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Archive for Rational Mechanics and Analysis

, Volume 230, Issue 3, pp 785–838 | Cite as

Equilibria Configurations for Epitaxial Crystal Growth with Adatoms

  • Marco Caroccia
  • Riccardo Cristoferi
  • Laurent Dietrich
Article

Abstract

The behavior of a surface energy \({\mathcal{F}(E,u)}\), where E is a set of finite perimeter and \({u\in L^1(\partial^{*} E, \mathbb{R}_+)}\), is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms, the freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process. Regular critical points, the existence and uniqueness of minimizers are discussed and the relaxation of \({\mathcal{F}}\) in a general setting under the L1 convergence of sets and the vague convergence of measures is characterized. This is part of an ongoing project aimed at an analytical study of diffuse interface approximations of the associated evolution equations.

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References

  1. 1.
    Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. I. Am. Math. Soc. Transl. 2(21), 341–354 (1962)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)zbMATHGoogle Scholar
  3. 3.
    Bonacini, M.: Epitaxially strained elastic films: the case of anisotropic surface energies. ESAIM Control Optim. Calc. Var. 19(1), 167–189 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonacini, M.: Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var. 8(2), 117–153 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bonnetier, E., Chambolle, A.: Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62(4), 1093–1121 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Braides, A.: \(\Gamma \)-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002)Google Scholar
  7. 7.
    Burger, M.: Surface diffusion including adatoms. Commun. Math. Sci. 4(1), 1–51 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buttazzo, G., Freddi, L.: Functionals defined on measures and applications to non equi-uniformly elliptic variational problems. Ann. Mat. Pura Appl. 159, 133–149 (1992)CrossRefzbMATHGoogle Scholar
  9. 9.
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Research Notes in Mathematics Series, vol. 207. Longman Scientific & Technical, Harlow; copublished in the United States with Wiley, New York, 1989Google Scholar
  10. 10.
    Capriani, G.M., Julin, V., Pisante, G.: A quantitative second order minimality criterion for cavities in elastic bodies. SIAM J. Math. Anal. 45(3), 1952–1991 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dal Maso, G.: An introduction to \(\Gamma \)-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc, Boston, MA (1993)Google Scholar
  12. 12.
    De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58(6), 842–850 (1975)Google Scholar
  13. 13.
    De Lellis, C.: Rectifiable sets, densities and tangent measures. European Mathematical Society (EMS), Zürich, Zurich Lectures in Advanced Mathematics (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Falconer, K.J.: Dimensions of intersections and distance sets for polyhedral norms. Real Anal. Exchange, 30(2), 719–726 (2004/05)Google Scholar
  15. 15.
    Fonseca, I.: Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh Sect. A 120(1–2), 99–115 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fonseca, I., Fusco, N., Leoni, G., Millot, V.: Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. (9), 96(6), 591–639 (2011)Google Scholar
  17. 17.
    Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal. 186(3), 477–537 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ration. Mech. Anal. 205(2), 425–466 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fried, E., Gurtin, M.E.: A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain- boundaries and epitaxy. Adv. Appl. Mech. 40, 1–177 (2004)CrossRefGoogle Scholar
  20. 20.
    Fusco, N., Morini, M.: Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal. 203(1), 247–327 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, second edn. Springer, Berlin, 1983Google Scholar
  22. 22.
    Maggi, F.: Sets of finite perimeter and geometric variational problems, An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge, 2012Google Scholar
  23. 23.
    Mattila, P.: On the Hausdorff dimension and capacities of intersections. Mathematika 32, 213–217 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pollard, D.: A user's guide to measure theoretic probability, vol. 8. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  25. 25.
    Rätz, A., Voigt, A.: A diffuse-interface approximation for surface diffusion including adatoms. Nonlinearity 20(1), 177–192 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stöcker, C., Voigt, A.: A level set approach to anisotropic surface evolution with free adatoms. SIAM J. Appl. Math. 69(1), 64–80 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Taylor, J.E.: II–Mean curvature and weighted mean curvature. Acta Metall. Mater. 40(7), 1475–1485 (1992)ADSCrossRefGoogle Scholar
  28. 28.
    Taylor, J.E.: Some mathematical challenges in materials science. Bull. Am. Math. Soc. (N.S.), 40(1), 69–87 (2003). Mathematical challenges of the 21st century (Los Angeles, CA, 2000)Google Scholar
  29. 29.
    Taylor, J.E., Cahn, J.W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77(1–2), 183–197 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  3. 3.Lycée Fabert, Bâtiment Toqueville (CPGE)MetzFrance

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