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

Abstract

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.

Auxiliary Results

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

Proof

Let (v, H) 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

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

(A.1)

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

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

(A.2)

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:

• (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)])

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

  (A.3)

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

• (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.

• (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

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

(A.4)

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 $\varepsilon$, $\delta$)

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

(A.5)

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

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

(A.6)

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

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

(A.7)

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

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

(A.8)

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

$$\begin{aligned} \mathcal {L}^d(\omega _z)\leqq 4 c k^{-1} \mathcal {H}^{d-1}(J_{v} \cap {Q_z^k}) \leqq 4 c \theta k^{-d}, \end{aligned}$$

(A.9a)

$$\begin{aligned} \Vert v-a_z\Vert _{L^{p}({\tilde{q}_z^k}{\setminus } \omega _z)} \leqq 4 ck^{-1} \Vert e(v)\Vert _{L^p({Q_z^k})}. \end{aligned}$$

(A.9b)

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

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

(A.10)

and

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

(A.11)

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)])

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

(A.12)

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

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

(A.13)

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

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

(A.14)

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). $\quad \square$