Local minimizers and slow motion for the mass preserving Allen–Cahn equation in higher dimensions

Leoni, Giovanni; Murray, Ryan

doi:10.1090/proc/13988

Local minimizers and slow motion for the mass preserving Allen–Cahn equation in higher dimensions
HTML articles powered by AMS MathViewer

by Giovanni Leoni and Ryan Murray PDF

Proc. Amer. Math. Soc. 147 (2019), 5167-5182 Request permission

Abstract:

This paper completely resolves the asymptotic development of order $2$ by $\Gamma$-convergence of the mass-constrained Cahn–Hilliard functional. Important new results on the slow motion of interfaces for the mass preserving Allen–Cahn equation and the Cahn–Hilliard equations in higher dimension are obtained as an application.

References

E. Acerbi, N. Fusco, and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013), no. 2, 515–557. MR 3077924, DOI 10.1007/s00220-013-1733-y
Nicholas D. Alikakos and Giorgio Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. I. Spectral estimates, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1397–1447. MR 1294466, DOI 10.1080/03605309408821059
S. M Allen and J. W Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), no. 6, 1085–1095.
Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
Gabriele Anzellotti and Sisto Baldo, Asymptotic development by $\Gamma$-convergence, Appl. Math. Optim. 27 (1993), no. 2, 105–123. MR 1202527, DOI 10.1007/BF01195977
Lia Bronsard and Robert V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), no. 8, 983–997. MR 1075075, DOI 10.1002/cpa.3160430804
Jack Carr, Morton E. Gurtin, and Marshall Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984), no. 4, 317–351. MR 759767, DOI 10.1007/BF00280031
J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon ^2u_{xx}-f(u)$, Comm. Pure Appl. Math. 42 (1989), no. 5, 523–576. MR 997567, DOI 10.1002/cpa.3160420502
L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Materials Research 32 (2002), no. 1, 113–140.
Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
Gianni Dal Maso, Irene Fonseca, and Giovanni Leoni, Second order asymptotic development for the anisotropic Cahn-Hilliard functional, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 1119–1145. MR 3385194, DOI 10.1007/s00526-015-0819-0
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), no. 1, 75–94. MR 1010961, DOI 10.1007/BF01048791
N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), no. 3, 941–980. MR 2456887, DOI 10.4007/annals.2008.168.941
E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753, DOI 10.1512/iumj.1983.32.32003
Christopher P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal. 26 (1995), no. 1, 21–34. MR 1311880, DOI 10.1137/S0036141092226053
Michael Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261–270. MR 862549, DOI 10.1007/BF00250810
Morton E. Gurtin and Hiroshi Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), no. 2, 301–317. MR 950604, DOI 10.1090/S0033-569X-1988-0950604-5
Vesa Julin and Giovanni Pisante, Minimality via second variation for microphase separation of diblock copolymer melts, J. Reine Angew. Math. 729 (2017), 81–117. MR 3680371, DOI 10.1515/crelle-2014-0117
Steven G. Krantz and Harold R. Parks, Distance to $C^{k}$ hypersurfaces, J. Differential Equations 40 (1981), no. 1, 116–120. MR 614221, DOI 10.1016/0022-0396(81)90013-9
Giovanni Leoni and Ryan Murray, Second-order $\Gamma$-limit for the Cahn-Hilliard functional, Arch. Ration. Mech. Anal. 219 (2016), no. 3, 1383–1451. MR 3448931, DOI 10.1007/s00205-015-0924-4
Francesco Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. MR 2976521, DOI 10.1017/CBO9781139108133
V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
Luciano Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142. MR 866718, DOI 10.1007/BF00251230
R. Murray, Some asymptotic results for phase transition models, Ph.D. Thesis, Carnegie Mellon University, 2016.
Ryan Murray and Matteo Rinaldi, Slow motion for the nonlocal Allen-Cahn equation in $n$ dimensions, Calc. Var. Partial Differential Equations 55 (2016), no. 6, Art. 147, 33. MR 3568052, DOI 10.1007/s00526-016-1086-4
Felix Otto and Maria G. Reznikoff, Slow motion of gradient flows, J. Differential Equations 237 (2007), no. 2, 372–420. MR 2330952, DOI 10.1016/j.jde.2007.03.007
J. S. Rowlinson, Translation of J. D. van der Waals’ “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”, J. Statist. Phys. 20 (1979), no. 2, 197–244. MR 523642, DOI 10.1007/BF01011513
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Peter Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), no. 3, 209–260. MR 930124, DOI 10.1007/BF00253122
Peter Sternberg and Kevin Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199–220. MR 1674097, DOI 10.4310/CAG.1999.v7.n1.a7
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3