Thermal capacity estimates on the Allen-Cahn equation

Sowers, Richard; Wu, Jang-Mei

doi:10.1090/S0002-9947-99-02399-5

Thermal capacity estimates on the Allen-Cahn equation
HTML articles powered by AMS MathViewer

by Richard B. Sowers and Jang-Mei Wu

Trans. Amer. Math. Soc. 351 (1999), 2553-2567

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

Published electronically: February 9, 1999

PDF | Request permission

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

R. Bañuelos and B. Øksendal, Exit times for elliptic diffusions and BMO, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 2, 273–287. MR 892696, DOI 10.1017/S0013091500028339
G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), no. 2, 439–469. MR 1205984, DOI 10.1137/0331021
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 207–210. MR 1030181
Kai Lai Chung, Lectures from Markov processes to Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer-Verlag, New York-Berlin, 1982. MR 648601, DOI 10.1007/978-1-4757-1776-1
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206, DOI 10.4310/jdg/1214446559
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc. 330 (1992), no. 1, 321–332. MR 1068927, DOI 10.1090/S0002-9947-1992-1068927-8
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal. 2 (1992), no. 2, 121–150. MR 1151756, DOI 10.1007/BF02921385
Lawrence C. Evans and Joel Spruck, Motion of level sets by mean curvature. IV, J. Geom. Anal. 5 (1995), no. 1, 77–114. MR 1315658, DOI 10.1007/BF02926443
L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097–1123. MR 1177477, DOI 10.1002/cpa.3160450903
W. K. Hayman and Ch. Pommerenke, On analytic functions of bounded mean oscillation, Bull. London Math. Soc. 10 (1978), no. 2, 219–224. MR 500932, DOI 10.1112/blms/10.2.219
G. A. Hunt, Markoff processes and potentials. III, Illinois J. Math. 2 (1958), 151–213. MR 107097
Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. MR 1237490
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Carl Mueller, A characterization of BMO and BMO$_{\rho }$, Studia Math. 72 (1982), no. 1, 47–57. MR 665891, DOI 10.4064/sm-72-1-47-57
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1991. MR 1083357, DOI 10.1007/978-3-662-21726-9
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: convergence, (to appear), Journal of Geometric Analysis.
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, II: development of the interface, (to appear), Journal of Geometric Analysis.
David A. Stegenga, A geometric condition which implies BMOA, Michigan Math. J. 27 (1980), no. 2, 247–252. MR 568645
Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
N. A. Watson, Thermal capacity, Proc. London Math. Soc. (3) 37 (1978), no. 2, 342–362. MR 507610, DOI 10.1112/plms/s3-37.2.342