rho}$, Studia Mathematica 72 (1982), 47-57. MR 84j:42032
  • [RY] D. Revuz, and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, New York, 1991. MR 92d:60053
  • [Sonera] H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: convergence, (to appear), Journal of Geometric Analysis.
  • [Sonerb] H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, II: development of the interface, (to appear), Journal of Geometric Analysis.
  • [Stegenga] D. A. Stegenga, A geometric condition which implies BMOA, Michigan Mathematical Journal 27 (1980), 247-252. MR 81h:30039
  • [Stroock] D. W. Stroock, Probability Theory, An Analytic View, Cambridge University Press, New York, 1993. MR 95f:60003
  • [Watson] N. A. Watson, Thermal capacity, Proc. London Mathematical Society (3) 37 (1978), 342-362. MR 80b:31005

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 31B35, 35K57, 60J45

Retrieve articles in all journals with MSC (1991): 31B35, 35K57, 60J45


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

American Mathematical Society