In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer of the total energy. Following the approach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Preprint], we obtain these results by means of a minimality criterion based on the positivity of the second variation.
Classification : 74G55, 74K35, 74G65, 49Q20, 49K10
Mots clés : epitaxially strained crystalline films, anisotropic surface energy, second order minimality conditions, second variation
@article{COCV_2013__19_1_167_0,
author = {Bonacini, Marco},
title = {Epitaxially strained elastic films: the case of anisotropic surface energies},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {167--189},
publisher = {EDP-Sciences},
volume = {19},
number = {1},
year = {2013},
doi = {10.1051/cocv/2012003},
mrnumber = {3023065},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012003/}
}
TY - JOUR AU - Bonacini, Marco TI - Epitaxially strained elastic films: the case of anisotropic surface energies JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 DA - 2013/// SP - 167 EP - 189 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012003/ UR - https://www.ams.org/mathscinet-getitem?mr=3023065 UR - https://doi.org/10.1051/cocv/2012003 DO - 10.1051/cocv/2012003 LA - en ID - COCV_2013__19_1_167_0 ER -
Bonacini, Marco. Epitaxially strained elastic films: the case of anisotropic surface energies. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 167-189. doi : 10.1051/cocv/2012003. http://www.numdam.org/articles/10.1051/cocv/2012003/
[1] , and , Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000). | MR 1857292 | Zbl 0957.49001
[2] and , Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. | MR 1898515 | Zbl 1001.49017
[3] , and , A relaxation result for energies defined on pairs set-function and applications. ESAIM : COCV 13 (2007) 717-734. | Numdam | MR 2351400 | Zbl 1149.49017
[4] , and , A second order minimality condition for the Mumford-Shah functional. Calc. Var. Partial Differential Equations 33 (2008) 37-74. | MR 2413101 | Zbl 1191.49018
[5] and , C∞ regularity of the free boundary for a two-dimensional optimal compliance problem. Calc. Var. Partial Differential Equations 18 (2003) 77-94. | MR 2001883 | Zbl 1026.49031
[6] and , Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal. 39 (2007) 77-102. | MR 2318376
[7] and , Regularity properties of equilibrium configurations of epitaxially strained elastic films. Submitted paper (2011)
[8] , The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991) 125-145. | MR 1116536 | Zbl 0725.49017
[9] and , A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh 119A (1991) 125-136. | MR 1130601 | Zbl 0752.49019
[10] , , and , Equilibrium configurations of epitaxially strained crystalline films : existence and regularity results. Arch. Rational Mech. Anal. 186 (2007) 477-537. | MR 2350364 | Zbl 1126.74029
[11] , , and , Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011) 591-639. | MR 2851683 | Zbl 1285.74003
[12] and , Equilibrium configurations of epitaxially strained elastic films : second order minimality conditions and qualitative properties of solutions. Arch. Rational Mech. Anal. 203 (2012) 247-327. | MR 2864412 | Zbl 1281.74024
[13] , A generalization of Goła¸b's theorem and applications to fracture mechanics. Math. Models Methods Appl. Sci. 12 (2002) 1245-1267. | MR 1927024 | Zbl 1092.74041
[14] , The stress driven instability in elastic crystals : mathematical models and physical manifestations. J. Nonlinear Sci. 3 (1993) 35-83. | MR 1216987 | Zbl 0843.73040
[15] , and , On optimal regularity of free boundary problems and a conjecture of De Giorgi. Comm. Pure Appl. Math. 58 (2005) 1051-1076. | MR 2143526 | Zbl 1082.35168
[16] , Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978) 568-588. | MR 493671 | Zbl 0392.49022
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