Abstract
These lecture notes are based on a series of lectures that I gave at the ERC School on Free Discontinuity Problems, at the De Giorgi Center, Pisa, Italy, July 7–July 11, 2014. The school was organized by Aldo Pratelli and Nicola Fusco. Although the basic structure of the lecture notes follows the one given at the school, I am adding here more material and a lot of the details that I skipped in class.
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References
E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Communications in Mathematical Physics 322 (2013), 515–557.
R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Dynamics of discrete screw dislocations on glide directions, preprint, 2014.
R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach, Archive for Rational Mechanics and Analysis 214 (2014), 269–330.
L. Ambrosio, N. Gigli and G. SavarÉ, “Gradient Flows: in Metric Spaces and in the Space of Probability Measures”, Springer Science & Business Media, 2008.
L. Ambrosio, M. Novaga and E. Paolini, Some regularity results for minimal crystals, ESAIM: Control, Optimisation and Calculus of Variations 8 (2002), 69–103.
S. Angenent and M. E. Gurtin Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface, Archive for Rational Mechanics and Analysis 108 (1989), 323–391.
P. Bella, M. Goldman and B. Zwicknagl Study of island formation in epitaxially strained films on unbounded domains, Archive for Rational Mechanics and Analysis (2014), 1–55.
T. Blass, I. Fonseca, G. Leoni and M. Morandotti, Dynamics for systems of screw dislocations, SIAM Journal on Applied Mathematics 75 (2015), 393–419.
T. Blass and M. Morandotti, Renormalized energy and Peach-Köhler forces for screw dislocations with antiplane shear, 2014, arXiv preprint arXiv: 1410.6200.
M. Bonacini, Epitaxially strained elastic films: the case of anisotropic surface energies, ESAIM: Control, Optimisation and Calculus of Variations 19 (2013), 167–189.
M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions, Advances in Calculus of Variations 8 (2015), 117–153.
M. Bonacini and R. Cristoferi, Local and global minimality results for a nonlocal isoperimetric problem on ℝN, SIAM Journal on Mathematical Analysis 46 (2014), 2310–2349.
M. Bonacini and M. Morini, Stable regular critical points of the Mumford-Shah functional are local minimizers In: “Annales de l’Institut Henri Poincare (C) Non Linear Analysis”, Elsevier, 2014.
A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications, ESAIM: Control, Optimisation and Calculus of Variations 13 (2007), 717–734.
F. Cagnetti, M. G. Mora and M. Morini A second order minimality condition for the Mumford-Shah functional Calculus of Variations and Partial Differential Equations 33 (2008), 37–74.
J. W. Cahn and J. E. Taylor, Overview no. 113 surface motion by surface diffusion, Acta metallurgica et materialia 42 (1994), 1045–1063.
G. M. Capriani, V. Julin and G. Pisante, A quantitative second order minimality criterion for cavities in elastic bodies, SIAM Journal on Mathematical Analysis 45 (2013), 1952–1991.
P. Cermelli and T. Armano, Noncrystallographic motion of a dislocation as a fine mixture of rectilinear paths, SIAM Journal on Applied Mathematics 64 (2004), 2121–2143.
P. Cermelli and M. E. Gurtin, The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip, Archive for rational mechanics and analysis 148 (1999), 3–52.
P. Cermelli and G. Leoni, Renormalized energy and forces on dislocations, SIAM journal on mathematical analysis 37 (2005), 1131–1160.
A. Chambolle and E. Bonnetier, Computing the equilibrium configuration of epitaxially strained crystalline films, SIAM Journal on Applied Mathematics 62 (2002), 1093–1121.
A. Chambolle and C. J. Larsen, C∞ regularity of the free boundary for a two-dimensional optimal compliance problem, Calculus of Variations and Partial Differential Equations 18 (2003), 77–94.
A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint, SIAM Journal on Mathematical Analysis 39 (2007), 77–102.
S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Archive for rational mechanics and analysis 175 (2005), 287–300.
L. De Luca, A. Garroni and M. Ponsiglione, Γ-convergence analysis of systems of edge dislocations: the self energy regime, Archive for Rational Mechanics and Analysis 206 (2012), 885–910.
B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films, In: “Topics in Modern Regularity Theory”, Springer, 2012, 169–204.
A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature, SIAM Journal on Applied Mathematics 52 (1992), 1111–1119.
K. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Physical Review B 75 (2007), 064107.
A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Communications in Mathematical Physics 336 (2015), 441–507.
M. Focardi and A. Garroni, A 1d macroscopic phase field model for dislocations and a second order γ-limit, Multiscale Modeling & Simulation 6 (2007), 1098–1124.
I. Fonseca, The Wulff theorem revisited, 432 (1991), 125–145.
I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies, Journal de mathematiques pures et appliquees 96 (2011), 591–639.
I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results, Archive for Rational Mechanics and Analysis 186 (2007), 477–537.
I. Fonseca, N. Fusco, G. Leoni and M. Morini Motion of elastic thin films by anisotropic surface diffusion with curvature regularization, Archive for Rational Mechanics and Analysis 205 (2012), 425–466.
I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films, preprint, 2015.
I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization, Analysis & PDE 8 (2015), 373–423.
I. Fonseca and S. MÜller, A uniqueness proof for the Wulff theorem, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 119 (1991), 125–136.
I. Fonseca, A. Pratelli and B. Zwicknagl, Shapes of epi-taxially grown quantum dots, Archive for Rational Mechanics and Analysis 214 (2014), 359–401.
F. Frank and J. H. van der Merwe, One-dimensional dislocations. I. Static theory Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1949), 205–216.
L. B. Freund and S. Suresh, “Thin Film Materials: Stress, Defect Formation and Surface Evolution”, Cambridge University Press, 2004.
K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality, Annals of Mathematics (1947), 441–471.
N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions, Archive for Rational Mechanics and Analysis 203 (2012), 247–327.
H. Gao and W. D. Nix, Surface roughening of heteroepitaxial thin films, Annual Review of Materials Science 29 (1999), 173–209.
A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc. (JEMS) 12 (2010), 1231–1266.
M. Goldman and B. Zwicknagl, Scaling law and reduced models for epitaxially strained crystalline films, SIAM Journal on Mathematical Analysis 46 (2014), 1–24.
P. Grisvard, Singularités en elasticité, Archive for rational mechanics and analysis 107 (1989), 157–180.
P. Grisvard, “Elliptic Problems in Nonsmooth Domains”, Vol. 69, SIAM, 2011.
M. E. Gurtin, “Multiphase Thermomechanics with Interfacial Structure 1. Heat Conduction and the Capillary Balance Law”, Springer, 1999.
M. E. Gurtin, H. M. Soner and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, Journal of differential equations 119 (1995), 54–108.
M. Haataja, J. MÜller, A. Rutenberg and M. Grant, Dislocations and morphological instabilities: Continuum modeling of misfitting heteroepitaxial films, Physical Review B 65 (2002), 165414.
C. Herring, Some theorems on the free energies of crystal surfaces, Physical Review 82 (1951), 87.
J. P. Hirth and J. Lothe, “Theory of Dislocations”, John Wiley & Sons, 1982.
T. Hudson and C. Ortner, Existence and stability of a screw dislocation under anti-plane deformation, Archive for Rational Mechanics and Analysis 213 (2014), 887–929.
D. Hull and D. J. Bacon, “Introduction to Dislocations”, Butter-worth-Heinemann, 2001.
D. Jesson, S. Pennycook, J.-M. Baribeau and D. Houghton, Direct imaging of surface cusp evolution during strainedlayer epitaxy and implications for strain relaxation, Physical review letters 71 (1993), 1744.
V. Julin and G. Pisante, Minimality via second variation for microphase separation of diblock copolymer melts, Journal für die reine und angewandte Mathematik (Crelles Journal), 2013.
P. Kohlert, K. Kassner and C. Misbah, Large-amplitude behavior of the Grinfeld instability: a variational approach, The European Physical Journal B-Condensed Matter and Complex Systems 35 (2003), 493–504.
R. Kukta and L. Freund, Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate, Journal of the Mechanics and Physics of Solids 45 (1997), 1835–1860.
G. Leoni, “A First Course in Sobolev Spaces”, Vol. 105, American Mathematical Society Providence, RI, 2009.
G. Lieberman, Regularized distance and its applications, Pacific journal of Mathematics 117 (1985), 329–352.
M. G. Mora, M. Peletier and L. Scardia, Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement, 2014, preprint arXiv:1409.4236.
S. MÜller, L. Scardia and C. I. Zeppieri, Geometric rigidity for incompatible fields and an application to strain-gradient plasticity, Indiana University Mathematics Journal, 2014.
W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics 28 (1957), 333–339.
S. Nicaise, About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. I: Regularity of the solutions, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 19 (1992), 327–361.
S. Nicaise and A.-M. SÄndig, General interface problems: I, Mathematical Methods in the Applied Sciences 17 (1994), 395–429.
J. A. Nitsche, On Korn’s second inequality, RAIRO-Analyse numérique 15 (1981), 237–248.
M. Novaga and E. Paolini, Regularity results for boundaries in ℝ2 with prescribed anisotropic curvature, Annali di Matematica Pura ed Applicata 184 (2005), 239–261.
D. Ornstein, A non-inequality for differential operators in the L1 norm, Archive for Rational Mechanics and Analysis 11 (1962), 40–49.
E. Orowan, Zur kristallplastizität. iii, Zeitschrift für Physik 89 (1934), 634–659.
P. Piovano, Evolution of elastic thin films with curvature regularization via minimizing movements, Calculus of Variations and Partial Differential Equations 49 (2014), 337–367.
M. Polanyi, Über eine art gitterstörung, die einen kristall plastisch machen könnte, Zeitschrift für Physik 89 (1934), 660–664.
C. Reina and S. Conti, Kinematic description of crystal plasticity in the finite kinematic framework: a micromechanical understanding of F = FeFp, Journal of the Mechanics and Physics of Solids 67 (2014), 40–61.
A. RÖssle, Corner singularities and regularity of weak solutions for the two-dimensional Lamé equations on domains with angular corners, Journal of elasticity 60 (2000), 57–75.
M. Siegel, M. Miksis and P. Voorhees, Evolution of material voids for highly anisotropic surface energy, Journal of the Mechanics and Physics of Solids 52 (2004), 1319–1353.
B. Spencer, Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski-Krastanow islands, Physical Review B 59 (1999), 2011.
B. Spencer and J. Tersoff, Equilibrium shapes and properties of epitaxially strained islands, Physical Review Letters 79 (1997), 4858.
B. Spencer and J. Tersoff, Asymmetry and shape transitions of epitaxially strained islands on vicinal surfaces, Applied Physics Letters 96 (2010), 073114.
B. J. Spencer, Asymptotic solutions for the equilibrium crystal shape with small corner energy regularization, Physical Review E 69 (2004), 011603.
E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Vol. 2. Princeton University Press, 1970.
G. I. Taylor, The mechanism of plastic deformation of crystals. Part I. Theoretical. Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 1934, 362–387.
J. E. Taylor, Crystalline variational problems, Bulletin of the American Mathematical Society 84 (1978), 568–588.
J. Tersoff and F. Le Goues, Competing relaxation mechanisms in strained layers, Physical review letters 72 (1994), 3570.
J. Tersoff and R. Tromp, Shape transition in growth of strained islands: spontaneous formation of quantum wires, Physical review letters 70 (1993), 2782.
T. W. Ting, Generalized Korn’s inequalities, Tensor 25 (1972), 295–302.
P. van Meurs, A. Muntean and M. Peletier, Upscaling of dislocation walls in finite domains, European Journal of Applied Mathematics 25 (2014), 749–781.
V. Volterra, Sur l’équilibre des corps élastiques multiplement connexes, In: “Annales scientifiques de l’Ecole Normale Superieure”, Vol. 24, Société mathématique de France, 1907, 401–517.
W. Yang and D. Srolovitz, Cracklike surface instabilities in stressed solids, Physical review letters 71 (1993), 1593.
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Leoni, G. (2016). Variational models for epitaxial growth. In: Fusco, N., Pratelli, A. (eds) Free Discontinuity Problems. Publications of the Scuola Normale Superiore, vol 19. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-593-6_2
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