Surface evolution of elastically stressed films

Irene Fonseca; Giovanni Leoni

doi:10.4171/mag/6

An overview of recent analytical developments in the study of epitaxial growth is presented. Quasistatic equilibrium is established, regularity of solutions is addressed, and the evolution of epitaxially strained elastic films is treated using minimizing movements.

In this paper, we give a brief overview of recent analytical developments in the study of the deposition of a crystalline film onto a substrate, with the atoms of the film occupying the substrate’s natural lattice positions. This process is called epitaxial growth. Here we are interested in heteroepitaxy, that is, epitaxy when the film and the substrate have different crystalline structures. At the onset of the deposition, the film’s atoms tend to align themselves with those of the substrate because the energy gain associated with the chemical bonding effect is greater than the film’s strain due to the mismatch between the lattice parameters. As the film continues to grow, the stored strain energy per unit area of the interface increases with the film thickness, rendering the film’s flat layer morphologically unstable or metastable after the thickness reaches a critical value. As a result, the film’s free surface becomes corrugated, and the material agglomerates into clusters or isolated islands on the substrate. The formation of islands in systems such as In-GaAs/GaAs or SiGe/Si has essential high-end technology applications, such as modern semiconductor electronic and optoelectronic devices (quantum dots laser). The Stranski–Krastanow (SK) growth mode occurs when the islands are separated by a thin wetting layer, while the Volmer–Weber (VW) growth mode refers to the case when the substrate is exposed between islands.

In what follows, we adopt the variational model considered by Spencer in [⁠41⁠ B. J. Spencer, Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski–Krastanow islands. Phys. Rev. B, 59, 2011 (1999) ] (see also [⁠36⁠ R. V. Kukta and L. B. Freund, Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate. J. Mech. Phys. Solids45, 1835–1860 (1997) , ⁠42⁠ B. J. Spencer and J. Tersoff, Equilibrium shapes and properties of epitaxially strained islands. Phys. Rev. Lett.79, 4858 (1997) ], and the references contained therein). To be precise, the free energy functional associated with the physical system is given by

\int_{Ω_{h}} W (E (u)) d x + \int_{Γ_{h}} ψ (ν) d H^{2} .

Here $h : Q \to [0, \infty)$ is the function whose graph $Γ_{h}$ describes the profile of the film, assumed to be $Q$ -periodic, with $Q := (0, b)^{2} \subset R^{2}$ , for some $b > 0$ , $Ω_{h}$ is the region occupied by the film, i.e., writing $x = (x, y, z)$ ,

Ω_{h} := {(x, y, z) \in Q \times R : 0 < z < h (x, y)},

$u : Ω_{h} \to R^{3}$ is displacement of the material, $E (u) := \frac{1}{2} (D u + D^{T} u)$ is the symmetric part of $D u$ . Also, the elastic energy density $W : M_{sym}^{3 \times 3} \to [0, + \infty)$ is a positive definite quadratic form defined on the space of $3 \times 3$ symmetric matrices

W (A) := \frac{1}{2} C A : A,

with $C$ a positive definite fourth-order tensor, so that $W (A) > 0$ for all $A \in M_{sym}^{3 \times 3} ∖ {0}$ , $ψ : R^{3} \to [0, \infty)$ is an anisotropic surface energy density evaluated at the unit normal $ν$ to $Γ_{h}$ , and $H^{2}$ denotes the two-dimensional Hausdorff measure. We suppose that $ψ$ is positively one-homogeneous and of class $C^{2}$ away from the origin, so that, in particular,

\frac{1}{c} ∣ ξ ∣ \leq ψ (ξ) \leq c ∣ ξ ∣ for all ξ \in R^{3},

for some constant $c > 0$ .

The substrate and the film admit different natural states corresponding to the mismatch between their respective crystalline structures. To be precise, a natural state for the substrate is given by $u \equiv 0$ , while a natural state for the film is given by $u \equiv A_{0} x$ for some nonzero $3 \times 3$ matrix $A_{0}$ . Our models will reflect this mismatch, either by setting the elastic bulk energy as $\int_{Ω_{h}} W (E (u) (x) - E_{0} (x)) d x$ , where

E_{0} (x) := {\frac{A _{0} + A _{0}^{T}}{2} 0 if z > 0, if z \leq 0,

or by imposing the Dirichlet boundary condition $u (x, y, 0) \equiv A_{0} (x, y, 0)$ .

In the two-dimensional static case, existence of equilibrium solutions and their qualitative properties, including regularity, were studied in [⁠3⁠ M. Bonacini, Epitaxially strained elastic films: The case of anisotropic surface energies. ESAIM Control Optim. Calc. Var.19, 167–189 (2013) , ⁠4⁠ M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var.8, 117–153 (2015) , ⁠5⁠ E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math.62, 1093–1121 (2002) , ⁠15⁠ E. Davoli and P. Piovano, Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019) , ⁠16⁠ E. Davoli and P. Piovano, Derivation of a heteroepitaxial thin-film model. Interfaces Free Bound.22, 1–26 (2020) , ⁠17⁠ B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films. In Topics in modern regularity theory, CRM Series 13, Ed. Norm., Pisa, 169–204 (2012) , ⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) , ⁠24⁠ I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth. Nonlinear Anal.154, 88–121 (2017) , ⁠26⁠ N. Fusco, Equilibrium configurations of epitaxially strained thin films. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.21, 341–348 (2010) , ⁠29⁠ N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012) , ⁠33⁠ S. Y. Kholmatov and P. Piovano, A unified model for stress-driven rearrangement instabilities. Arch. Ration. Mech. Anal.238, 415–488 (2020) ]. The variational techniques and analytical arguments developed in these papers have been used to treat other materials phenomena, such as voids and cavities in elastic solids [⁠9⁠ G. M. Capriani, V. Julin and G. Pisante, A quantitative second order minimality criterion for cavities in elastic bodies. SIAM J. Math. Anal.45, 1952–1991 (2013) , ⁠19⁠ I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. (9)96, 591–639 (2011) ].

The scaling regimes of the minimal energy in epitaxial growth were identified in [⁠2⁠ P. Bella, M. Goldman and B. Zwicknagl, Study of island formation in epitaxially strained films on unbounded domains. Arch. Ration. Mech. Anal.218, 163–217 (2015) , ⁠30⁠ M. Goldman and B. Zwicknagl, Scaling law and reduced models for epitaxially strained crystalline films. SIAM J. Math. Anal.46, 1–24 (2014) ] in terms of the parameters of the problem. The shape of the islands under the constraint of faceted profiles was addressed in [⁠25⁠ I. Fonseca, A. Pratelli and B. Zwicknagl, Shapes of epitaxially grown quantum dots. Arch. Ration. Mech. Anal.214, 359–401 (2014) ]. A variational model that takes into account the formation of misfit dislocations was introduced in [⁠23⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films. J. Math. Pures Appl. (9)111, 126–160 (2018) ].

The effect of atoms freely diffusing on the surface (called adatoms) was studied in [⁠10⁠ M. Caroccia, R. Cristoferi and L. Dietrich, Equilibria configurations for epitaxial crystal growth with adatoms. Arch. Ration. Mech. Anal.230, 785–838 (2018) ], where the model involves only surface energies.

A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates was undertaken in [⁠35⁠ L. C. Kreutz and P. Piovano, Microscopic validation of a variational model of epitaxially strained crystalline films. SIAM J. Math. Anal.53, 453–490 (2021) , ⁠38⁠ P. Piovano and I. Velčić, Microscopical justification of solid-state wetting and dewetting. arXiv:2010.08787 (2020) ].

The three-dimensional static case was studied in [⁠6⁠ A. Braides, A. Chambolle and M. Solci, A relaxation result for energies defined on pairs set-function and applications. ESAIM Control Optim. Calc. Var.13, 717–734 (2007) , ⁠12⁠ A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal.39, 77–102 (2007) ] in the case in which the symmetrized gradient $E (u)$ is replaced by the gradient (see also [⁠4⁠ M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var.8, 117–153 (2015) ]). More recently, new developments in the theory of $GSB D$ , i.e., generalized special functions of bounded deformation (see [⁠13⁠ V. Crismale and M. Friedrich, Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity. Arch. Ration. Mech. Anal.237, 1041–1098 (2020) , ⁠14⁠ G. Dal Maso, Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS)15, 1943–1997 (2013) ], and the references therein) have led to considerable progress on the relaxation of the functional (⁠1⁠) in the three dimensional case (see [⁠13⁠ V. Crismale and M. Friedrich, Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity. Arch. Ration. Mech. Anal.237, 1041–1098 (2020) ]). The regularity of equilibrium solutions remains an open problem. A local minimality sufficiency criterion, based on the strict positivity of the second variation, was established in [⁠4⁠ M. Bonacini, Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var.8, 117–153 (2015) ], based on the work [⁠29⁠ N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012) ].

To study the morphological evolution of anisotropic epitaxially strained films, we assume that the surface evolves by surface diffusion under the influence of a chemical potential $μ$ . To be precise, according to the Einstein–Nernst relation, the evolution is governed by the volume preserving equation

V = C Δ_{_{Γ}} μ,

where $C > 0$ , $V$ denotes the normal velocity of the evolving interface $Γ$ , $Δ_{_{Γ}}$ stands for the tangential laplacian, and the chemical potential $μ$ is given by the first variation of the underlying free-energy functional. In our context, this becomes (assuming $C = 1$ )

V = Δ_{Γ} [div_{Γ} (D ψ (ν)) + W (E (u))],

where $div_{Γ}$ stands for the tangential divergence along $Γ_{h (\cdot, t)}$ , and $u (\cdot, t)$ is the elastic equilibrium in $Ω_{h (\cdot, t)}$ , i.e., the minimizer of the elastic energy under the prescribed periodicity and boundary conditions (see (⁠7⁠) below).

If the surface energy density $ψ$ is highly anisotropic, there may be directions $ν$ for which

D^{2} ψ (ν) [τ, τ] > 0 for all τ ⊥ ν, τ \neq = 0

fails, see for instance [⁠18⁠ A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math.52, 1111–1119 (1992) , ⁠40⁠ M. Siegel, M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004) ]. In this case, the evolution equation (⁠4⁠) is backward parabolic, and to overcome the ill-posedness of the problem we consider the following singular perturbation of the surface energy

\int_{Γ_{h}} (ψ (ν) + \frac{ε}{p} ∣ H ∣^{p}) d H^{2},

where $p > 2$ , $H$ stands for the sum $κ_{1} + κ_{2}$ of the principal curvatures of $Γ_{h}$ , and $ε$ is a small positive constant (see [⁠18⁠ A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math.52, 1111–1119 (1992) , ⁠31⁠ M. E. Gurtin and M. E. Jabbour, 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) , ⁠32⁠ C. Herring, Some theorems on the free energies of crystal surfaces. Phys. Rev.82, 87 (1951) ]). The restriction $p > 2$ in $R^{3}$ is motivated by the fact that the profile $h$ of the film will belong to $W^{2, p} (Q)$ , where $Q \subset R^{2}$ , so that $W^{2, p} (Q)$ is continuously embedded into $C^{1, \frac{p - 2}{p}} (Q)$ . This regularity is strongly used to prove existence of solutions. In contrast, in $R^{2}$ we can assume $p \geq 2$ since $W^{2, 2} ((0, b))$ is embedded in $C^{1, 1} ([0, b])$ .

The regularized free-energy functional becomes

\int_{Ω_{h}} W (E (u)) d x + \int_{Γ_{h}} (ψ (ν) + \frac{ε}{p} ∣ H ∣^{p}) d H^{2},

and (⁠3⁠) is replaced by

V = Δ_{Γ} [div_{Γ} (D ψ (ν)) + W (E (u)) - ε (Δ_{Γ} (∣ H ∣^{p - 2} H) - ∣ H ∣^{p - 2} H (κ_{1}^{2} + κ_{2}^{2} - \frac{1}{p} H^{2}))] .

Coupling this evolution equation on the profile of the film with the elastic equilibrium elliptic system holding in the film, and parametrizing $Γ$ using $h : R^{2} \times [0, T_{0}] \to (0, \infty)$ , we obtain the following Cauchy system of equations with initial and natural boundary conditions:

⎩ ⎨ ⎧ \frac{1}{J} \frac{\partial h}{\partial t} = Δ_{Γ} [div_{Γ} (D ψ (ν)) + W (E (u)) \frac{1}{p} - ε (Δ_{Γ} (∣ H ∣^{p - 2} H) - ∣ H ∣^{p - 2} H (κ_{1}^{2} + κ_{2}^{2} - \frac{1}{p} H^{2}))] in R^{2} \times (0, T_{0}), div C E (u) = 0 in Ω_{h}, C E (u) [ν] = 0 on Γ_{h}, u (x, y, 0, t) = A_{0} (x, y, 0), h (\cdot, t) and D u (\cdot, t) are Q -periodic, h (\cdot, 0) = h_{0},

where $J := 1 + ∣ D h ∣^{2}$ and $h_{0} \in H_{loc}^{2} (R^{2})$ is a $Q$ -periodic function.

One can find in the literature sixth-order evolution equations of this type (see, e.g., [⁠31⁠ M. E. Gurtin and M. E. Jabbour, 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) ] for the case without elasticity, see [⁠40⁠ M. Siegel, M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004) ] for the evolution of voids in elastically stressed materials, and [⁠7⁠ M. Burger, F. Haußer, C. Stöcker and A. Voigt, A level set approach to anisotropic flows with curvature regularization. J. Comput. Phys.225, 183–205 (2007) , ⁠39⁠ A. Rätz, A. Ribalta and A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys.214, 187–208 (2006) ]).

We use the gradient flow structure of (⁠7⁠) with respect to a suitable $H^{- 1}$ -metric (see, e.g., [⁠8⁠ J. W. Cahn and J. E. Taylor, Overview no. 113, surface motion by surface diffusion. Acta Metall. Mater.42, 1045–1063 (1994) ]) to solve the equation via a minimizing movement scheme (see [⁠1⁠ L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5)19, 191–246 (1995) ]), i.e., we discretize the problem in time and solve suitable minimum incremental problems.

If instead of $H^{- 1}$ we used the gradient flow with respect to an $L^{2}$ -metric, we would obtain a fourth order evolution equation describing motion by evaporation-condensation (see [⁠8⁠ J. W. Cahn and J. E. Taylor, Overview no. 113, surface motion by surface diffusion. Acta Metall. Mater.42, 1045–1063 (1994) , ⁠31⁠ M. E. Gurtin and M. E. Jabbour, 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) , ⁠37⁠ P. Piovano, Evolution of elastic thin films with curvature regularization via minimizing movements. Calc. Var. Partial Differ. Equ.49, 337–367 (2014) ]).

The short time existence of solutions to (⁠7⁠) established in [⁠22⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE8, 373–423 (2015) ] is the first such result for geometric surface diffusion equations with elasticity in three-dimensions. In the recent paper [⁠28⁠ N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in three dimensions. Arch. Ration. Mech. Anal.237, 1325–1382 (2020) ] (see also [⁠27⁠ N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in the plane. Comm. Math. Phys.362, 571–607 (2018) ] for the two-dimensional case), the authors proved short-time existence of a smooth solution without the additional curvature regularization. They also showed asymptotic stability of strictly stable stationary sets.

The results summarized here can be found in the papers [⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) , ⁠21⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ration. Mech. Anal.205, 425–466 (2012) , ⁠22⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE8, 373–423 (2015) ].

1 2D quasistatic equilibrium of epitaxially strained elastic films

In the following sections we assume self-similarity with respect to a planar axis and reduce the context to a two-dimensional framework. To be precise, we suppose that the material fills the infinite strip

Ω_{h} := {x = (x, y) : 0 < x < b, y < h (x)},

where $h : [0, b] \to [0, \infty)$ is a Lipschitz function representing the free profile of the film, which occupies the open set

Ω_{h}^{+} := Ω_{h} \cap {y > 0} .

The line $y = 0$ corresponds to the film/substrate interface.

We assume that the mismatch strain corresponding to different natural states of the material in the substrate and in the film, respectively, is represented by

E_{0} (y) = {\hat{E}_{0} 0 if y \geq 0, if y < 0,

with $\hat{E}_{0} \neq = 0 > 0$ . We will suppose that the film and the substrate share material properties, with homogeneous elasticity positive definite fourth-order tensor $C$ . Hence, bearing in mind the mismatch, the elastic energy per unit area is given by $W (E - E_{0} (y))$ , where

W (E) := \frac{1}{2} E \cdot C [E]

for all symmetric matrices $E \neq = 0$ .

In turn, the interfacial energy density $ψ$ has a step discontinuity at $y = 0$ , i.e.,

ψ (y) := {γ_{film} γ_{sub} if y > 0, if y = 0,

where the property

γ_{sub} \geq γ_{film} > 0

will favor the SK growth mode over the VW mode. For the case $γ_{sub} < γ_{film}$ , and for different crystalline materials stress tensors $C$ for the substrate and for the film, we refer to [⁠15⁠ E. Davoli and P. Piovano, Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019) , ⁠16⁠ E. Davoli and P. Piovano, Derivation of a heteroepitaxial thin-film model. Interfaces Free Bound.22, 1–26 (2020) ].

The total energy of the system is given by

F (u, h) := \int_{Ω_{h}} W (E (u) - E_{0}) d x + \int_{Γ_{h}} ψ d s,

where $Γ_{h}$ represents the free surface of the film, that is,

Γ_{h} := \partial Ω_{h} \cap ((0, b) \times R) .

Since the functional $F$ is not lower semicontinuous, and thus, in general, does not admit minimizers, we are led to study its relaxation. Let

X := {(u, h) : h : [0, b] \to [0, \infty) Lipschitz, \int_{0}^{b} h d x = d, u \in H_{loc}^{1} (Ω_{h}; R^{2})}

and

X_{0} = {(u, h) : h : [0, b] \to [0, \infty) lower semicontinuous, [0, b] var h < \infty, \int_{0}^{b} h d x = d, u \in H_{loc}^{1} (Ω_{h}; R^{2})},

where $var_{[0, b]} h$ stands for the pointwise variation of the function $h$ . Note that $length Γ_{h}$ coincides with the pointwise variation of the function $x \in [0, b] \mapsto (x, h (x))$ , and so

var_{[0, b]} h \leq length Γ_{h} \leq b + var_{[0, b]} h .

For $(u, h) \in X_{0}$ define

G (u, h) := \int_{Ω_{h}} W (E (u) (x) - E_{0} (y)) d x + γ_{film} length Γ_{h} .

Theorem 1 (Existence).

The following equalities hold:

(u, h) \in X in f F (u, h) = (u, h) \in X in f G (u, h) = (u, h) \in X_{0} min G (u, h) .

We refer to [⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) ] for a proof.

Next we study regularity properties of minimizers of $G$ in $X_{0}$ . As customary in constrained variational problems, in order to have more flexibility in the choice of test functions, we prove that the volume constraint $\int_{0}^{b} h (x) d x = d$ can be replaced by a volume penalization.

Theorem 2 (Volume penalization).

Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠) with $\int_{0}^{b} h_{0} (x) d x = d$ . Then there exists $k_{0} \in N$ such that for every integer $k \geq k_{0}$ , $(u_{0}, h_{0})$ is a minimizer of the penalized functional

G_{k} (u, h) := \int_{Ω_{h}} W (E (u) - E_{0}) d x + γ_{film} length Γ_{h} + k \int_{0}^{b} h d x - d

over all $(u, h) \in X_{0}$ .

Proof. An argument similar to that of the proof of Theorem ⁠1⁠ guarantees that for every $k \in N$ there exists a minimizer $(v_{k}, f_{k})$ of $G_{k}$ . If $\int_{0}^{b} f_{k} d x = d$ for all $k$ sufficiently large, then

G (u_{0}, h_{0}) \leq G (v_{k}, f_{k}) = G_{k} (v_{k}, f_{k}) \leq G_{k} (u_{0}, h_{0}) = G (u_{0}, h_{0}) < \infty,

and so $(u_{0}, h_{0})$ is a minimizer of $G_{k}$ .

Assume now that there is a subsequence, not relabeled, such that $\int_{0}^{b} f_{k} d x \neq = d$ for all $k$ . If

\int_{0}^{b} f_{k} d x > d

for countably many $k$ , define

h_{k} := min {f_{k}, t_{k}},

where $t_{k} > 0$ has been chosen so that $\int_{0}^{b} h_{k} d x = d$ . Note that $length Γ_{h_{k}} \leq length Γ_{f_{k}}$ . Indeed, for every partition $x_{0} = 0 < \dots < x_{n} = b$ , we have that

(h_{k} (x_{i}) - h_{k} (x_{i - 1}))^{2} \leq (f_{k} (x_{i}) - f_{k} (x_{i - 1}))^{2}

for all $i = 1, \dots, n$ . Hence,

G (v_{k}, h_{k}) = G_{k} (v_{k}, h_{k}) < G_{k} (v_{k}, f_{k}),

which is a contradiction. Therefore, for all $k$ sufficiently large

\int_{0}^{b} f_{k} d x < d .

Since

G_{k} (v_{k}, f_{k}) \leq G_{k} (u_{0}, h_{0}) = G (u_{0}, h_{0}) < \infty,

it follows from (⁠18⁠) and (⁠20⁠) that $\int_{0}^{b} f_{k} d x \to d$ as $k \to \infty$ and that $sup_{k} length Γ_{f_{k}} < \infty$ . In turn, by (⁠16⁠), $∥ f_{k} ∥_{\infty} \leq c$ for some constant $c$ independent of $k$ .

Let $k_{1}$ be so large that $\int_{0}^{b} f_{k} d x > \frac{d}{2}$ for all $k \geq k_{1}$ . Then

t_{k} := \frac{d}{\int _{0}^{b} f _{k} d x} \in (0, 2)

and the function $h_{k} (x) := t_{k} f_{k} (x)$ , $x \in (0, b)$ , satisfies

\int_{0}^{b} h_{k} d x = d .

Consider a partition $0 = x_{0} < \dots < x_{ℓ} = b$ . Then

i = 1 \sum ℓ (x_{i} - x_{i - 1})^{2} + (h_{k} (x_{i}) - h_{k} (x_{i - 1}))^{2}

= i = 1 \sum ℓ (x_{i} - x_{i - 1})^{2} + t_{k}^{2} (f_{k} (x_{i}) - f_{k} (x_{i - 1}))^{2}

\leq t_{k} i = 1 \sum ℓ (x_{i} - x_{i - 1})^{2} + (f_{k} (x_{i}) - f_{k} (x_{i - 1}))^{2}

\leq t_{k} length Γ_{f_{k}},

where we used the fact that $t_{k} > 1$ . Hence,

length Γ_{h_{k}} \leq t_{k} length Γ_{f_{k}},

and so, by (⁠20⁠),

γ_{film} length Γ_{h_{k}} - γ_{film} length Γ_{f_{k}}

\leq (t_{k} - 1) γ_{film} length Γ_{f_{k}} \leq (t_{k} - 1) G_{k} (v_{k}, f_{k})

\leq (t_{k} - 1) G (u_{0}, h_{0}) .

We deduce that

γ_{film} length Γ_{h_{k}} \leq γ_{film} length Γ_{f_{k}} + (t_{k} - 1) G (u_{0}, h_{0}) .

For $(x, y^{'}) \in Ω_{h_{k}}$ define

w_{k} (x, y^{'}) := ((v_{k})_{1} (x, \frac{y ^{'}}{t _{k}}), \frac{1}{t _{k}} (v_{k})_{2} (x, \frac{y ^{'}}{t _{k}})) .

By a change of variables and (⁠10⁠), we have

\int_{Ω_{h_{k}}} W (E (w_{k}) (x, y^{'}) - E_{0} (y^{'})) d x d y^{'} = \frac{1}{t _{k}} \int_{Ω_{f_{k}}} W (E (v_{k}) (x) - E_{0} (y)) d x,

where $E (v_{k}) (x)$ is the $2 \times 2$ matrix whose entries are

E_{11} (v_{k}) (x) = E_{11} (v_{k}) (x), E_{12} (v_{k}) (x) = \frac{1}{t _{k}} E_{12} (v_{k}) (x),

E_{22} (v_{k}) (x) = \frac{1}{t _{k}^{2}} E_{22} (v_{k}) (x) .

Observe that

(∣ E (v_{k}) - E_{0} ∣ + ∣ E (v_{k}) - E_{0} ∣) ∣ E (v_{k}) - E (v_{k}) ∣

\leq c (t_{k} - 1) (∣ E (v_{k}) - E_{0} ∣ + ∣ E (v_{k}) - E_{0} ∣) ∣ E (v_{k}) ∣

\leq c (t_{k} - 1) (∣ E (v_{k}) ∣ + ∣ E_{0} ∣) (∣ E (v_{k}) - E_{0} ∣ + ∣ E_{0} ∣)

\leq c (t_{k} - 1) (∣ E (v_{k}) - E_{0} ∣ + ∣ E_{0} ∣)^{2} .

Since $W (E)$ is a positive definite quadratic form over the $2 \times 2$ symmetric matrices (see (⁠11⁠)), we have that

∣ W (E) - W (E_{1}) ∣ \leq c (∣ E ∣ + ∣ E_{1} ∣) ∣ E - E_{1} ∣

for all $2 \times 2$ symmetric matrices $E$ and $E_{1}$ . Hence by (⁠1⁠), (⁠10⁠) and (LABEL:601)

\int_{Ω_{h_{k}}} W (E (w_{k}) (x, y^{'}) - E_{0} (y^{'})) d x^{'} - \int_{Ω_{f_{k}}} W (E (v_{k}) (x) - E_{0} (y)) d x

= \frac{1}{t _{k}} \int_{Ω_{f_{k}}} [W (E (v_{k}) (x) - E_{0} (y)) - W (E (v_{k}) (x) - E_{0} (y))] d x

\leq c \int_{Ω_{f_{k}}} (∣ E (v_{k}) - E_{0} ∣ + ∣ E (v_{k}) - E_{0} ∣) (∣ E (v_{k}) - E (v_{k}) ∣) d x

\leq c (t_{k} - 1) \int_{Ω_{f_{k}}} (∣ E (v_{k}) - E_{0} ∣ + ∣ E_{0} ∣)^{2} d x

\leq c (t_{k} - 1) (G_{k} (v_{k}, f_{k}) + ∣ \hat{E}_{0} ∣^{2}) \leq c (t_{k} - 1) (G (u_{0}, h_{0}) + ∣ \hat{E}_{0} ∣^{2}),

where $c$ depends only on the ellipticity constants of $W$ and $sup_{k} ∥ f_{k} ∥_{\infty}$ . By (⁠20⁠), (⁠21⁠), and (⁠24⁠), we have that

G (u_{0}, h_{0})

\leq G (w_{k}, h_{k}) \leq G (v_{k}, f_{k}) + (t_{k} - 1) [(c + 1) G (u_{0}, h_{0}) + c ∣ \hat{E}_{0} ∣^{2}]

= G_{k} (v_{k}, f_{k}) + (t_{k} - 1) [(c + 1) G (u_{0}, h_{0}) + c ∣ \hat{E}_{0} ∣^{2}]

- k (d - \int_{0}^{b} f_{k} d x)

= G_{k} (v_{k}, f_{k}) + (t_{k} - 1) [(c + 1) G (u_{0}, h_{0}) + c ∣ \hat{E}_{0} ∣^{2}]

- (t_{k} - 1) k \int_{0}^{b} f_{k} d x

\leq G (u_{0}, h_{0}) + (t_{k} - 1) [(c + 1) G (u_{0}, h_{0}) + c ∣ \hat{E}_{0} ∣^{2} - k \frac{d}{2}] .

Thus, if

k \geq \frac{2}{d} [(c + 1) G (u_{0}, h_{0}) + c ∣ \hat{E}_{0} ∣^{2}] + 1,

we get a contradiction, and this completes the proof. ∎

To prove the regularity of the free boundary we use the following internal sphere condition.

Theorem 3 (Internal Sphere’s Condition).

Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). Then there exists $r_{0} > 0$ with the property that for every $z_{0} \in \overline{Γ}_{h_{0}}$ there exists an open ball $B (x_{0}, r_{0})$ , with $B (x_{0}, r_{0}) \cap ((0, b) \times R) \subseteq Ω_{h_{0}}$ , such that

\partial B (x_{0}, r_{0}) \cap \overline{Γ}_{h_{0}} = {z_{0}} .

This result was first proved in a slightly different context by Chambolle and Larsen [⁠11⁠ A. Chambolle and C. J. Larsen, C∞ regularity of the free boundary for a two-dimensional optimal compliance problem. Calc. Var. Partial Differential Equations18, 77–94 (2003) ] (see also [⁠9⁠ G. M. Capriani, V. Julin and G. Pisante, A quantitative second order minimality criterion for cavities in elastic bodies. SIAM J. Math. Anal.45, 1952–1991 (2013) , ⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) ]). The argument is entirely two-dimensional and its extension to three dimensions is open.

Remark 4. Note that if $ν_{0} \in \partial B (0, 1)$ is the outward unit normal to $B (x_{0}, r_{0})$ at $z_{0}$ , then $x_{0} = z_{0} - r_{0} ν_{0}$ . Thus, the set

N_{z_{0}} := {ν \in \partial B (0, 1) : B (z_{0} - r_{0} ν, r_{0}) \cap ((0, b) \times R) \subseteq Ω_{h_{0}}}

is nonempty.

In the next theorem we prove that $h_{0}$ admits a left and right derivative at all but countably many points.

Theorem 5 (Left and Right Derivatives of

h

).

Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). Then $\overline{Γ}_{h_{0}}$ admits a left and a right tangent at every point $z$ not of the form $z = (x, h_{0} (x))$ with $x \in S$ , where

S := {x \in (0, b) : h_{0} (x) < t \to x lim inf h_{0} (t)} .

Define

Γ_{cusps} := {z \in \overline{Γ}_{h_{0}} : \pm e_{1} \in N_{z}}

and

Γ_{cuts} := {(x, y) : x \in (0, b) \cap S, h_{0} (x) \leq y \leq t \to x lim inf h_{0} (t)},

where $N_{z}$ is the set defined in (⁠25⁠) and $S$ is the set defined in (⁠26⁠).

Theorem 6 (Cusps and Cuts).

Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). Then the sets $Γ_{cusps}$ and $Γ_{cuts}$ contain at most finitely many vertical segments.

Remark 7. If $- e_{1} \in N_{z_{0}}$ , then since $B ((x_{0} + r_{0}, y_{0}), r_{0}) \cap ((0, b) \times R) \subseteq Ω_{h_{0}}$ and $h_{0}$ is lower semicontinuous, for all $x > x_{0}$ sufficiently close to $x_{0}$ , we have that

h_{0} (x) \geq y_{0} + r_{0}^{2} - (x - (x_{0} + r_{0}))^{2},

and so

\frac{h _{0} ( x ) - y _{0}}{x - x _{0}} \geq \frac{2 r _{0} - ( x - x _{0} )}{x - x _{0}} \to \infty

as $x \to x_{0}^{+}$ . By Theorem ⁠5⁠ it follows that $\overline{Γ}_{h_{0}}$ admits a right vertical tangent at $z_{0}$ . Similarly, if $e_{1} \in N_{z_{0}}$ then for $x < x_{0}$ , $\overline{Γ}_{h_{0}}$ admits a left vertical tangent at $z_{0}$ . In particular, if $\pm e_{1} \in N_{z_{0}}$ and $h_{0}$ is continuous at $x_{0}$ , then

(h_{0})_{-}^{'} (x_{0}) = - \infty, (h_{0})_{+}^{'} (x_{0}) = \infty.

The next theorem shows that, except for cut and cusp points, $\overline{Γ}_{h_{0}}$ is locally Lipschitz.

Theorem 8.

Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). If $z_{0} \in \overline{Γ}_{h_{0}} ∖ (Γ_{cuts} \cup Γ_{cusps})$ , then $\overline{Γ}_{h_{0}}$ is Lipschitz in a neighborhood of $z_{0}$ .

In order to improve the regularity results for $h$ , we restrict our attention to the linearly isotropic case in which

W (E) = \frac{1}{2} λ [tr (E)]^{2} + μ tr (E^{2}),

where $λ$ and $μ$ are the (constant) Lamé moduli with

μ > 0, μ + λ > 0.

Note that in this range, the quadratic form $W$ is coercive. We also assume that the matrix $\hat{E}_{0}$ in (⁠10⁠) takes the form

\hat{E}_{0} = (e_{0} 0 00)

for some $e_{0} > 0$ , which measures the mismatch between the lattices of the two materials.

Since $h_{0}$ is now Lipschitz with left and right derivatives at all but, at most, a finite number of points, we can now obtain classical decay estimates for the solution $u_{0}$ . In turn, these will exclude corners in the graph $Γ_{h_{0}}$ of $h_{0}$ .

Theorem 9 (Decay Estimate).

Assume (⁠30⁠) and (⁠32⁠). Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). Suppose that $\overline{Γ}_{h_{0}}$ has a corner at some point $z_{0} \in \overline{Γ}_{h_{0}} ∖ (Γ_{cusps} \cup Γ_{cuts})$ . Then there exist a constant $c > 0$ , a radius $r_{0}$ , and an exponent $\frac{1}{2} < α < 1$ such that

\int_{B (z_{0}, r) \cap Ω_{h_{0}}} ∣ \nabla u_{0} ∣^{2} d x \leq c r^{2 α}

for all $0 < r < r_{0} .$

Using the previous decay estimate, it can be shown that for $(u_{0}, Ω) \in X$ the upper boundary $\overline{Γ}_{h_{0}}$ is of class $C^{1}$ away from $Γ_{cusps} \cup Γ_{cuts}$ .

Theorem 10 (

C^{\infty}

Regularity of

Γ

).

Assume (⁠30⁠) and (⁠32⁠). Let $(u_{0}, h_{0}) \in X_{0}$ be a minimizer of the functional $G$ defined in (⁠17⁠). Then $\overline{Γ}_{h_{0}} ∖ (Γ_{cusps} \cup Γ_{cuts})$ is of class $C^{1}$ .

Theorem ⁠10⁠ can be significantly improved. Indeed, using another blow-up argument it is possible to show that $\overline{Γ}_{h_{0}} ∖ (Γ_{cusps} \cup Γ_{cuts})$ is of class $C^{1, α}$ for all $0 < α < \frac{1}{2}$ . In turn, this implies that $u_{0}$ is of class $C^{1, β}$ for some $β > 0$ away from the $x$ -axis and from $Γ_{cusps} \cup Γ_{cuts}$ . By a classical bootstrap argument, one can obtain $C^{\infty}$ regularity and then use results of [⁠34⁠ 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, 1051–1076 (2005) ] by Koch, Morini and the second author to prove analyticity of $\overline{Γ}_{h_{0}} ∖ (Γ_{cusps} \cup Γ_{cuts})$ away from the $x$ -axis. We refer to [⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) ] for more details.

2 Evolution of epitaxially strained elastic films: The $2$ D case

The evolution of epitaxially strained elastic films depends strongly on the possible anisotropy of the surface energy density. For this reason, in (⁠17⁠) we replace the isotropic surface energy $γ_{film} length Γ_{h}$ by

\int_{Γ_{h}} ψ (ν) d H^{1},

where $ψ : R^{2} \to [0, \infty)$ is a positively one-homogeneous function of class $C^{2}$ away from the origin. Also, the mismatch between the substrate and film crystalline structures is represented by the Dirichlet condition (see (⁠32⁠))

u (x, 0) = (e_{0} x, 0) for all x \in (0, b) .

As discussed in the introduction, strong anisotropy of $ψ$ may lead to the ill-posedness of the evolution law, and thus we add a higher order regularizing term. To be precise, for $ε > 0$ small the energy under study becomes

I (u, h) := \int_{Ω_{h}} W (E (u)) d x + \int_{Γ_{h}} (ψ (ν) + \frac{ε}{2} k^{2}) d H^{1},

where $k$ denotes the curvature of $Γ_{h}$ and $ν$ is the outer unit normal to $Ω_{h}$ .

We consider periodicity conditions. Hence, given a positive $b$ -periodic function $h : R \to [0, + \infty)$ , with locally finite pointwise variation, we set

Ω_{h}^{#} := {x = (x, y) : x \in R, 0 < y < h (x)},

and

Γ_{h}^{#} := {x = (x, y) : x \in R, y = h (x)} .

Given $h \in W_{♯}^{2, 2} ((0, b); R^{2})$ , where $W_{♯}^{2, 2} ((0, b); R^{2})$ is the space of $b$ periodic functions in $W_{loc}^{2, 2} (R; R^{2})$ , we denote

X_{#} (Ω_{h}; R^{2}) := {u \in L_{loc}^{2} (Ω_{h}^{#}; R^{2}) : u (x, y) = u (x + b, y) for (x, y) \in Ω_{h}^{#}, E (u) ∣_{Ω_{h}} \in L^{2} (Ω_{h}; R^{2})}

and

X_{e_{0}} := {(u, h) : h \in W_{♯}^{2, 2} ((0, b); R^{2}), u \in e_{0} (\cdot, 0) + L D_{#} (Ω_{h}; R^{2}), and u (x, 0) = (e_{0} x, 0) for all x \in R} .

We next introduce the incremental minimum problems used to define the discrete time evolutions. This will lead to the existence of solutions for the evolution equation (⁠40⁠) below via minimizing movements. Let $(u_{0}, h_{0}) \in X_{e_{0}}$ be such that

h_{0} > 0,

and $u_{0}$ minimizes the elastic energy in $Ω_{h_{0}}$ among all $u$ with $(u, h_{0}) \in X_{e_{0}}$ . Given $T > 0$ , $N \in N$ , we set $Δ T := \frac{T}{N}$ . For $i = 1, \dots, N$ , we define inductively $(u_{i, N}, h_{i, N})$ as a solution of the minimum problem

min {I (u, h) + \frac{1}{2Δ T} \int_{Γ_{h_{i - 1, N}}} (\int_{0}^{x} (h (ζ) - h_{i - 1, N} (ζ)) d ζ)^{2} d H^{1} (x, y) : (u, h) \in X_{e_{0}}, ∥ h^{'} ∥_{\infty} \leq Λ_{0}, \int_{0}^{b} h d x = \int_{0}^{b} h_{0} d x, \int_{Γ_{h_{i - 1, N}}} \int_{0}^{x} (h (ζ) - h_{i - 1, N} (ζ)) d ζ d H^{1} (x, y) = 0},

where $∥ h_{0}^{'} ∥_{\infty} < Λ_{0}$ .

Then for $x \in R$ and $(i - 1) Δ T \leq t \leq i Δ T$ , $i = 1, \dots, N$ , we set

h_{N} (x, t) := h_{i - 1, N} (x) + \frac{1}{Δ T} (t - (i - 1) Δ T) (h_{i, N} (x) - h_{i - 1, N} (x)),

and we let $u_{N} (\cdot, t)$ be the elastic equilibrium corresponding to $h_{N} (\cdot, t)$ , i.e., the minimizer of the elastic energy in $Ω_{h_{N} (\cdot, t)}$ among all $u$ such that $(h_{N} (\cdot, t), u) \in X_{e_{0}}$ .

We remark the incremental minimum problem can be written as

min {I (u, h) + \frac{1}{2Δ T} \frac{h - h _{i - 1, N}}{1 + h _{i - 1, N}^{'2}}_{H^{- 1} (Γ_{i - 1, N})}^{2} : (u, h) \in X_{e_{0}}, ∥ h^{'} ∥_{\infty} \leq Λ_{0}, \int_{0}^{b} h d x = \int_{0}^{b} h_{0} d x, \int_{Γ_{h_{i - 1, N}}} \int_{0}^{x} (h (ζ) - h_{i - 1, N} (ζ)) d ζ d H^{1} (x, y) = 0} .

We now show that the incremental minimum problem (⁠36⁠) admits a solution.

Theorem 11.

For every $i = 1, \dots, N$ , the minimum problem (⁠36⁠) admits a solution $(u_{i, N}, h_{i, N}) \in X_{e_{0}}$ .

Proof. Let ${(u_{n}, h_{n})} \subset X_{e_{0}}$ be a minimizing sequence for (⁠36⁠). Since $\int_{0}^{b} h_{n} d x = \int_{0}^{b} h_{0} d x$ ,

n sup \int_{0}^{b} \frac{( h _{n}^{''} ) ^{2}}{1 + ( h _{n}^{'} ) ^{2}} d x < \infty

and $∥ h_{n}^{'} ∥_{\infty} \leq Λ_{0}$ , it follows that $∥ h_{n} ∥_{W^{2, 2}} \leq C$ for some constant $C > 0$ and for all $n$ . Then, up to a subsequence (not relabelled), we may assume that $h_{n} ⇀ h$ weakly in $W_{♯}^{2, 2} ((0, b); R^{2})$ , and thus strongly in $C^{1} (R; R^{2})$ . As a consequence,

\int_{Γ_{h}} (ψ (ν) + \frac{ε}{2} k^{2}) d H^{1} \leq n \to \infty lim inf \int_{Γ_{h_{n}}} (ψ (ν) + \frac{ε}{2} k_{n}^{2}) d H^{1},

and

\int_{Γ_{h_{i - 1, N}}} (\int_{0}^{x} (h (ζ) - h_{i - 1, N} (ζ)) d ζ)^{2} d H^{1} = n \to \infty lim \int_{Γ_{h_{i - 1, N}}} (\int_{0}^{x} (h_{n} (ζ) - h_{i - 1, N} (ζ)) d ζ)^{2} d H^{1} .

Finally, since $sup_{n} \int_{Ω_{h_{n}}} ∣ E (u_{n}) ∣^{2} d x < \infty$ , reasoning as in [⁠20⁠ I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007) , Proposition 2.2], from the $C^{1}$ convergence of ${h_{n}}$ to $h$ and Korn’s inequality we deduce that there exists $u \in H_{loc}^{1} (Ω_{h}^{#}; R^{2})$ such that $(u, h) \in X_{e_{0}}$ and, up to a subsequence, $u_{n} ⇀ u$ weakly in $H_{loc}^{1} (Ω_{h}^{#}; R^{2})$ . Therefore, we have that

\int_{Ω_{h}} W (E (u)) d x \leq n \to \infty lim inf \int_{Ω_{h_{n}}} W (E (u_{n})) d x,

which, together with (⁠38⁠) and (⁠39⁠), allows us to conclude that $(u, h)$ is a minimizer. ∎

Next, we show that solutions of the discrete time evolution problems converge to a function $h = h (x, t)$ , which is a weak solution of the following geometric evolution equation,

\frac{\partial h}{\partial t} = [\frac{1}{J} (ε (\frac{h _{xx}}{J ^{5}})_{xx} + \frac{5 ε}{2} (\frac{h _{xx}^{2}}{J ^{7}} h_{x})_{x} + (ψ_{x} (- h_{x}, 1))_{x} + W (E (u)))_{x}]_{x}

for a short time interval $[0, T_{0}]$ , where $0 < T_{0} \leq T$ , where $T_{0}$ depends on $(u_{0}, h_{0})$ . Here $J := 1 + (h_{x})^{2}$ . Since $∥ h_{0}^{'} ∥_{\infty} < Λ_{0}$ , for all $t$ sufficiently small we have that $∥ \frac{\partial h}{\partial t} ∥_{\infty} < Λ_{0}$ , and so we are allowed to take admissible variations of $h$ to obtain (⁠40⁠).

Theorem 12.

There exist $T_{0} \in (0, T]$ and $C > 0$ depending only $(h_{0}, u_{0})$ such that:

$h_{N} \to h$ in $C^{0, β} ([0, T_{0}]; C^{1, α} ([0, b]))$ for every $α \in (0, \frac{1}{2})$ , and $β \in (0, (1 - 2 α) /32)$ ,
$E (u_{N} (\cdot, h_{N})) \to E (u (\cdot, h))$ in $C^{0, β} ([0, T_{0}]; C^{1, α} ([0, b]))$ for every $α \in (0, \frac{1}{2})$ , and $0 \leq β < (1 - 2 α) /32$ , where $u (\cdot, t)$ is the elastic equilibrium in $Ω_{h (\cdot, t)}$ ,

and $h$ is a weak solution to (⁠40⁠) with initial data $h_{0}$ . Moreover, if $ψ \in C^{3} (R^{2} ∖ {0})$ then $h (\cdot, t) \in H_{#}^{5} (0, b)$ for almost every $t \in [0, T_{0}]$ , and $h$ is the unique solution.

For linearly isotropic energy densities of the form (⁠30⁠), where $λ$ and $μ$ satisfy (⁠31⁠), and for sufficiently regular surface energy densities, we can prove asymptotic stability of the flat configuration $h_{flat} \equiv d / b$ when $d$ is sufficiently small. Consider the Grinfeld function $K$ defined by

K (y) := n \in N max \frac{1}{n} J (n y), y \geq 0,

where

J (y) := \frac{y + ( 3 - 4 ν _{p} ) sinh y cosh y}{4 ( 1 - ν _{p} ) ^{2} + y ^{2} + ( 3 - 4 ν _{p} ) sinh ^{2} y},

and $ν_{p}$ is the Poisson modulus of the elastic material, i.e.,

ν_{p} := \frac{λ}{2 ( λ + μ )} .

It turns out that $K$ is strictly increasing and continuous, $K (y) \leq C y$ , and $lim_{y \to + \infty} K (y) = 1$ , for some positive constant $C$ .

Theorem 13.

Assume that $W$ takes the form (⁠30⁠), where $λ$ and $μ$ satisfy (⁠31⁠), and that $ψ \in C^{3} (R^{2} ∖ {0})$ satisfies $\partial_{11}^{2} ψ (0, 1) > 0$ and

D^{2} ψ (ξ) [τ, τ] > 0 for all τ ⊥ ξ, τ \neq = 0

for every $ξ \in S^{1}$ . Let $d_{loc} : (0, \infty) \to (0, \infty]$ be defined as $d_{loc} (b) := \infty$ if $0 < b \leq \frac{π}{4} \frac{( 2 μ + λ ) \partial _{11}^{2} ψ ( 0 , 1 )}{e _{0}^{2} μ ( μ + λ )}$ , and as the solution to

K (\frac{2 π d _{loc} ( b )}{b}) = \frac{π}{4} \frac{( 2 μ + λ ) \partial _{11}^{2} ψ ( 0 , 1 )}{e _{0}^{2} μ ( μ + λ )} \frac{1}{b}

otherwise. Then, for all $d \in (0, d_{loc} (b))$ the flat configuration $h_{flat} \equiv d / b$ is asymptotically stable, that is, there exists $δ > 0$ such that if $h_{0} \in W_{♯}^{2, 2} ((0, b); R^{2})$ with $\int_{0}^{b} h_{0} d x = d$ and $∥ h_{0} - h_{flat} ∥_{W^{2, 2}} \leq δ$ , then the solution $h$ to (⁠40⁠) with initial datum $h_{0}$ exists for all times and

∥ h (\cdot, t) - h_{flat} ∥_{W^{2, 2}} \to 0

as $t \to \infty$ .

Acknowledgements The research of I. Fonseca was partially funded by the National Science Foundation under Grants No. DMS-1411646 and DMS-1906238, and the one of G. Leoni by DMS-1714098.

Irene Fonseca’s main contributions have been on the variational study of ferroelectric and magnetic materials, composites, thin structures, phase transitions, and on the mathematical analysis of image segmentation, denoising, detexturing, registration and recolorization in computer vision. She continues to explore modern methods in the calculus of variations motivated by problems issuing from materials science and imaging science. She is a Fellow of the American Mathematical Society, and of the Society for Industrial and Applied Mathematics. She was SIAM President in 2013 and 2014. She is a Grand Officer of the “Ordem Militar de Sant’Iago da Espada” (a Portuguese decoration). fonseca@andrew.cmu.edu Giovanni Leoni is a professor of mathematics at Carnegie Mellon University, His areas of expertise are the calculus of variations, geometric measure theory, and partial differential equations with applications to engineering, materials science, and mechanics. He is the recipient of the award “Premio Giuseppe Bartolozzi” of the Italian Mathematical Society for best Italian mathematician under 34, 2001, and of the Julius Ashkin Teaching Award and of the Richard Moore Award of the Mellon College of Sciences. He wrote two books Modern methods in the calculus of variations: $L^{p}$ spaces (with I. Fonseca) and A first course in Sobolev spaces. giovanni@andrew.cmu.edu

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M. Caroccia, R. Cristoferi and L. Dietrich, Equilibria configurations for epitaxial crystal growth with adatoms. Arch. Ration. Mech. Anal.230, 785–838 (2018)
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A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal.39, 77–102 (2007)
V. Crismale and M. Friedrich, Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity. Arch. Ration. Mech. Anal.237, 1041–1098 (2020)
G. Dal Maso, Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS)15, 1943–1997 (2013)
E. Davoli and P. Piovano, Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019)
E. Davoli and P. Piovano, Derivation of a heteroepitaxial thin-film model. Interfaces Free Bound.22, 1–26 (2020)
B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films. In Topics in modern regularity theory, CRM Series 13, Ed. Norm., Pisa, 169–204 (2012)
A. Di Carlo, M. E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math.52, 1111–1119 (1992)
I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. (9)96, 591–639 (2011)
I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007)
I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ration. Mech. Anal.205, 425–466 (2012)
I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE8, 373–423 (2015)
I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films. J. Math. Pures Appl. (9)111, 126–160 (2018)
I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth. Nonlinear Anal.154, 88–121 (2017)
I. Fonseca, A. Pratelli and B. Zwicknagl, Shapes of epitaxially grown quantum dots. Arch. Ration. Mech. Anal.214, 359–401 (2014)
N. Fusco, Equilibrium configurations of epitaxially strained thin films. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.21, 341–348 (2010)
N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in the plane. Comm. Math. Phys.362, 571–607 (2018)
N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in three dimensions. Arch. Ration. Mech. Anal.237, 1325–1382 (2020)
N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012)
M. Goldman and B. Zwicknagl, Scaling law and reduced models for epitaxially strained crystalline films. SIAM J. Math. Anal.46, 1–24 (2014)
M. E. Gurtin and M. E. Jabbour, 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)
C. Herring, Some theorems on the free energies of crystal surfaces. Phys. Rev.82, 87 (1951)
S. Y. Kholmatov and P. Piovano, A unified model for stress-driven rearrangement instabilities. Arch. Ration. Mech. Anal.238, 415–488 (2020)
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, 1051–1076 (2005)
L. C. Kreutz and P. Piovano, Microscopic validation of a variational model of epitaxially strained crystalline films. SIAM J. Math. Anal.53, 453–490 (2021)
R. V. Kukta and L. B. Freund, Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate. J. Mech. Phys. Solids45, 1835–1860 (1997)
P. Piovano, Evolution of elastic thin films with curvature regularization via minimizing movements. Calc. Var. Partial Differ. Equ.49, 337–367 (2014)
P. Piovano and I. Velčić, Microscopical justification of solid-state wetting and dewetting. arXiv:2010.08787 (2020)
A. Rätz, A. Ribalta and A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys.214, 187–208 (2006)
M. Siegel, M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004)
B. J. Spencer, Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski–Krastanow islands. Phys. Rev. B, 59, 2011 (1999)
B. J. Spencer and J. Tersoff, Equilibrium shapes and properties of epitaxially strained islands. Phys. Rev. Lett.79, 4858 (1997)

Cite this article

Irene Fonseca, Giovanni Leoni, Surface evolution of elastically stressed films. Eur. Math. Soc. Mag. 119 (2021), pp. 31–39

DOI 10.4171/MAG/6

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[bib-bib15] E. Davoli and P. Piovano, Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci.29, 2183–2223 (2019)

[bib-bib16] E. Davoli and P. Piovano, Derivation of a heteroepitaxial thin-film model. Interfaces Free Bound.22, 1–26 (2020)

[bib-bib17] B. De Maria and N. Fusco, Regularity properties of equilibrium configurations of epitaxially strained elastic films. In Topics in modern regularity theory, CRM Series 13, Ed. Norm., Pisa, 169–204 (2012)

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[bib-bib19] I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. (9)96, 591–639 (2011)

[bib-bib20] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal.186, 477–537 (2007)

[bib-bib21] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ration. Mech. Anal.205, 425–466 (2012)

[bib-bib22] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE8, 373–423 (2015)

[bib-bib23] I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films. J. Math. Pures Appl. (9)111, 126–160 (2018)

[bib-bib24] I. Fonseca, G. Leoni and M. Morini, Equilibria and dislocations in epitaxial growth. Nonlinear Anal.154, 88–121 (2017)

[bib-bib25] I. Fonseca, A. Pratelli and B. Zwicknagl, Shapes of epitaxially grown quantum dots. Arch. Ration. Mech. Anal.214, 359–401 (2014)

[bib-bib26] N. Fusco, Equilibrium configurations of epitaxially strained thin films. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.21, 341–348 (2010)

[bib-bib27] N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in the plane. Comm. Math. Phys.362, 571–607 (2018)

[bib-bib28] N. Fusco, V. Julin and M. Morini, The surface diffusion flow with elasticity in three dimensions. Arch. Ration. Mech. Anal.237, 1325–1382 (2020)

[bib-bib29] N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal.203, 247–327 (2012)

[bib-bib30] M. Goldman and B. Zwicknagl, Scaling law and reduced models for epitaxially strained crystalline films. SIAM J. Math. Anal.46, 1–24 (2014)

[bib-bib31] M. E. Gurtin and M. E. Jabbour, 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)

[bib-bib32] C. Herring, Some theorems on the free energies of crystal surfaces. Phys. Rev.82, 87 (1951)

[bib-bib33] S. Y. Kholmatov and P. Piovano, A unified model for stress-driven rearrangement instabilities. Arch. Ration. Mech. Anal.238, 415–488 (2020)

[bib-bib34] 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, 1051–1076 (2005)

[bib-bib35] L. C. Kreutz and P. Piovano, Microscopic validation of a variational model of epitaxially strained crystalline films. SIAM J. Math. Anal.53, 453–490 (2021)

[bib-bib36] R. V. Kukta and L. B. Freund, Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate. J. Mech. Phys. Solids45, 1835–1860 (1997)

[bib-bib37] P. Piovano, Evolution of elastic thin films with curvature regularization via minimizing movements. Calc. Var. Partial Differ. Equ.49, 337–367 (2014)

[bib-bib38] P. Piovano and I. Velčić, Microscopical justification of solid-state wetting and dewetting. arXiv:2010.08787 (2020)

[bib-bib39] A. Rätz, A. Ribalta and A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys.214, 187–208 (2006)

[bib-bib40] M. Siegel, M. J. Miksis and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353 (2004)

[bib-bib41] B. J. Spencer, Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski–Krastanow islands. Phys. Rev. B, 59, 2011 (1999)

[bib-bib42] B. J. Spencer and J. Tersoff, Equilibrium shapes and properties of epitaxially strained islands. Phys. Rev. Lett.79, 4858 (1997)

1 2D quasistatic equilibrium of epitaxially strained elastic films

2 Evolution of epitaxially strained elastic films: The 2D case

References

Cite this article

2 Evolution of epitaxially strained elastic films: The $2$ D case