Motion of Elastic Thin Films by Anisotropic Surface Diffusion with Curvature Regularization

1. Adams R.A., Fournier J.F.: Sobolev Spaces, 2nd edn. In: Pure and Applied Mathematics (Amsterdam) Vol. 140. Elsevier/Academic Press, Amsterdam (2003)

   Google Scholar 

2. Agmon S.: Lectures on Elliptic Boundary Value Problems. In: Van Nostrand Mathematical Studies Vol. 2. D . Van Nostrand Co. Inc., Princeton, N.J. (1965)

   Google Scholar 

3. Almgren F., Taylor J.E., Wang L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31, 387–438 (1993)

   Article  MathSciNet  MATH  Google Scholar 

4. Ambrosio L.: Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 5, 191–246 (1995)

   MathSciNet  Google Scholar 

5. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. In: Oxford Mathematical Monographs The Clarendon Press Oxford University Press, New York, (2000)

6. Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. 2nd edn. In: Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)

   Google Scholar 

7. Angenent S., Gurtin M.E.: Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108, 323–391 (1989)

   Article  MathSciNet  ADS  MATH  Google Scholar 

8. Bellettini G., Caselles V., Chambolle A., Novaga M.: Crystalline mean curvature flow of convex sets. Arch. Ration. Mech. Anal. 179, 109–152 (2006)

   Article  MathSciNet  MATH  Google Scholar 

9. Bellettini G., Mantegazza C., Novaga M.: Singular perturbations of mean curvature flow. J. Differ. Geom. 75, 403–431 (2007)

   MathSciNet  MATH  Google Scholar 

10. Bonnetier E., Chambolle A.: Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62, 1093–1121 (2002)

    Article  MathSciNet  MATH  Google Scholar 

11. Burger M., Hausser H., Stöcker C., Voigt A.: A level set approach to anisotropic flows with curvature regularization. J. Comput. Phys. 225, 183–205 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

12. Cahn J.W., Taylor J.E.: Overview N0-113—surface motion by surface-diffusion. Acta Metallurgica et Materialia 42, 1045–1063 (1994)

    Article  Google Scholar 

13. Caselles V., Chambolle A.: Anisotropic curvature-drive flow of convex sets. Nonlinear Anal. 65, 1547–1577 (2006)

    Article  MathSciNet  MATH  Google Scholar 

14. Chen X.: The Hele–Shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal. 123, 117–151 (1993)

    Article  MATH  Google Scholar 

15. Di Carlo A., Gurtin M.E., Podio-Guidugli P.: A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math. 52, 1111–1119 (1992)

    Article  MathSciNet  MATH  Google Scholar 

16. Elliott C.M., Garcke H.: Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7, 467–490 (1997)

    MathSciNet  MATH  Google Scholar 

17. Escher , J. , Mayer U.F., Simonett G.: The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal. 29, 1419–1433 (1998)

    Article  MathSciNet  MATH  Google Scholar 

18. Fonseca I., Fusco N., Leoni G., Millot V.: Material voids for anisotropic surface energies. J. Math. Pures Appl. 96, 591–639 (2011)

    MathSciNet  MATH  Google Scholar 

19. Fonseca I., Fusco N., Leoni G., Morini M.: Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal. 186, 477–537 (2007)

    Article  MathSciNet  MATH  Google Scholar 

20. Fung, Y.C.: A First Course in Continuum Mechanics. Prentice-Hall, Inc, Englewood Cliffs, N.J.,1969

21. 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, 247–327

22. Garcke H.: On Cahn–Hilliard systems with elasticity. Proc. R. Soc. Edinb. Sect. A 133, 307–331 (2003)

    Article  MathSciNet  MATH  Google Scholar 

23. Garcke H.: On a Cahn–Hilliard model for phase separation with elastic misfit. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 165–185 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

24. Gurtin M.E., Jabbour M.E.: Interface evolution in three dimensions with curvature-dependent energy and surface diffusion: interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Ration. Mech. Anal. 163, 171–208 (2002)

    Article  MathSciNet  MATH  Google Scholar 

25. Gurtin M.E., Soner H.M., Souganidis P.E.: Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles. J. Differ. Equ. 119, 54–108 (1995)

    Article  MathSciNet  MATH  Google Scholar 

26. Herring C.: Some theorems on the free energies of crystal surfaces. Phys. Rev. 82, 87–93 (1951)

    Article  ADS  MATH  Google Scholar 

27. Leoni G.: A First Course in Sobolev Spaces. In: Graduate Studies in Mathematics Vol 105. Am. Math. Soc. Providence, Rhode Island, 2009

28. Leoni G.: Interpolation for Intermediate Derivatives. http://www.ams.org/publications/authors/books/postpub/gsm-105

29. MantegazzaC. : Smooth geometric evolutions of hypersurfaces. Geom. Funct. Anal. 12, 138–182 (2002)

    Article  MathSciNet  Google Scholar 

30. Mullins, W.W.: Solid Surface Morphologies Governed by Capillarity. In: Metal Surfaces. American society for metals, 1963

31. Rätz , Ribalta, A., Voigt, A : Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comp. Phys. 214, 187–208 (2006)

    Article  ADS  MATH  Google Scholar 

32. Siegel M., Miksis M.J., Voorhees P.W.: Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids 52, 1319–1353 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references