The anisotropic Cahn–Hilliard equation: Regularity theory and strict separation properties

Harald Garcke; Patrik Knopf; Julia Wittmann

doi:10.3934/dcdss.2023146

Article Contents

2023, Volume 16, Issue 12: 3622-3660. Doi: 10.3934/dcdss.2023146

This issue Previous Article On the viscous Cahn–Hilliard system with two mobilities Next Article On some doubly-nonlinear parabolic equations posed in $ \mathbb{R}^{{d}} $

The anisotropic Cahn–Hilliard equation: Regularity theory and strict separation properties

1.
Faculty for Mathematics, University of Regensburg, Universitätsstr. 31, 93053 Regensburg, Germany

^*Corresponding author: Patrik Knopf
^*Corresponding author: Patrik Knopf

Received: May 1, 2023

Revised: July 1, 2023

Early access: August 2, 2023

Published online: December 1, 2023

The authors are partially supported by the RTG 2339 "Interfaces, Complex Structures, and Singular Limits" of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

The Cahn–Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn–Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35K61; Secondary: 74E10, 35Q99.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	M. Alfaro, H. Garcke, D. Hilhorst, H. Matano and R. Schätzle, Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen–Cahn equation, Proc. R. Soc. Edinb. A, 140 (2010), 673-706. doi: 10.1017/S0308210508000541.
[2]	H. W. Alt, Linear Functional Analysis - An Application-Oriented Introduction, Springer, London, 2016. doi: 10.1007/978-1-4471-7280-2.
[3]	L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives, Proc. Amer. Math. Soc., 108 (1990), 691-702. doi: 10.1090/S0002-9939-1990-0969514-3.
[4]	J. W. Barrett, H. Garcke and R. Nürnberg, On the stable discretization of strongly anisotropic phase field models with applications to crystal growth, ZAMM Z. Angew. Math. Mech., 93 (2013), 719-732. doi: 10.1002/zamm.201200147.
[5]	J. W. Barrett, H. Garcke and R. Nürnberg, Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., 34 (2014), 1289-1327. doi: 10.1093/imanum/drt044.
[6]	A. C. Barroso and I. Fonseca, Anisotropic singular perturbations–-the vectorial case, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 527-571. doi: 10.1017/S0308210500028778.
[7]	G. Bellettini, A. Braides and G. Riey, Variational approximation of anisotropic functionals on partitions, Ann. Mat. Pura Appl. (4), 184 (2005), 75-93. doi: 10.1007/s10231-003-0090-4.
[8]	G. Bouchitté, Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. Optim., 21 (1990), 289-314. doi: 10.1007/BF01445167.
[9]	L. Cherfils, A. Miranville and S. Peng, Higher-order anisotropic models in phase separation, Adv. Nonlinear Anal., 8 (2019), 278-302. doi: 10.1515/anona-2016-0137.
[10]	M. Dziwnik, A. Münch and B. Wagner, An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit, Nonlinearity, 30 (2017), 1465-1496. doi: 10.1088/1361-6544/aa5e5d.
[11]	C. M. Elliott and R. Schätzle, The limit of the anisotropic double-obstacle Allen–Cahn equation, Proc. R. Soc. Edinb. A, 126 (1996), 1217-1234. doi: 10.1017/S0308210500023374.
[12]	L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, second edition, 2010. doi: 10.1090/gsm/019.
[13]	P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, (2000), pages No. 48, 26 pp.
[14]	C. G. Gal, A. Giorgini and M. Grasselli, The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases, Discrete Contin. Dyn. Syst., 43 (2023), 2270-2304. doi: 10.3934/dcds.2023010.
[15]	H. Garcke, On Cahn-Hilliard systems with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331. doi: 10.1017/S0308210500002419.
[16]	H. Garcke, Curvature driven interface evolution, Jahresber. Dtsch. Math.-Ver., 115 (2013), 63-100. doi: 10.1365/s13291-013-0066-2.
[17]	H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach, SIAM J. Math. Anal., 52 (2020), 340-369. doi: 10.1137/19M1258840.
[18]	H. Garcke, P. Knopf, R. Nürnberg and Q. Zhao, A diffuse-interface approach for solid-state dewetting with anisotropic surface energies, J. Nonlinear Sci., 33 (2023), Paper No. 34, 56 pp. doi: 10.1007/s00332-023-09889-y.
[19]	H. Garcke, K. F. Lam, R. Nürnberg and A. Signori, Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies, Appl. Math. Optim., 87 (2023), Paper No. 44, 50 pp. doi: 10.1007/s00245-022-09939-z.
[20]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Berlin, Heidelberg, 2001.
[21]	C. Gräser, R. Kornhuber and U. Sack, Time discretizations of anisotropic Allen–Cahn equations, IMA J. Numer. Anal., 33 (2013), 1226-1244. doi: 10.1093/imanum/drs043.
[22]	T. Laux, K. Stinson and C. Ullrich, Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow, Preprint: arXiv: 2212.11939 [math.AP], 2022.
[23]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[24]	A. Miranville, The Cahn-Hilliard Equation, volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. Recent advances and applications. doi: 10.1137/1.9781611975925.
[25]	N. C. Owen and P. Sternberg, Nonconvex variational problems with anisotropic perturbations, Nonlinear Anal., 16 (1991), 705-719. doi: 10.1016/0362-546X(91)90177-3.
[26]	A. Rätz, A. Ribalta and A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model, J. Comput. Phys., 214 (2006), 187-208. doi: 10.1016/j.jcp.2005.09.013.
[27]	J. E. Taylor and J. W. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197. doi: 10.1007/BF02186838.
[28]	S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Secr. A Math. Phys. Eng. Sci., 465 (2009), 1337-1359. doi: 10.1098/rspa.2008.0385.
[29]	N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. doi: 10.1512/iumj.1968.17.17028.