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_{\Omega_{h}}W\bigl(\bm{E}(\bm{u})\bigr)\,d\bm{x}+\int_{\Gamma_{h}}\psi(\bm{\nu})\,d{\mathcal{H}}^{2}.

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

\Omega_{h}:=\bigl\{(x,y,z)\in Q\times\mathbb{R}:\,0<z<h(x,y)\bigr\},

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

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

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

\frac{1}{c}|\bm{\xi}|\leq\psi(\bm{\xi})\leq c|\bm{\xi}|\qquad\text{for all $\bm{\xi}\in\mathbb{R}^{3}$},

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 $\bm{u}\equiv\bm{0}$ , while a natural state for the film is given by $\bm{u}\equiv\bm{A}_{0}\bm{x}$ for some nonzero $3\times 3$ matrix $\bm{A}_{0}$ . Our models will reflect this mismatch, either by setting the elastic bulk energy as $\int_{\Omega_{h}}W(\bm{E}(\bm{u})(\bm{x})-\bm{E}_{0}(\bm{x}))\,d\bm{x}$ , where

\bm{E}_{0}(\bm{x}):=\begin{cases}\frac{\bm{A}_{0}+\bm{A}_{0}^{T}}{2}&\text{if }z>0,\\ \bm{0}&\text{if }z\leq 0,\end{cases}

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

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 $\bm{E}(\bm{u})$ 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 $GSBD$ , 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 $\mu$ . To be precise, according to the Einstein–Nernst relation, the evolution is governed by the volume preserving equation

V=C\Delta_{{}_{\Gamma}}\mu\,,

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

V=\Delta_{\Gamma}\bigl[\operatorname{div}_{\Gamma}\bigl(D\psi(\bm{\nu})\bigr)+W\bigl(\bm{E}(\bm{u})\bigr)\bigr]\,,

where $\operatorname{div}_{\Gamma}$ stands for the tangential divergence along $\Gamma_{h(\cdot,t)}$ , and $\bm{u}(\cdot,t)$ is the elastic equilibrium in $\Omega_{h(\cdot,t)}$ , i.e., the minimizer of the elastic energy under the prescribed periodicity and boundary conditions (see (⁠7⁠) below).

If the surface energy density $\psi$ is highly anisotropic, there may be directions $\bm{\nu}$ for which

D^{2}\psi(\bm{\nu})[\bm{\tau},\bm{\tau}]>0\qquad\text{for all $\bm{\tau}\perp\bm{\nu},~{}\bm{\tau}\neq$}\bm{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_{\Gamma_{h}}\Bigl(\psi(\bm{\nu})+\frac{\varepsilon}{p}|H|^{p}\Bigr)\,d\mathcal{H}^{2},

where $p>2$ , $H$ stands for the sum $\kappa_{1}+\kappa_{2}$ of the principal curvatures of $\Gamma_{h}$ , and $\varepsilon$ 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 $\mathbb{R}^{3}$ is motivated by the fact that the profile $h$ of the film will belong to $W^{2,p}(Q)$ , where $Q\subset\mathbb{R}^{2}$ , 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 $\mathbb{R}^{2}$ 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_{\Omega_{h}}W\bigl(\bm{E}(\bm{u})\bigr)\,d\bm{x}+\int_{\Gamma_{h}}\Bigl(\psi(\bm{\nu})+\frac{\varepsilon}{p}|H|^{p}\Bigr)\,d{\mathcal{H}}^{2},

and (⁠3⁠) is replaced by

V=\Delta_{\Gamma}\biggl[\operatorname{div}_{\Gamma}\bigl(D\psi(\bm{\nu})\bigr)+W\bigl(\bm{E}(\bm{u})\bigr)\\ -\varepsilon\biggl(\Delta_{\Gamma}(|H|^{p-2}H)-|H|^{p-2}H\Bigl(\kappa_{1}^{2}+\kappa_{2}^{2}-\frac{1}{p}H^{2}\Bigr)\biggr)\biggr].

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

\begin{cases}\displaystyle\frac{1}{J}\frac{\partial h}{\partial t}=\Delta_{\Gamma}\left[\operatorname{div}_{\Gamma}\bigl(D\psi(\bm{\nu})\bigr)+W\bigl(\bm{E}(\bm{u})\bigr)\vphantom{\frac{1}{p}}\right.\\ \displaystyle\qquad\quad\left.-\varepsilon\biggl(\Delta_{\Gamma}\bigl(|H|^{p-2}H\bigr)-|H|^{p-2}H\Bigl(\kappa_{1}^{2}+\kappa_{2}^{2}-\frac{1}{p}H^{2}\Bigr)\biggr)\right]\\ \hskip 142.26378pt\text{ in $\mathbb{R}^{2}\times(0,T_{0})$,}\\ \operatorname{div}\mathbb{C}\bm{E}(\bm{u})=0~{}\text{ in $\Omega_{h}$},\\[4.3pt] \mathbb{C}\bm{E}(\bm{u})[\bm{\nu}]=0~{}\text{ on $\Gamma_{h}$,}\quad\bm{u}(x,y,0,t)=\bm{A}_{0}(x,y,0),\\[4.3pt] h(\cdot,t)~{}\text{ and }~{}D\bm{u}(\cdot,t)~{}\text{ are $Q$-periodic,}\\[4.3pt] h(\cdot,0)=h_{0},\end{cases}

where $J:=\sqrt{1+|Dh|^{2}}$ and $h_{0}\in H_{\operatorname*{loc}}^{2}(\mathbb{R}^{2})$ 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

\Omega_{h}:=\bigl\{\bm{x}=(x,y):\,0<x<b,\,y<h(x)\bigr\},

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

\Omega_{h}^{+}:=\Omega_{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

\bm{E}_{0}(y)=\begin{cases}\bm{\hat{E}}_{0}&\text{if }y\geq 0,\\ 0&\text{if }y<0,\end{cases}

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

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

for all symmetric matrices $\bm{E}\neq\bm{0}$ .

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

\psi\left(y\right):=\begin{cases}\gamma_{\operatorname*{film}}&\text{if }y>0,\\ \gamma_{\operatorname*{sub}}&\text{if }y=0,\end{cases}

where the property

\gamma_{\operatorname*{sub}}\geq\gamma_{\operatorname*{film}}>0

will favor the SK growth mode over the VW mode. For the case $\gamma_{\operatorname*{sub}}<\gamma_{\operatorname*{film}}$ , and for different crystalline materials stress tensors $\mathbb{C}$ 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

\mathcal{F}(\bm{u},h):=\int_{\Omega_{h}}W\bigl(\bm{E}(\bm{u})-\bm{E}_{0}\bigr)\,d\bm{x}+\int_{\Gamma_{h}}\psi\,ds,

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

\Gamma_{h}:=\partial\Omega_{h}\cap\bigl((0,b)\times\mathbb{R}\bigr).

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

X:={\biggl\{}(\bm{u},h):~{}h:[0,b]\rightarrow[0,\infty)\text{ Lipschitz,}\\ \int_{0}^{b}h~{}dx=d,\quad\bm{u}\in H_{\operatorname*{loc}}^{1}\bigl(\Omega_{h};\mathbb{R}^{2}\bigr){\biggr\}}

and

X_{0}={\biggl\{}(\bm{u},h):~{}h:[0,b]\rightarrow[0,\infty)\text{ lower semicontinuous,}\\ \operatorname*{var}\nolimits_{[0,b]}h<\infty,\quad\int_{0}^{b}h~{}dx=d,\quad\bm{u}\in H_{\operatorname*{loc}}^{1}(\Omega_{h};\mathbb{R}^{2}){\biggr\}},

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

\operatorname{var}\nolimits_{[0,b]}h\leq\operatorname*{length}\Gamma_{h}\leq b+\operatorname{var}\nolimits_{[0,b]}h.

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

\mathcal{G}(\bm{u},h):=\int_{\Omega_{h}}W\bigl(\bm{E}(\bm{u})(\bm{x})-\bm{E}_{0}(y)\bigr)\,d\bm{x}+\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{h}.

Theorem 1 (Existence).

The following equalities hold:

\inf_{(\bm{u},h)\in X}\mathcal{F}(\bm{u},h)=\inf_{(\bm{u},h)\in X}\mathcal{G}(\bm{u},h)=\min_{(\bm{u},h)\in X_{0}}\mathcal{G}(\bm{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 $\mathcal{G}$ 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 $\int_{0}^{b}h(x)~{}dx=d$ can be replaced by a volume penalization.

Theorem 2 (Volume penalization).

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

\mathcal{G}_{k}(\bm{u},h):=\int_{\Omega_{h}}\!\!W\bigl(\bm{E}(\bm{u})-\bm{E}_{0}\bigr)\,d\bm{x}+\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{h}+k\left|\int_{0}^{b}h~{}dx-d\right|

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

Proof. An argument similar to that of the proof of Theorem ⁠1⁠ guarantees that for every $k\in\mathbb{N}$ there exists a minimizer $(\bm{v}_{k},f_{k})$ of $\mathcal{G}_{k}$ . If $\int_{0}^{b}f_{k}~{}dx=d$ for all $k$ sufficiently large, then

\mathcal{G}(\bm{u}_{0},h_{0})\leq\mathcal{G}(\bm{v}_{k},f_{k})=\mathcal{G}_{k}(\bm{v}_{k},f_{k})\leq\mathcal{G}_{k}(\bm{u}_{0},h_{0})=\mathcal{G}(\bm{u}_{0},h_{0})<\infty,

and so $(\bm{u}_{0},h_{0})$ is a minimizer of $\mathcal{G}_{k}$ .

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

\int_{0}^{b}f_{k}~{}dx>d

for countably many $k$ , define

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

where $t_{k}>0$ has been chosen so that $\int_{0}^{b}h_{k}~{}dx=d$ . Note that $\operatorname*{length}\Gamma_{h_{k}}\leq\operatorname*{length}\Gamma_{f_{k}}$ . Indeed, for every partition $x_{0}=0<\cdots<x_{n}=b$ , we have that

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

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

\mathcal{G}(\bm{v}_{k},h_{k})=\mathcal{G}_{k}(\bm{v}_{k},h_{k})<\mathcal{G}_{k}(\bm{v}_{k},f_{k}),

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

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

Since

\mathcal{G}_{k}(\bm{v}_{k},f_{k})\leq\mathcal{G}_{k}(\bm{u}_{0},h_{0})=\mathcal{G}(\bm{u}_{0},h_{0})<\infty,

it follows from (⁠18⁠) and (⁠20⁠) that $\int_{0}^{b}f_{k}~{}dx\rightarrow d$ as $k\rightarrow\infty$ and that $\sup_{k}\operatorname*{length}\Gamma_{f_{k}}<\infty$ . In turn, by (⁠16⁠), $\|f_{k}\|_{\infty}\leq c$ for some constant $c$ independent of $k$ .

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

t_{k}:=\frac{d}{\int_{0}^{b}f_{k}~{}dx}\in(0,2)

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

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

Consider a partition $0=x_{0}<\cdots<x_{\ell}=b$ . Then

\displaystyle\sum_{i=1}^{\ell}\sqrt{(x_{i}-x_{i-1})^{2}+\bigl(h_{k}(x_{i})-h_{k}(x_{i-1})\bigr)^{2}}

\displaystyle\hskip 40.0pt=\sum_{i=1}^{\ell}\sqrt{(x_{i}-x_{i-1})^{2}+t_{k}^{2}\bigl(f_{k}(x_{i})-f_{k}(x_{i-1})\bigr)^{2}}

\displaystyle\hskip 40.0pt\leq t_{k}\sum_{i=1}^{\ell}\sqrt{(x_{i}-x_{i-1})^{2}+\bigl(f_{k}(x_{i})-f_{k}(x_{i-1})\bigr)^{2}}

\displaystyle\hskip 40.0pt\leq t_{k}\operatorname*{length}\Gamma_{f_{k}},

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

\operatorname*{length}\Gamma_{h_{k}}\leq t_{k}\operatorname*{length}\Gamma_{f_{k}},

and so, by (⁠20⁠),

\displaystyle\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{h_{k}}-\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{f_{k}}

\displaystyle\hskip 60.0pt\leq(t_{k}-1)\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{f_{k}}\leq(t_{k}-1)\mathcal{G}_{k}(\bm{v}_{k},f_{k})

\displaystyle\hskip 60.0pt\leq(t_{k}-1)\mathcal{G}(\bm{u}_{0},h_{0}).

We deduce that

\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{h_{k}}\leq\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{f_{k}}+(t_{k}-1)\mathcal{G}(\bm{u}_{0},h_{0}).

For $(x,y^{\prime})\in\Omega_{h_{k}}$ define

\bm{w}_{k}(x,y^{\prime}):=\left((\bm{v}_{k})_{1}\Bigl(x,\frac{y^{\prime}}{t_{k}}\Bigr),\frac{1}{t_{k}}(\bm{v}_{k})_{2}\Bigl(x,\frac{y^{\prime}}{t_{k}}\Bigr)\right).

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

\int_{\Omega_{h_{k}}}W\bigl(\bm{E}(\bm{w}_{k})(x,y^{\prime})-\bm{E}_{0}(y^{\prime})\bigr)~{}dxdy^{\prime}\\[-4.3pt] =\frac{1}{t_{k}}\int_{\Omega_{f_{k}}}W\bigl(\mathbf{\widetilde{\bm{E}}}(\bm{v}_{k})(\bm{x})-\bm{E}_{0}(y)\bigr)~{}d\bm{x},

where $\mathbf{\widetilde{\bm{E}}}(\bm{v}_{k})(\bm{x})$ is the $2\times 2$ matrix whose entries are

\displaystyle\mathbf{\widetilde{\bm{E}}}_{11}(\bm{v}_{k})(\bm{x})=\bm{E}_{11}(\bm{v}_{k})(\bm{x}),\qquad\mathbf{\widetilde{\bm{E}}}_{12}(\bm{v}_{k})(\bm{x})=\frac{1}{t_{k}}\bm{E}_{12}(\bm{v}_{k})(\bm{x}),

\displaystyle\mathbf{\widetilde{\bm{E}}}_{22}(\bm{v}_{k})(\bm{x})=\frac{1}{t_{k}^{2}}\bm{E}_{22}(\bm{v}_{k})(\bm{x})\,.

Observe that

\displaystyle\bigl(|\mathbf{\widetilde{E}}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|\bigr)|\mathbf{\widetilde{E}}(\bm{v}_{k})-\bm{E}(\bm{v}_{k})|

\displaystyle\leq c(t_{k}-1)\bigl(|\mathbf{\widetilde{E}}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|\bigr)|\bm{E}(\bm{v}_{k})|

\displaystyle\leq c(t_{k}-1)\bigl(|\bm{E}(\bm{v}_{k})|+|\bm{E}_{0}|\bigr)(|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}_{0}|)

\displaystyle\leq c(t_{k}-1)\bigl(|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}_{0}|\bigr)^{2}.

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

|W(\bm{E})-W(\bm{E}_{1})|\leq c\left(|\bm{E}|+|\bm{E}_{1}|\right)|\bm{E}-\bm{E}_{1}|

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

\displaystyle\int_{\Omega_{h_{k}}}W\bigl(\bm{E}(\bm{w}_{k})(x,y^{\prime})-\bm{E}_{0}(y^{\prime})\bigr)~{}d\bm{x}\mathbf{{}^{\prime}}-\int_{\Omega_{f_{k}}}\!\!\!W\bigl(\bm{E}(\bm{v}_{k})(\bm{x})-\bm{E}_{0}(y)\bigr)~{}d\bm{x}

\displaystyle\quad=\frac{1}{t_{k}}\int_{\Omega_{f_{k}}}\left[W\bigl(\mathbf{\widetilde{E}}(\bm{v}_{k})(\bm{x})-\bm{E}_{0}(y)\bigr)-W\bigl(\bm{E}(\bm{v}_{k})(\bm{x})-\bm{E}_{0}(y)\bigr)\right]~{}d\bm{x}

\displaystyle\quad\leq c\int_{\Omega_{f_{k}}}\bigl(|\mathbf{\widetilde{E}}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|\bigr)\bigl(|\mathbf{\widetilde{E}}(\bm{v}_{k})-\bm{E}(\bm{v}_{k})|\bigr)~{}d\bm{x}

\displaystyle\quad\leq c(t_{k}-1)\int_{\Omega_{f_{k}}}(|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}_{0}|)^{2}~{}d\bm{x}

\displaystyle\quad\leq c(t_{k}-1)(\mathcal{G}_{k}(\bm{v}_{k},f_{k})+|\bm{\hat{E}}_{0}|^{2})\leq c(t_{k}-1)\bigl(\mathcal{G}(\bm{u}_{0},h_{0})+|\bm{\hat{E}}_{0}|^{2}\bigr),

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

\displaystyle\mathcal{G}(\bm{u}_{0},h_{0})

\displaystyle\quad\leq\mathcal{G}(\bm{w}_{k},h_{k})\leq\mathcal{G}\left(\bm{v}_{k},f_{k}\right)+(t_{k}-1)\left[(c+1)\mathcal{G}\left(\bm{u}_{0},h_{0}\right)+c|\bm{\hat{E}}_{0}|^{2}\right]

\displaystyle\quad=\mathcal{G}_{k}(\bm{v}_{k},f_{k})+(t_{k}-1)\left[(c+1)\mathcal{G}(\bm{u}_{0},h_{0})+c|\bm{\hat{E}}_{0}|^{2}\right]

\displaystyle\hskip 142.26378pt-k\Bigl(d-\int_{0}^{b}f_{k}~{}dx\Bigr)

\displaystyle\quad=\mathcal{G}_{k}(\bm{v}_{k},f_{k})+(t_{k}-1)\left[(c+1)\mathcal{G}(\bm{u}_{0},h_{0})+c|\bm{\hat{E}}_{0}|^{2}\right]

\displaystyle\hskip 142.26378pt-(t_{k}-1)k\int_{0}^{b}f_{k}~{}dx

\displaystyle\quad\leq\mathcal{G}(\bm{u}_{0},h_{0})+(t_{k}-1)\left[(c+1)\mathcal{G}\left(\bm{u}_{0},h_{0}\right)+c|\bm{\hat{E}}_{0}|^{2}-k\frac{d}{2}\right].

Thus, if

k\geq\frac{2}{d}\left[(c+1)\mathcal{G}(\bm{u}_{0},h_{0})+c|\bm{\hat{E}}_{0}|^{2}\right]+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 $(\bm{u}_{0},h_{0})\in X_{0}$ be a minimizer of the functional $\mathcal{G}$ defined in (⁠17⁠). Then there exists $r_{0}>0$ with the property that for every $\bm{z}_{0}\in\overline{\Gamma}_{h_{0}}$ there exists an open ball $B(\bm{x}_{0},r_{0})$ , with $B(\bm{x}_{0},r_{0})\cap((0,b)\times\mathbb{R})\subseteq\Omega_{h_{0}}$ , such that

\partial B(\bm{x}_{0},r_{0})\cap\overline{\Gamma}_{h_{0}}=\{\bm{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 $\bm{\nu}_{0}\in\partial B(\mathbf{0},1)$ is the outward unit normal to $B(\bm{x}_{0},r_{0})$ at $\bm{z}_{0}$ , then $\bm{x}_{0}=\bm{z}_{0}-r_{0}\bm{\nu}_{0}$ . Thus, the set

N_{\bm{z}_{0}}:=\bigl\{\bm{\nu}\in\partial B(\mathbf{0},1):~{}B(\bm{z}_{0}-r_{0}\bm{\nu},r_{0})\cap\bigl((0,b)\times\mathbb{R}\bigr)\subseteq\Omega_{h_{0}}\bigr\}

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 $(\bm{u}_{0},h_{0})\in X_{0}$ be a minimizer of the functional $\mathcal{G}$ defined in (⁠17⁠