Skip to main content

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity


We extend the results about the existence of minimizers, relaxation, and approximation proven by Bonnetier and Chambolle (SIAM J Appl Math 62:1093–1121, 2002), Chambolle and Solci (SIAM J Math Anal 39:77–102, 2007) for an energy related to epitaxially strained crystalline films, and by Braides et al. (ESAIM Control Optim Calc Var 13:717–734, 2007) for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for \((d{-}1)\)-rectifiable sets that are jumps of \({ GSBD}^p\) functions, called \(\sigma ^p_{\mathrm{sym}}\)-convergence.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111, 291–322, 1990

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238, 1997

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Asaro, R.J., Tiller, W.A.: Interface morphology development during stress corrosion cracking: Part I: via surface diffusion. Metall. Trans. 3, 1789–1796, 1972

    Google Scholar 

  5. Bella, P., Goldman, M., Zwicknagl, B.: Study of island formation in epitaxially strained films on unbounded domains. Arch. Ration. Mech. Anal. 218, 163–217, 2015

    MathSciNet  MATH  Google Scholar 

  6. Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in \({\rm SBD}(\Omega )\). Math. Z. 228, 337–351, 1998

    MathSciNet  MATH  Google Scholar 

  7. Bonacini, M.: Epitaxially strained elastic films: the case of anisotropic surface energies. ESAIM Control Optim. Calc. Var. 19, 167–189, 2013

    MathSciNet  MATH  Google Scholar 

  8. Bonacini, M.: Stability of equilibrium configurations for elastic films in two and three dimensions. Adv. Calc. Var. 8, 117–153, 2015

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Braides, A.: Approximation of free-discontinuity problems, vol. 1694. Lecture Notes in Mathematics. Springer, Berlin 1998

  11. Braides, A., Chambolle, A., Solci, M.: A relaxation result for energies defined on pairs set-function and applications. ESAIM Control Optim. Calc. Var. 13, 717–734, 2007

    MathSciNet  MATH  Google Scholar 

  12. Cagnetti, F., Colombo, M., De Philippis, G., Maggi, F.: Essential connectedness and the rigidity problem for Gaussian symmetrization. J. Eur. Math. Soc. (JEMS)19, 395–439, 2017

    MathSciNet  MATH  Google Scholar 

  13. Cagnetti, F., Scardia, L.: An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains. J. Math. Pures Appl. 9(95), 349–381, 2011

    MathSciNet  MATH  Google Scholar 

  14. Capriani, G.M., Julin, V., Pisante, G.: A quantitative second order minimality criterion for cavities in elastic bodies. SIAM J. Math. Anal. 45, 1952–1991, 2013

    MathSciNet  MATH  Google Scholar 

  15. Chambolle, A., Conti, S., Francfort, G.A.: Korn-Poincaré inequalities for functions with a small jump set. Indiana Univ. Math. J. 65, 1373–1399, 2016

    MathSciNet  MATH  Google Scholar 

  16. Chambolle, A., Conti, S., Iurlano, F.: Approximation of functions with small jump sets and existence of strong minimizers of Griffith’s energy. J. Math. Pures Appl. 9(128), 119–139, 2019

    MathSciNet  MATH  Google Scholar 

  17. Chambolle, A., Crismale, V.: A Density Result in \({{\rm GSBD}}^p\) with Applications to the Approximation of Brittle Fracture Energies. Arch. Ration. Mech. Anal. 232, 1329–1378, 2019

    MathSciNet  MATH  Google Scholar 

  18. Chambolle, A., Crismale, V.: Phase-field approximation of a class of cohesive fracture energies with an activation threshold. Adv. Calc. Var.

  19. Chambolle, A., Crismale, V.: Compactness and lower semicontinuity in \(GSBD\). J. Eur. Math. Soc. (JEMS). Preprint arXiv:1802.03302. 2018.

  20. Chambolle, A., Solci, M.: Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal. 39, 77–102, 2007

    MathSciNet  MATH  Google Scholar 

  21. Conti, S., Focardi, M., Iurlano, F.: Which special functions of bounded deformation have bounded variation? Proc. Roy. Soc. Edinburgh Sect. A 2016

  22. Conti, S., Focardi, M., Iurlano, F.: Integral representation for functionals defined on \(SBD^p\) in dimension two. Arch. Ration. Mech. Anal. 223, 1337–1374, 2017

    MathSciNet  MATH  Google Scholar 

  23. Conti, S., Focardi, M., Iurlano, F.: Existence of strong minimizers for the Griffith static fracture model in dimension two. Ann. Inst. H. Poincaré Anal. Non Linéaire, 36, 455-474, 2019

  24. Conti, S., Focardi, M., Iurlano, F.: Approximation of fracture energies with \(p\)-growth via piecewise affine finite elements. ESAIM Control Optim. Calc. Var., 25, Art.34. (2019)

  25. Crismale, V.: On the approximation of \(SBD\) functions and some applications. SIAM J. Math. Anal. 51, 5011–5048, 2019

    MathSciNet  MATH  Google Scholar 

  26. Dal Maso, G.: An introduction to \(\Gamma \)-convergence, vol. 8. Progress in Nonlinear Differential Equations and their ApplicationsBirkhäuser Boston Inc, Boston, MA 1993

  27. Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS)15, 1943–1997, 2013

    MathSciNet  MATH  Google Scholar 

  28. Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225, 2005

    MathSciNet  MATH  Google Scholar 

  29. Davoli, E., Piovano, P.: Derivation of a heteroepitaxial thin-film model, Interfaces Free Bound.

  30. Davoli, E., Piovano, P.: Analytical validation of the Young–Dupré law for epitaxially-strained thin films. Math. Models Methods Appl. Sci. 29, 2183–2223, 2019

    MathSciNet  MATH  Google Scholar 

  31. De Giorgi, E., Ambrosio, L.: New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82(1988), 199–210, 1989.

  32. Focardi, M., Gelli, M.S.: Asymptotic analysis of Mumford-Shah type energies in periodically perforated domains. Interfaces Free Bound. 9, 107–132, 2007

    MathSciNet  MATH  Google Scholar 

  33. Focardi, M., Gelli, M.S.: Relaxation of free-discontinuity energies with obstacles. ESAIM Control Optim. Calc. Var. 14, 879–896, 2008

    MathSciNet  MATH  Google Scholar 

  34. Fonseca, I., Fusco, N., Leoni, G., Millot, V.: Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 9(96), 591–639, 2011

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE8, 373–423, 2015

    MathSciNet  MATH  Google Scholar 

  37. Fonseca, I., Pratelli, A., Zwicknagl, B.: Shapes of epitaxially grown quantum dots. Arch. Ration. Mech. Anal. 214, 359–401, 2014

    MathSciNet  MATH  Google Scholar 

  38. Friedrich, M.: A derivation of linearized Griffith energies from nonlinear models. Arch. Ration. Mech. Anal. 225, 425–467, 2017

    MathSciNet  MATH  Google Scholar 

  39. Friedrich, M.: A Korn-type inequality in SBD for functions with small jump sets. Math. Models Methods Appl. Sci. 27, 2461–2484, 2017

    MathSciNet  MATH  Google Scholar 

  40. Friedrich, M.: A Piecewise Korn inequality in SBD and applications to embedding and density results. SIAM J. Math. Anal. 50, 3842–3918, 2018

    MathSciNet  MATH  Google Scholar 

  41. Friedrich, M.: A compactness result in \(GSBV^p\) and applications to \(\Gamma \)-convergence for free discontinuity problems. Calc. Var. Partial Differential Equations, 58, Art. 86, 31. 2019

  42. Friedrich, M., Solombrino, F.: Quasistatic crack growth in 2d-linearized elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire, 35, 27–64, 2018

  43. Fusco, N., Julin, V., Morini, M.: The surface diffusion flow with elasticity in the plane. Commun. Math. Phys. 362, 571–607, 2018

    MathSciNet  MATH  Google Scholar 

  44. Fusco, N., Julin, V., Morini, M.: The surface diffusion flow with elasticity in three dimensions. Preprint CVGMT 4082. 2018

  45. Fusco, N., Morini, M.: Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Ration. Mech. Anal. 203, 247–327, 2012

    MathSciNet  MATH  Google Scholar 

  46. Gao, H., Nix, W.: Surface roughening of heteroepitaxial thin films. Ann. Rev. Mater. Sci. 29, 173–209, 1999

    ADS  Google Scholar 

  47. Giacomini, A., Ponsiglione, M.: A \(\Gamma \)-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications. Arch. Ration. Mech. Anal. 180, 399–447, 2006

    MathSciNet  MATH  Google Scholar 

  48. Goldman, M., Zwicknagl, B.: Scaling law and reduced models for epitaxially strained crystalline films. SIAM J. Math. Anal. 46, 1–24, 2014

    MathSciNet  MATH  Google Scholar 

  49. Grinfeld, M.A.: Instability of the separation boundary between a non-hydrostatically stressed elastic body and a melt. Sov. Phys. Dokl. 31, 831–834, 1986

    ADS  Google Scholar 

  50. Grinfeld, M.A.: The stress driven instability in elastic crystals: mathematical models and physical manifestations. J. Nonlinear Sci. 3, 35–83, 1993

    ADS  MathSciNet  MATH  Google Scholar 

  51. Iurlano, F.: A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51, 315–342, 2014

    MathSciNet  MATH  Google Scholar 

  52. Kholmatov, S., Piovano, P.: A unified model for stress-driven rearrangement instabilities. Preprint CVGMT 4228, 2019.

  53. Kreutz, L., Piovano, P.: Microscopic validation of a variational model of epitaxially strained crystalline films, Preprint CVGMT 4197, 2019.

  54. Nitsche, J.A.: On Korn’s second inequality. RAIRO Anal. Numér. 15, 237–248, 1981

    MathSciNet  MATH  Google Scholar 

  55. Piovano, P.: Evolution of elastic thin films with curvature regularization via minimizing movements. Calc. Var. Partial Differ. Equ. 49, 337–367, 2014

    MathSciNet  MATH  Google Scholar 

  56. Siegel, M., Miksis, M.J., Voorhees, P.W.: Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Solids52, 1319–1353, 2004

    ADS  MathSciNet  MATH  Google Scholar 

  57. Simon, L.: Lectures on geometric measure theory, vol. 3. Proceedings of the Centre for Mathematical Analysis. Australian National University, Australian National University, Centre for Mathematical Analysis, Canberra 1983

  58. Spencer, B.J.: Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for Stranski–Krastanow islands. Phys. Rev. B59, 2011–2017, 1999

    ADS  Google Scholar 

  59.  Temam, R.: Mathematical problems in plasticity, Gauthier-Villars, Paris, 1985. Translation of Problèmes mathématiques en plasticité. Gauthier-Villars, Paris, 1983.

Download references


Vito Crismale is supported by the Marie Skłodowska-Curie Standard European Fellowship No. 793018. Manuel Friedrich acknowledges support by the DFG Project FR 4083/1-1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics–Geometry–Structure.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Vito Crismale.

Ethics declarations

Conflict of interest and ethical statement

The authors declare that they have no conflict of interest and guarantee the compliance with the Ethics Guidelines of the journal.

Additional information

Communicated by I. Fonseca

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Auxiliary Results

Auxiliary Results

In this appendix, we prove two technical approximation results employed in Sections 5 and 6, based on tools from [17].


Let (vH) be given as in the statement of the lemma. Clearly, it suffices to prove the following statement: for every \(\eta >0\), there exists \(( {v}^\eta , H^\eta ) \in L^p(\varOmega ;\mathbb {R}^d){\times }\mathfrak {M}(\varOmega )\) with the regularity and the properties required in the statement of the lemma (in particular, \( {v}^\eta = u_0\) in a neighborhood \(V^\eta \subset \varOmega \) of \(\partial _D \varOmega \)), such that, for a universal constant C, one has \(\bar{d}( v^\eta , v )\leqq C\eta \) (cf. (3.13) for \(\bar{d}\)), \(\mathcal {L}^d(H\triangle H^\eta )\leqq C\eta \), and

$$\begin{aligned} {\overline{F}}'_{\mathrm {Dir}}( {v}^\eta , H^\eta ) \leqq {\overline{F}}'_{\mathrm {Dir}}(v,H) + C\eta . \end{aligned}$$

We start by recalling the main steps of the construction in [17, Theorem 5.5] and we refer to [17] for details (see also [18, Section 4, first part]). Based on this, we then explain how to construct \(( {v}^\eta , H^\eta )\) simultaneously, highlighting particularly the steps needed for constructing \(H^\eta \).

Let \(\varepsilon >0\), to be chosen small with respect to \(\eta \). By using the assumptions on \(\partial \varOmega \) given before (2.4), a preliminary step is to find cubes \((Q_j)_{j=1}^{ J }\) with pairwise disjoint closures and hypersurfaces \((\varGamma _j)_{j=1}^J\) with the following properties: each \(Q_j\) is centered at \(x_j \in \partial _N \varOmega \) with sidelength \(\varrho _j\), \( \mathrm{dist} (Q_j, {\partial _D \varOmega })> d_\varepsilon >0 \) with \(\lim _{\varepsilon \rightarrow 0} d_\varepsilon =0\), and

$$\begin{aligned} \mathcal {H}^{d-1}({\partial _N \varOmega }{\setminus }\widehat{Q}) + \mathcal {L}^d(\widehat{Q}) \leqq \varepsilon ,\qquad \text {for }\widehat{Q}:= \bigcup \nolimits _{j=1}^J {\overline{Q}}_j. \end{aligned}$$

Moreover, each \(\varGamma _j\) is a \(C^1\)-hypersurface with \(x_j \in \varGamma _j \subset {\overline{Q}}_j\),

$$\begin{aligned} \begin{aligned} \mathcal {H}^{d-1}\big (({\partial _N \varOmega }\triangle \varGamma _j)\,\cap \, \overline{Q_j} \big ) \leqq \varepsilon (2\varrho _j)^{d-1}\leqq \, \frac{\varepsilon }{1-\varepsilon } \mathcal {H}^{d-1}({\partial _N \varOmega }\cap \overline{Q_j}), \end{aligned} \end{aligned}$$

and \(\varGamma _j\) is a \(C^1\)-graph with respect to \(\nu _{{\partial \varOmega }}(x_j)\) with Lipschitz constant less than \(\varepsilon /2\). (We can say that \({\partial _N \varOmega }\cap Q_j\) is “almost” the intersection of \(Q_j\) with the hyperplane passing through \(x_j\) with normal \(\nu _{{\partial \varOmega }}(x_j)\).) We can also guarantee that

$$\begin{aligned} \mathcal {H}^{d-1}\big ((\partial ^* H \cup J_u) \cap \varOmega \cap \widehat{Q}\big ) \leqq \varepsilon , \ \ \ \ \ \ \ \ \mathcal {H}^{d-1}\big ((\partial ^* H \cup J_u) \cap \partial Q_j \big )= 0 \end{aligned}$$

for all \(j=1,\ldots ,J\). To each \(Q_j\), we associate the following rectangles:

$$\begin{aligned} R_{j}:= & {} \Big \{x_{j}+\sum \nolimits _{i=1}^{d-1} y_i\, b_{j,i}+y_d\, \nu _{j} :y_i\in (-\,\varrho _{j},\varrho _{j}),\, y_d \in (-\,3\varepsilon \varrho _{j}-t, -\varepsilon \varrho _{j}) \Big \},\\ R'_{j}:= & {} \Big \{x_{j}+\sum \nolimits _{i=1}^{ d-1 } y_i\, b_{j,i}+y_d\, \nu _{j} :y_i\in (-\,\varrho _{j},\varrho _{j}),\, y_d \in (-\,\varepsilon \varrho _{j}, \varepsilon \varrho _{j}+t) \Big \}, \end{aligned}$$

and \(\widehat{R}_{j}:=R_{j} \cup R'_{j}\), where \(\nu _{j}=-\,\nu _{{\partial \varOmega }}(x_{j})\) denotes the generalized outer normal, \((b_{j,i})_{i=1}^{d-1}\) is an orthonormal basis of \((\nu _{j})^\perp \), and \(t>0\) is small with respect to \(\eta \). We remark that \(\varGamma _j \subset R'_j\) and that \(R_j\) is a small strip adjacent to \(R_j'\), which is included in \(\varOmega \cap Q_j\). (We use here the notation \(_j\) in place of \(_{h,N}\) adopted in [17, Theorem 5.5].)

After this preliminary part, the approximating function \(u^\eta \) was constructed in [17, Theorem 5.5] starting from a given function u through the following three steps:

  1. (i)

    definition of an extension \(\widetilde{u} \in GSBD^p(\varOmega + B_t(0))\) which is obtained by a reflection argument la Nitsche [54] inside \(\widehat{R}_j\), equal to u in \(\varOmega {\setminus }\bigcup _j \widehat{R}_j\), and equal to \(u_0\) elsewhere. This can be done such that, for t and \(\varepsilon \) small, there holds (see below [17, (5.13)])

    $$\begin{aligned} \int \limits _{(\varOmega +B_t(0)) {\setminus }\varOmega } |e(u_0)|^p \, \mathrm {d} x+ \int \limits _{\widehat{R}} |e(\widetilde{u})|^p \, \mathrm {d} x+ \int \limits _{{R}} |e(u)|^p \, \mathrm {d} x+ \mathcal {H}^{d-1}\big (J_{\widetilde{u}} \cap \widehat{R}\big ) \leqq \eta , \end{aligned}$$

    where \(R:= \bigcup _{j=1}^J {R}_j \) and \(\widehat{R}:= \bigcup _{j=1}^J \widehat{R}_j \cap (\varOmega +B_t(0))\).

  2. (ii)

    application of Theorem 3.4 on the function \(\widetilde{u}^\delta := \widetilde{u}\circ (O_{\delta ,x_0})^{-1} + u_0 - u_0 \circ (O_{\delta ,x_0})^{-1}\) (for some \(\delta \) sufficiently small) to get approximating functions \(\widetilde{u}^\delta _n\) with the required regularity which are equal to \(u_0 *\psi _n \) in a neighborhood of \({\partial _D \varOmega }\) in \(\varOmega \), where \(\psi _n\) is a suitable mollifier. Here, assumption (2.4) is crucial.

  3. (iii)

    correcting the boundary values by defining \(u^\eta \) as \( u^\eta := \widetilde{u}^\delta _n + u_0 - u_0 *\psi _n \), for \(\delta \) and 1/n small enough.

After having recalled the main steps of the construction in [17, Theorem 5.5], let us now construct \( {v}^\eta \) and \(H^\eta \) at the same time, following the lines of the steps (i), (ii), and (iii) above. The main novelty is the analog of step (i) for the approximating sets, while the approximating functions are constructed in a very similar way. For this reason, we do not recall more details from [17, Theorem 5.5].

Step (i). Step (i) for \( {v}^\eta \) is the same done before for \(u^\eta \), starting from v in place of u. Hereby, we get a function \(\widetilde{v} \in GSBD^p(\varOmega + B_t(0)) \).

For the construction of \(H^\eta \), we introduce a set \(\widetilde{H} \subset \varOmega +B_t(0)\) as follows: in \(R'_j\), we define a set \(H'_j\) by a simple reflection of the set \(H \cap R_j\) with respect to the common hyperface between \(R_j\) and \(R'_j\). Then, we let \( \widetilde{H}:= H \cup \bigcup _{j=1}^J (H'_j \cap (\varOmega +B_t(0)))\). Since H has finite perimeter, also \(\widetilde{H}\) has finite perimeter. By (A.2) we get \(\mathcal {H}^{d-1}(\partial ^*\widetilde{H} \cap \widehat{R} )\leqq \eta /3 \) for \(\varepsilon \) small, where as before \(\widehat{R}:= \bigcup _{j=1}^J \widehat{R}_j \cap (\varOmega +B_t(0))\). We choose \(\delta \), \(\varepsilon \), and t so small that

$$\begin{aligned} \mathcal {H}^{d-1} \Big (O_{\delta ,x_0}\Big ( \bigcup \nolimits _{j=1}^J \partial R_j' {\setminus } \partial R_j \Big ) \cap \varOmega \Big ) \leqq \frac{\eta }{ 3 }. \end{aligned}$$

We let \( {H}^\eta :=O_{\delta ,x_0}(\widetilde{H})\). Then, we get \(\mathcal {L}^d({H}^\eta \triangle H)\leqq \eta \) for \(\varepsilon \), t, and \(\delta \) small enough. By (A.1), (A.4), and \(\mathcal {H}^{d-1}(\partial ^*\widetilde{H} \cap \widehat{R} )\leqq \eta /3 \) we also have (again take suitable \(\varepsilon \), \(\delta \))

$$\begin{aligned} \int \nolimits _{\partial ^*{H}^\eta } \varphi (\nu _{{H}^\eta }) \, \mathrm {d} \mathcal {H}^{d-1}\leqq \int \nolimits _{ \partial ^* H \cap (\varOmega \cup {\partial _D \varOmega })} \varphi (\nu _H) \, \mathrm {d} \mathcal {H}^{d-1}+ \eta . \end{aligned}$$

Moreover, in view of (2.4) and \( \mathrm{dist} (Q_j, {\partial _D \varOmega })> d_\varepsilon >0 \) for all j, \({H}^\eta \) does not intersect a suitable neighborhood of \({\partial _D \varOmega }\). Define \(\widetilde{v}^\delta := \widetilde{v}\circ (O_{\delta ,x_0})^{-1} + u_0 - u_0 \circ (O_{\delta ,x_0})^{-1}\) and observe that the function \(\widetilde{v}^\delta \chi _{({H}^\eta )^0}\) coincides with \(u_0\) in a suitable neighborhood of \({\partial _D \varOmega }\). By (A.5), by the properties recalled for \(\widetilde{u}\), see (A.3), and the fact that \(v = v \chi _{H^0}\), it is elementary to check that

$$\begin{aligned} {\overline{F}}'_{\mathrm {Dir}}(\widetilde{v}^\delta \chi _{({H}^\eta )^0}, {H}^\eta ) \leqq {\overline{F}}'_{\mathrm {Dir}}(v\chi _{H^0}, H) + C \eta = {\overline{F}}'_{\mathrm {Dir}}(v, H) + C \eta . \end{aligned}$$

Notice that here it is important to take the same \(\delta \) both for \(\widetilde{v}^\delta \) and \({H}^\eta \), that is to “dilate” the function and the set at the same time.

Step 2. We apply Theorem 3.4 to \(\widetilde{v}^\delta \chi _{({H}^\eta )^0}\), to get approximating functions \(\widetilde{v}^\delta _n\) with the required regularity. For n sufficiently large, we obtain \(\bar{d} (\widetilde{v}^\delta _n \chi _{({H}^\eta )^0}, \widetilde{v}^\delta \chi _{({H}^\eta )^0})\leqq \eta \) and

$$\begin{aligned} |{\overline{F}}'_{\mathrm {Dir}}(\widetilde{v}^\delta _n \chi _{({H}^\eta )^0}, {H}^\eta ) - {\overline{F}}'_{\mathrm {Dir}}(\widetilde{v}^\delta \chi _{({H}^\eta )^0}, {H}^\eta )| \leqq \eta . \end{aligned}$$

Step 3. Similar to item (ii) above, we obtain \(\widetilde{v}^\delta _n= u_0 *\psi _n \) in a neighborhood of \({\partial _D \varOmega }\). Therefore, it is enough to define \( {v}^\eta \) as \( {v}^\eta := \widetilde{v}^\delta _n + u_0 - u_0 *\psi _n \). Then by (A.6) and Step 2 we obtain \(\bar{d} ( v^\eta , v )\leqq C\eta \) and \({\overline{F}}'_{\mathrm {Dir}}( {v}^\eta , H^\eta ) \leqq {\overline{F}}'_{\mathrm {Dir}}(v,H) + C\eta \) for n sufficiently large. \(\quad \square \)

We now proceed with the proof of Lemma 6.6 which relies strongly on [17, Theorem 3.1]. Another main ingredient is the following Korn–Poincaré inequality in \({ GSBD}^p\), see [15, Proposition 3].

Proposition A.1

Let \(Q =(-\,r,r)^d\), \(Q'=(-\,r/2, r/2)^d\), \(u\in GSBD^p(Q)\), \(p\in [1,\infty )\). Then there exist a Borel set \(\omega \subset Q'\) and an affine function \(a:\mathbb {R}^d\rightarrow \mathbb {R}^d\) with \(e(a)=0\) such that \(\mathcal {L}^d(\omega )\leqq cr \mathcal {H}^{d-1}(J_u)\) and

$$\begin{aligned} \int \nolimits _{Q'{\setminus } \omega }(|u-a|^{p}) ^{1^*} \, \mathrm {d} x\leqq cr^{(p-1)1^*}\Bigg (\int \nolimits _Q|e(u)|^p\, \mathrm {d} x\Bigg )^{1^*}. \end{aligned}$$

If additionally \(p>1\), then there exists \(q>0\) (depending on p and d) such that, for a given mollifier \(\varphi _r\in C_c^{\infty }(B_{r/4}), \varphi _r(x)=r^{-d}\varphi _1(x/r)\), the function \( w=u \chi _{Q'{\setminus } \omega }+a\chi _\omega \) obeys

$$\begin{aligned} \int \nolimits _{Q''}|e( w *\varphi _r)-e(u)*\varphi _r|^p\, \mathrm {d} x\leqq c\left( \frac{\mathcal {H}^{d-1}(J_u)}{ r^{d-1} }\right) ^q \int \nolimits _Q|e(u)|^p\, \mathrm {d} x, \end{aligned}$$

where \(Q''=(-\,r/4,r/4)^d\). The constant in (A.7) depends only on p and d, the one in (A.8) also on \(\varphi _1\).

Proof of Lemma 6.6

We recall the definition of the hypercubes

$$\begin{aligned} \begin{aligned} q_z^k&:=z+(-\,k^{-1},k^{-1})^d,\qquad {\tilde{q}}_z^k:= z+(-\,2k^{-1},2k^{-1})^d,\\ Q_z^k&:=z+(-\,5k^{-1},5k^{-1})^d, \end{aligned} \end{aligned}$$

where in addition to the notation in (6.18), we have also defined the hypercubes \({\tilde{q}_z^k}\). In contrast to [17, Theorem 3.1], the cubes \({Q_z^k}\) have sidelength \(10k^{-1}\) instead of \(8k^{-1}\). This, however, does not affect the estimates. We point out that at some points in [17, Theorem 3.1] cubes of the form \(z+(-\,8k^{-1},8k^{-1})^d\) are used. By a slight alternation of the argument, however, it suffices to take cubes \(Q^k_z\). In particular it is enough to show the inequality [17, (3.19)] for a cube \(Q_j\) (of sidelength \(10k^{-1}\)) in place of \(\widetilde{Q}_j\) (of sidelength \(16k^{-1}\)), which may be done by employing rigidity properties of affine functions. Let us fix a smooth radial function \(\varphi \) with compact support on the unit ball \(B_1(0)\subset \mathbb {R}^d\), and define \(\varphi _k(x):=k^d\varphi (kx)\). We choose \(\theta < (16c)^{-1}\), where c is the constant in Proposition A.1 (cf. also [17, Lemma 2.12]). Recall (6.19) and set

$$\begin{aligned} \mathcal {N}'_k:=\{ z \in (2k^{-1}) \mathbb {Z}^d :{q_z^k}\cap (U)^k {\setminus }V \ne \emptyset \}. \end{aligned}$$

We apply Proposition A.1 for \(r = 4k^{-1}\), for any \(z \in \mathcal {N}'_k\) by taking v as the reference function and \(z+(-\,4k^{-1}, 4k^{-1})^d\) as Q therein. (In the following, we may then use the bigger cube \({Q_z^k}\) in the estimates from above.) Then, there exist \(\omega _z \subset {\tilde{q}_z^k}\) and \(a_z:\mathbb {R}^d\rightarrow \mathbb {R}^d\) affine with \(e(a_z)=0\) such that by (6.30), (A.7), and Hölder’s inequality it holds that

$$\begin{aligned} \mathcal {L}^d(\omega _z)\leqq 4 c k^{-1} \mathcal {H}^{d-1}(J_{v} \cap {Q_z^k}) \leqq 4 c \theta k^{-d}, \end{aligned}$$
$$\begin{aligned} \Vert v-a_z\Vert _{L^{p}({\tilde{q}_z^k}{\setminus } \omega _z)} \leqq 4 ck^{-1} \Vert e(v)\Vert _{L^p({Q_z^k})}. \end{aligned}$$

Moreover, by (6.30) and (A.8) it holds that

$$\begin{aligned}&\int \nolimits _{{q_z^k}}|e(\hat{v}_z*\varphi _k)-e(v)*\varphi _k|^p\, \mathrm {d} x\leqq c\left( \mathcal {H}^{d-1}(J_v \cap {Q_z^k})\,k^{d-1}\right) ^q \int \nolimits _{{Q_z^k}}|e(v)|^p\, \mathrm {d} x\\&\quad \leqq c \theta ^q \int \nolimits _{{Q_z^k}}|e(v)|^p\, \mathrm {d} x\end{aligned}$$

for \(\hat{v}_z:= v\chi _{{\tilde{q}_z^k}{\setminus } \omega _z}+a_z \chi _{\omega _z}\) and a suitable \(q>0\) depending on p and d. Let us set

$$\begin{aligned} \omega ^k:= \bigcup \nolimits _{ z \in \mathcal {N}_k' } \, \omega _{z}. \end{aligned}$$

We order (arbitrarily) the nodes \(z \in \mathcal {N}_k'\), and denote the set by \((z_j)_{j\in J}\). We define

$$\begin{aligned} \widetilde{w}_k:= {\left\{ \begin{array}{ll} v \quad &{}\text {in }\big (\bigcup _{z \in \mathcal {N}'_k} {Q_z^k}\big ) {\setminus } \omega ^k,\\ a_{z_j}\quad &{}\text {in }\omega _{z_j}{\setminus } \bigcup _{i<j}\omega _{z_i}, \end{array}\right. } \end{aligned}$$


$$\begin{aligned} w_k:= \widetilde{w}_k *\varphi _k \quad \text {in }(U)^k {\setminus }V. \end{aligned}$$

We have that \(w_k\) is smooth since \((U)^k {\setminus }V + \mathrm {supp} \,\varphi _k \subset \bigcup _{z \in \mathcal {N}'_k} {\tilde{q}_z^k}\subset U \) (recall (6.19)) and \( v|_{{\tilde{q}_z^k}{\setminus }\omega ^k} \in L^p({\tilde{q}_z^k}{\setminus }\omega ^k; \mathbb {R}^d )\) for any \(z \in \mathcal {N}'_k\), by (A.9b).

We define the sets \(G^k_1:=\{ z \in \mathcal {N}'_k :\mathcal {H}^{d-1}(J_v \cap {Q_z^k})\leqq k^{1/2 - d}\}\) and \(G^k_2:= \mathcal {N}'_k {\setminus }G^k_2\). By \(\widetilde{G}^k_1\) and \(\widetilde{G}^k_2\), respectively, we denote their “neighbors”, see [17, (3.11)] for the exact definition. We let

$$\begin{aligned} \widetilde{\varOmega }^k_{g,2}:= \bigcup \nolimits _{z \in \widetilde{G}^k_2} \, {Q_z^k}. \end{aligned}$$

It holds that (cf. [17, (3.8), (3.9), (3.12)])

$$\begin{aligned} \lim _{k\rightarrow \infty } \big ( \mathcal {L}^d(\omega ^k) + \mathcal {L}^d(\widetilde{\varOmega }^k_{g,2})\big ) = 0. \end{aligned}$$

At this point, we notice that the set \(E_k\) in [17, (3.8)] reduces to \(\omega ^k\) since in our situation all nodes are “good” (see (6.30) and [17, (3.2)]) and therefore \(\widetilde{\varOmega }^k_b\) therein is empty.

The proof of (3.1a), (3.1d), (3.1b) in [17, Theorem 3.1] may be followed exactly, with the modifications described just above and the suitable slight change of notation. More precisely, by [17, equation below (3.22)] we obtain

$$\begin{aligned} \Vert w_k-v\Vert _{L^p( ((U)^k {\setminus }V) {\setminus }\omega ^k)} \leqq Ck^{-1} \Vert e(v)\Vert _{L^p(U)}\, \end{aligned}$$

for a constant \(C>0\) depending only on d and p, and [17, equation before (3.26)] gives

$$\begin{aligned}&\int \nolimits _{\omega ^k} \psi (|w_k-v|) \, \mathrm {d} x\leqq C \Big ( \int \nolimits _{\omega ^k \cup \widetilde{\varOmega }^k_{g,2}} \big (1+\psi (|v|)\big ) \, \mathrm {d} x+ k^{-1/2}\nonumber \\&\quad \int \nolimits _U \big ( 1+\psi (|v|)\big ) \, \mathrm {d} x+ k^{-p}\int \nolimits _U |e(v)|^p \,\, \mathrm {d} x\Big ), \end{aligned}$$

where \(\psi (t) = t \wedge 1\). Combining (A.13)-(A.14), using (A.12), and recalling that \(\psi \) is sublinear, we obtain (6.31a). Note that the sequence \(R_k \rightarrow 0\) can be chosen independently of \(v \in \mathcal {F}\) since \(\psi (|v|) + |e(v)|^p\) is equiintegrable for \(v \in \mathcal {F}\).

Moreover, recalling (A.10)-(A.11), we sum [17, (3.34)] for \(z=z_j \in \widetilde{G}^k_2\) and [17, (3.35)] for \(z=z_j \in \widetilde{G}^k_1\) to obtain

$$\begin{aligned} \int \nolimits _{(U)^k {\setminus } V} |e(w_k)|^p \, \, \mathrm {d} x\leqq \int \nolimits _U |e(v)|^p \, \, \mathrm {d} x+ Ck^{-q'/2} \int \nolimits _U |e(v)|^p \, \, \mathrm {d} x+ C\int \nolimits _{\widetilde{\varOmega }^k_{g,2}} |e(v)|^p \, \, \mathrm {d} x\end{aligned}$$

for some \(q' >0\). This along with (A.12) and the equiintegrability of \(|e(v)|^p\) shows (6.31b). \(\quad \square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Crismale, V., Friedrich, M. Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity. Arch Rational Mech Anal 237, 1041–1098 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: