Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T02:24:42.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Epitaxially strained elastic films: the case of anisotropicsurface energies

Published online by Cambridge University Press:  01 March 2012

Marco Bonacini
Marco Bonacini*
Affiliation:
SISSA, Via Bonomea 265, 34136 Trieste, Italy. [email protected]
Get access

Abstract

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 determinationof equilibrium configurations. We show that the threshold effect that describes thestability 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 showthat the flat configuration is always a local minimizer of the total energy. Following theapproach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strainedelastic films: second order minimality conditions and qualitative properties of solutions.Preprint], we obtain these results by means of a minimality criterion based on thepositivity of the second variation.

Keywords

Epitaxially strained crystalline filmsanisotropic surface energysecond order minimality conditionssecond variation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 167 - 189
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000).
Bonnetier, E. and Chambolle, A., Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093–1121. Google Scholar
Braides, A., Chambolle, A. and Solci, M., A relaxation result for energies defined on pairs set-function and applications. ESAIM : COCV 13 (2007) 717–734. Google Scholar
Cagnetti, F., Mora, M.G. and Morini, M., A second order minimality condition for the Mumford-Shah functional. Calc. Var. Partial Differential Equations 33 (2008) 37–74. Google Scholar
Chambolle, A. and Larsen, C.J., C ∞ regularity of the free boundary for a two-dimensional optimal compliance problem. Calc. Var. Partial Differential Equations 18 (2003) 77–94. Google Scholar
Chambolle, A. and Solci, M., Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal. 39 (2007) 77–102. Google Scholar
B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films. Submitted paper (2011)
Fonseca, I., The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991) 125–145. Google Scholar
Fonseca, I. and Müller, S., A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh 119A (1991) 125–136. Google Scholar
Fonseca, I., Fusco, N., Leoni, G. and Morini, M., Equilibrium configurations of epitaxially strained crystalline films : existence and regularity results. Arch. Rational Mech. Anal. 186 (2007) 477–537. Google Scholar
Fonseca, I., Fusco, N., Leoni, G. and Millot, V., Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011) 591–639. Google Scholar
Fusco, N. and Morini, M., Equilibrium configurations of epitaxially strained elastic films : second order minimality conditions and qualitative properties of solutions. Arch. Rational Mech. Anal. 203 (2012) 247–327. Google Scholar
Giacomini, A., A generalization of Goła¸b’s theorem and applications to fracture mechanics. Math. Models Methods Appl. Sci. 12 (2002) 1245–1267. Google Scholar
Grinfeld, M.A., The stress driven instability in elastic crystals : mathematical models and physical manifestations. J. Nonlinear Sci. 3 (1993) 35–83. Google Scholar
Koch, H., Leoni, G. and Morini, M., On optimal regularity of free boundary problems and a conjecture of De Giorgi. Comm. Pure Appl. Math. 58 (2005) 1051–1076. Google Scholar
Taylor, J., Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978) 568–588. Google Scholar