no. 1
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.

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.

DOI : 10.1051/cocv/2012003
Classification : 74G55, 74K35, 74G65, 49Q20, 49K10
Mots clés : epitaxially strained crystalline films, anisotropic surface energy, second order minimality conditions, second variation 
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     title = {Epitaxially strained elastic films: the case of anisotropic surface energies},
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     pages = {167--189},
     publisher = {EDP-Sciences},
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     year = {2013},
     doi = {10.1051/cocv/2012003},
     mrnumber = {3023065},
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     url = {http://archive.numdam.org/articles/10.1051/cocv/2012003/}
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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://archive.numdam.org/articles/10.1051/cocv/2012003/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000). | MR | Zbl

[2] E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. | MR | Zbl

[3] A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications. ESAIM : COCV 13 (2007) 717-734. | Numdam | MR | Zbl

[4] F. Cagnetti, M.G. Mora and M. Morini, A second order minimality condition for the Mumford-Shah functional. Calc. Var. Partial Differential Equations 33 (2008) 37-74. | MR | Zbl

[5] A. Chambolle and C.J. Larsen, C∞ regularity of the free boundary for a two-dimensional optimal compliance problem. Calc. Var. Partial Differential Equations 18 (2003) 77-94. | MR | Zbl

[6] A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal. 39 (2007) 77-102. | MR

[7] B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films. Submitted paper (2011)

[8] I. Fonseca, The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991) 125-145. | MR | Zbl

[9] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh 119A (1991) 125-136. | MR | Zbl

[10] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films : existence and regularity results. Arch. Rational Mech. Anal. 186 (2007) 477-537. | MR | Zbl

[11] I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011) 591-639. | MR | Zbl

[12] N. Fusco and M. Morini, 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 | Zbl

[13] A. Giacomini, A generalization of Goła¸b's theorem and applications to fracture mechanics. Math. Models Methods Appl. Sci. 12 (2002) 1245-1267. | MR | Zbl

[14] M.A. Grinfeld, The stress driven instability in elastic crystals : mathematical models and physical manifestations. J. Nonlinear Sci. 3 (1993) 35-83. | MR | Zbl

[15] H. Koch, G. Leoni and M. Morini, On optimal regularity of free boundary problems and a conjecture of De Giorgi. Comm. Pure Appl. Math. 58 (2005) 1051-1076. | MR | Zbl

[16] J. Taylor, Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978) 568-588. | MR | Zbl

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