EMS Magazine2021 / No. 119
MAG119DOI 10.4171/MAG-6

Surface evolution of elastically stressed films

Irene Fonseca and Giovanni Leoni
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. Solids 45, 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

ΩhW(E(u))dx+Γhψ(ν)dH2.\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[0,)h:Q\rightarrow[0,\infty) is the function whose graph Γh\Gamma_{h} describes the profile of the film, assumed to be QQ-periodic, with Q:=(0,b)2R2Q:=(0,b)^{2}\subset\mathbb{R}^{2}, for some b>0b>0, Ωh\Omega_{h} is the region occupied by the film, i.e., writing x=(x,y,z)\bm{x}=(x,y,z),

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

u:ΩhR3\bm{u}:\Omega_{h}\rightarrow\mathbb{R}^{3} is displacement of the material, E(u):=12(Du+DTu)\bm{E}(\bm{u}):=\frac{1}{2}(D\bm{u}+D^{T}\bm{u}) is the symmetric part of DuD\bm{u}. Also, the elastic energy density W:Msym3×3[0,+)W:\mathbb{M}_{\mathrm{sym}}^{3\times 3}\rightarrow[0,+\infty) is a positive definite quadratic form defined on the space of 3×33\times 3 symmetric matrices

W(A):=12CA:A,W(A):=\frac{1}{2}\mathbb{C}A:A,

with C\mathbb{C} a positive definite fourth-order tensor, so that W(A)>0W(A)>0 for all AMsym3×3{0}A\in\mathbb{M}_{\mathrm{sym}}^{3\times 3}\setminus\{0\}, ψ:R3[0,)\psi:\mathbb{R}^{3}\rightarrow[0,\infty) is an anisotropic surface energy density evaluated at the unit normal ν\bm{\nu} to Γh\Gamma_{h}, and H2\mathcal{H}^{2} denotes the two-dimensional Hausdorff measure. We suppose that ψ\psi is positively one-homogeneous and of class C2C^{2} away from the origin, so that, in particular,

1cξψ(ξ)cξfor all ξR3,\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>0c>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 u0\bm{u}\equiv\bm{0}, while a natural state for the film is given by uA0x\bm{u}\equiv\bm{A}_{0}\bm{x} for some nonzero 3×33\times 3 matrix A0\bm{A}_{0}. Our models will reflect this mismatch, either by setting the elastic bulk energy as ΩhW(E(u)(x)E0(x))dx\int_{\Omega_{h}}W(\bm{E}(\bm{u})(\bm{x})-\bm{E}_{0}(\bm{x}))\,d\bm{x}, where

E0(x):={A0+A0T2if z>0,0if z0,\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 u(x,y,0)A0(x,y,0)\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 E(u)\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 GSBDGSBD, 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\mu. To be precise, according to the Einstein–Nernst relation, the evolution is governed by the volume preserving equation

V=CΔΓμ,V=C\Delta_{{}_{\Gamma}}\mu\,,

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

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

where divΓ\operatorname{div}_{\Gamma} stands for the tangential divergence along Γh(,t)\Gamma_{h(\cdot,t)}, and u(,t)\bm{u}(\cdot,t) is the elastic equilibrium in Ωh(,t)\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 ψ\psi is highly anisotropic, there may be directions ν\bm{\nu} for which

D2ψ(ν)[τ,τ]>0for all τν, τ0D^{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. Solids 52, 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

Γh(ψ(ν)+εpHp)dH2,\int_{\Gamma_{h}}\Bigl(\psi(\bm{\nu})+\frac{\varepsilon}{p}|H|^{p}\Bigr)\,d\mathcal{H}^{2},

where p>2p>2, HH stands for the sum κ1+κ2\kappa_{1}+\kappa_{2} of the principal curvatures of Γh\Gamma_{h}, and ε\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>2p>2 in R3\mathbb{R}^{3} is motivated by the fact that the profile hh of the film will belong to W2,p(Q)W^{2,p}(Q), where QR2Q\subset\mathbb{R}^{2}, so that W2,p(Q)W^{2,p}(Q) is continuously embedded into C1,p2p(Q)C^{1,\frac{p-2}{p}}(Q). This regularity is strongly used to prove existence of solutions. In contrast, in R2\mathbb{R}^{2} we can assume p2p\geq 2 since W2,2((0,b))W^{2,2}((0,b)) is embedded in C1,1([0,b])C^{1,1}([0,b]).

The regularized free-energy functional becomes

ΩhW(E(u))dx+Γh(ψ(ν)+εpHp)dH2,\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=ΔΓ[divΓ(Dψ(ν))+W(E(u))ε(ΔΓ(Hp2H)Hp2H(κ12+κ221pH2))].\displaystyle V=\Delta_{\Gamma}\biggl[\operatorname{div}_{\Gamma}\bigl(D\psi(\bm{\nu})\bigr)+W\bigl(\bm{E}(\bm{u})\bigr)\\ \displaystyle-\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 Γ\Gamma using h:R2×[0,T0](0,)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:

{1Jht=ΔΓ[divΓ(Dψ(ν))+W(E(u))1pε(ΔΓ(Hp2H)Hp2H(κ12+κ221pH2))] in R2×(0,T0),divCE(u)=0  in Ωh,CE(u)[ν]=0  on Γh,u(x,y,0,t)=A0(x,y,0),h(,t)  and  Du(,t)  are Q-periodic,h(,0)=h0,\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 $\Omega_{h}$},\\ \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),\\ h(\cdot,t)~{}\text{ and }~{}D\bm{u}(\cdot,t)~{}\text{ are $Q$-periodic,}\\ h(\cdot,0)=h_{0},\end{cases}

where J:=1+Dh2J:=\sqrt{1+|Dh|^{2}} and h0Hloc2(R2)h_{0}\in H_{\operatorname*{loc}}^{2}(\mathbb{R}^{2}) is a QQ-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. Solids 52, 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 H1H^{-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 H1H^{-1} we used the gradient flow with respect to an L2L^{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. PDE 8, 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. PDE 8, 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

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

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

Ωh+:=Ωh{y>0}.\Omega_{h}^{+}:=\Omega_{h}\cap\{y>0\}.

The line y=0y=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

E0(y)={E^0if y0,0if y<0,\bm{E}_{0}(y)=\begin{cases}\bm{\hat{E}}_{0}&\text{if }y\geq 0,\\ 0&\text{if }y<0,\end{cases}

with E^00>0\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 C\mathbb{C}. Hence, bearing in mind the mismatch, the elastic energy per unit area is given by W(EE0(y))W(\bm{E}-\bm{E}_{0}(y)), where

W(E):=12EC[E]W\left(\bm{E}\right):=\frac{1}{2}\bm{E}\cdot\mathbb{C}\left[\bm{E}\right]

for all symmetric matrices E0\bm{E}\neq\bm{0}.

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

ψ(y):={γfilmif y>0,γsubif y=0,\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

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

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

F(u,h):=ΩhW(E(u)E0)dx+Γhψds,\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 Γh\Gamma_{h} represents the free surface of the film, that is,

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

Since the functional F\mathcal{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][0,) Lipschitz,0bh dx=d,uHloc1(Ωh;R2)}\displaystyle X:={\biggl\{}(\bm{u},h):~{}h:[0,b]\rightarrow[0,\infty)\text{ Lipschitz,}\\ \displaystyle\int_{0}^{b}h~{}dx=d,\quad\bm{u}\in H_{\operatorname*{loc}}^{1}\bigl(\Omega_{h};\mathbb{R}^{2}\bigr){\biggr\}}

and

X0={(u,h): h:[0,b][0,) lower semicontinuous,var[0,b]h<,0bh dx=d,uHloc1(Ωh;R2)},\displaystyle X_{0}={\biggl\{}(\bm{u},h):~{}h:[0,b]\rightarrow[0,\infty)\text{ lower semicontinuous,}\\ \displaystyle\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 var[0,b]h\operatorname*{var}\nolimits_{[0,b]}h stands for the pointwise variation of the function hh. Note that lengthΓh\operatorname*{length}\Gamma_{h} coincides with the pointwise variation of the function x[0,b](x,h(x))x\in[0,b]\mapsto(x,h(x)), and so

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

For (u,h)X0(\bm{u},h)\in X_{0} define

G(u,h):=ΩhW(E(u)(x)E0(y))dx+γfilmlengthΓh.\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(u,h)XF(u,h)=inf(u,h)XG(u,h)=min(u,h)X0G(u,h).\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 G\mathcal{G} in X0X_{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 0bh(x) dx=d\int_{0}^{b}h(x)~{}dx=d can be replaced by a volume penalization.

Theorem 2 (Volume penalization).

Let (u0,h0)X0(\bm{u}_{0},h_{0})\in X_{0} be a minimizer of the functional G\mathcal{G} defined in (17) with 0bh0(x) dx=d\int_{0}^{b}h_{0}(x)~{}dx=d. Then there exists k0Nk_{0}\in\mathbb{N} such that for every integer kk0k\geq k_{0}, (u0,h0)(\bm{u}_{0},h_{0}) is a minimizer of the penalized functional

Gk(u,h):=Ωh ⁣ ⁣W(E(u)E0)dx+γfilmlengthΓh+k0bh dxd\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 (u,h)X0(\bm{u},h)\in X_{0}.

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

G(u0,h0)G(vk,fk)=Gk(vk,fk)Gk(u0,h0)=G(u0,h0)<,\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 (u0,h0)(\bm{u}_{0},h_{0}) is a minimizer of Gk\mathcal{G}_{k}.

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

0bfk dx>d\int_{0}^{b}f_{k}~{}dx>d

for countably many kk, define

hk:=min{fk,tk},h_{k}:=\min\{f_{k},t_{k}\},

where tk>0t_{k}>0 has been chosen so that 0bhk dx=d\int_{0}^{b}h_{k}~{}dx=d. Note that lengthΓhklengthΓfk\operatorname*{length}\Gamma_{h_{k}}\leq\operatorname*{length}\Gamma_{f_{k}}. Indeed, for every partition x0=0<<xn=bx_{0}=0<\cdots<x_{n}=b, we have that

(hk(xi)hk(xi1))2(fk(xi)fk(xi1))2\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,,ni=1,\ldots,n. Hence,

G(vk,hk)=Gk(vk,hk)<Gk(vk,fk),\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 kk sufficiently large

0bfk dx<d.\int_{0}^{b}f_{k}~{}dx<d.

Since

Gk(vk,fk)Gk(u0,h0)=G(u0,h0)<,\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 0bfk dxd\int_{0}^{b}f_{k}~{}dx\rightarrow d as kk\rightarrow\infty and that supklengthΓfk<\sup_{k}\operatorname*{length}\Gamma_{f_{k}}<\infty. In turn, by (16), fkc\|f_{k}\|_{\infty}\leq c for some constant cc independent of kk.

Let k1k_{1} be so large that 0bfk dx>d2\int_{0}^{b}f_{k}~{}dx>\frac{d}{2} for all kk1k\geq k_{1}. Then

tk:=d0bfk dx(0,2)t_{k}:=\frac{d}{\int_{0}^{b}f_{k}~{}dx}\in(0,2)

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

0bhk dx=d.\int_{0}^{b}h_{k}~{}dx=d.

Consider a partition 0=x0<<x=b0=x_{0}<\cdots<x_{\ell}=b. Then

i=1(xixi1)2+(hk(xi)hk(xi1))2\displaystyle\sum_{i=1}^{\ell}\sqrt{(x_{i}-x_{i-1})^{2}+\bigl(h_{k}(x_{i})-h_{k}(x_{i-1})\bigr)^{2}}
            =i=1(xixi1)2+tk2(fk(xi)fk(xi1))2\displaystyle            =\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}}
            tki=1(xixi1)2+(fk(xi)fk(xi1))2\displaystyle            \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}}
            tklengthΓfk,\displaystyle            \leq t_{k}\operatorname*{length}\Gamma_{f_{k}},

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

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

and so, by (20),

γfilmlengthΓhkγfilmlengthΓfk\displaystyle\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{h_{k}}-\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{f_{k}}
                  (tk1)γfilmlengthΓfk(tk1)Gk(vk,fk)\displaystyle                  \leq(t_{k}-1)\gamma_{\operatorname*{film}}\operatorname*{length}\Gamma_{f_{k}}\leq(t_{k}-1)\mathcal{G}_{k}(\bm{v}_{k},f_{k})
                  (tk1)G(u0,h0).\displaystyle                  \leq(t_{k}-1)\mathcal{G}(\bm{u}_{0},h_{0}).

We deduce that

γfilmlengthΓhkγfilmlengthΓfk+(tk1)G(u0,h0).\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)Ωhk(x,y^{\prime})\in\Omega_{h_{k}} define

wk(x,y):=((vk)1(x,ytk),1tk(vk)2(x,ytk)).\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

ΩhkW(E(wk)(x,y)E0(y)) dxdy=1tkΩfkW(E~(vk)(x)E0(y)) dx,\displaystyle\int_{\Omega_{h_{k}}}W\bigl(\bm{E}(\bm{w}_{k})(x,y^{\prime})-\bm{E}_{0}(y^{\prime})\bigr)~{}dxdy^{\prime}\\ \displaystyle=\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 E~(vk)(x)\mathbf{\widetilde{\bm{E}}}(\bm{v}_{k})(\bm{x}) is the 2×22\times 2 matrix whose entries are

E~11(vk)(x)=E11(vk)(x),E~12(vk)(x)=1tkE12(vk)(x),\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}),
E~22(vk)(x)=1tk2E22(vk)(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

(E~(vk)E0+E(vk)E0)E~(vk)E(vk)\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})|
c(tk1)(E~(vk)E0+E(vk)E0)E(vk)\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})|
c(tk1)(E(vk)+E0)(E(vk)E0+E0)\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}|)
c(tk1)(E(vk)E0+E0)2.\displaystyle\leq c(t_{k}-1)\bigl(|\bm{E}(\bm{v}_{k})-\bm{E}_{0}|+|\bm{E}_{0}|\bigr)^{2}.

Since W(E)W(\bm{E}) is a positive definite quadratic form over the 2×22\times 2 symmetric matrices (see (11)), we have that

W(E)W(E1)c(E+E1)EE1|W(\bm{E})-W(\bm{E}_{1})|\leq c\left(|\bm{E}|+|\bm{E}_{1}|\right)|\bm{E}-\bm{E}_{1}|

for all 2×22\times 2 symmetric matrices E\bm{E} and E1\bm{E}_{1}. Hence by (1), (10) and (23)

ΩhkW(E(wk)(x,y)E0(y)) dxΩfk ⁣ ⁣ ⁣W(E(vk)(x)E0(y)) dx\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}
=1tkΩfk[W(E~(vk)(x)E0(y))W(E(vk)(x)E0(y))] dx\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}
cΩfk(E~(vk)E0+E(vk)E0)(E~(vk)E(vk)) dx\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}
c(tk1)Ωfk(E(vk)E0+E0)2 dx\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}
c(tk1)(Gk(vk,fk)+E^02)c(tk1)(G(u0,h0)+E^02),\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 cc depends only on the ellipticity constants of WW and supkfk\sup_{k}\|f_{k}\|_{\infty}. By (20), (21), and (24), we have that

G(u0,h0)\displaystyle\mathcal{G}(\bm{u}_{0},h_{0})
G(wk,hk)G(vk,fk)+(tk1)[(c+1)G(u0,h0)+cE^02]\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]
=Gk(vk,fk)+(tk1)[(c+1)G(u0,h0)+cE^02]\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]
                                           k(d0bfk dx)\displaystyle                                           -k\Bigl(d-\int_{0}^{b}f_{k}~{}dx\Bigr)
=Gk(vk,fk)+(tk1)[(c+1)G(u0,h0)+cE^02]\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]
                                           (tk1)k0bfk dx\displaystyle                                           -(t_{k}-1)k\int_{0}^{b}f_{k}~{}dx
G(u0,h0)+(tk1)[(c+1)G(u0,h0)+cE^02kd2].\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

k2d[(c+1)G(u0,h0)+cE^02]+1,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 (u0,h0)X0(\bm{u}_{0},h_{0})\in X_{0} be a minimizer of the functional G\mathcal{G} defined in (17). Then there exists r0>0r_{0}>0 with the property that for every z0Γh0\bm{z}_{0}\in\overline{\Gamma}_{h_{0}} there exists an open ball B(x0,r0)B(\bm{x}_{0},r_{0}), with B(x0,r0)((0,b)×R)Ωh0B(\bm{x}_{0},r_{0})\cap((0,b)\times\mathbb{R})\subseteq\Omega_{h_{0}}, such that

B(x0,r0)Γh0={z0}.\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 Equations 18, 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 ν0B(0,1)\bm{\nu}_{0}\in\partial B(\mathbf{0},1) is the outward unit normal to B(x0,r0)B(\bm{x}_{0},r_{0}) at z0\bm{z}_{0}, then x0=z0r0ν0\bm{x}_{0}=\bm{z}_{0}-r_{0}\bm{\nu}_{0}. Thus, the set

Nz0:={νB(0,1): B(z0r0ν,r0)((0,b)×R)Ωh0}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 h0h_{0} admits a left and right derivative at all but countably many points.

Theorem 5 (Left and Right Derivatives of hh).

Let (u0,h0)X0(\bm{u}_{0},h_{0})\in X_{0} be a minimizer of the functional G\mathcal{G} defined in (17). Then Γh0\overline{\Gamma}_{h_{0}} admits a left and a right tangent at every point z\bm{z} not of the form z=(x,h0(x))\bm{z}=(x,h_{0}(x)) with xSx\in S, where

S:={x(0,b):h0(x)<lim inftxh0(t)}.S:=\Bigl\{x\in(0,b):\,h_{0}(x)<\liminf_{t\rightarrow x}h_{0}(t)\Bigr\}.

Define

Γcusps:={zΓh0:±e1Nz}\Gamma_{\operatorname*{cusps}}:=\Bigl\{\bm{z}\in\overline{\Gamma}_{h_{0}}:\,\pm\bm{e}_{1}\in N_{\bm{z}}\Bigr\}

and

Γcuts:={(x,y):x(0,b)S,h0(x)ylim inftxh0(t)},\Gamma_{\operatorname*{cuts}}:=\Bigl\{(x,y):\,x\in(0,b)\cap S,\,h_{0}(x)\leq y\leq\liminf_{t\rightarrow x}h_{0}(t)\Bigr\},

where NzN_{\bm{z}} is the set defined in (25) and SS is the set defined in (26).

Theorem 6 (Cusps and Cuts).

Let (u0,h0)X0(\bm{u}_{0},h_{0})\in X_{0} be a minimizer of the functional G\mathcal{G} defined in (17). Then the sets Γcusps\Gamma_{\operatorname*{cusps}} and Γcuts\Gamma_{\operatorname*{cuts}} contain at most finitely many vertical segments.

Remark 7. If e1Nz0-\bm{e}_{1}\in N_{\bm{z}_{0}}, then since B((x0+r0,y0),r0)((0,b)×R)Ωh0B((x_{0}+r_{0},y_{0}),r_{0})\cap((0,b)\times\mathbb{R})\subseteq\Omega_{h_{0}} and h0h_{0} is lower semicontinuous, for all x>x0x>x_{0} sufficiently close to x0x_{0}, we have that

h0(x)y0+r02(x(x0+r0))2,h_{0}(x)\geq y_{0}+\sqrt{r_{0}^{2}-\bigl(x-(x_{0}+r_{0})\bigr)^{2}},

and so

h0(x)y0xx02r0(xx0)xx0\frac{h_{0}(x)-y_{0}}{x-x_{0}}\geq\frac{\sqrt{2r_{0}-(x-x_{0})}}{\sqrt{x-x_{0}}}\rightarrow\infty

as xx0+x\rightarrow x_{0}^{+}. By Theorem 5 it follows that Γh0\overline{\Gamma}_{h_{0}} admits a right vertical tangent at z0\bm{z}_{0}. Similarly, if e1Nz0\bm{e}_{1}\in N_{\bm{z}_{0}} then for x<x0x<x_{0}, Γh0\overline{\Gamma}_{h_{0}} admits a left vertical tangent at z0\bm{z}_{0}. In particular, if ±e1Nz0\pm\bm{e}_{1}\in N_{\bm{z}_{0}} and h0h_{0} is continuous at x0x_{0}, then

(h0)(x0)=,(h0)+(x0)=.(h_{0})_{-}^{\prime}(x_{0})=-\infty,\text{\quad}(h_{0})_{+}^{\prime}(x_{0})=\infty.