On the simultaneous homogenization and dimension reduction in elasticity and locality of $\varGamma$-closure

Abstract

We provide a framework for simultaneous homogenization and dimension reduction in the setting of linearized elasticity as well as non-linear elasticity for the derivation of homogenized von Kármán plate and bending rod models. The framework encompasses even perforated domains and domains with oscillatory boundary, provided that the corresponding extension operator can be constructed. Locality property of $\varGamma$-closure is established, i.e. every energy density obtained by the homogenization process can be in almost every point obtained as the limit of periodic energy densities.

Appendix

Theorem 4.1

(Korn’s inequalities, [17]) Let $p>1, \varOmega \subset {\mathbb {R}}^3$ and $\varGamma \subset \partial \varOmega$ of positive measure, then the following inequalities hold:

$$\begin{aligned} \Vert {\varvec{\psi }}\Vert _{W^{1,p}}^p&\le C_K\left( \Vert {\varvec{\psi }}\Vert _{L^p}^p + \Vert {\text {sym}}\,\nabla {\varvec{\psi }}\Vert _{L^p}^p\right) , \quad \forall {\varvec{\psi }}\in W^{1,p}(\varOmega ,{\mathbb {R}}^3), \end{aligned}$$

(59)

$$\begin{aligned} \Vert {\varvec{\psi }}\Vert _{W^{1,p}}^p&\le C_K^{\varGamma }(\Vert {{\varvec{\psi }}}\Vert ^p_{L^p(\varGamma )}+\left\| {\text {sym}}\,\nabla {\varvec{\psi }}\Vert _{L^p}^p\right) , \quad \forall {\varvec{\psi }}\in W^{1,p}(\varOmega ,{\mathbb {R}}^3), \end{aligned}$$

(60)

where positive constants $C_K$ and $C_K^{\varGamma }$ depend only on $p, \varOmega$ and $\varGamma$.

Theorem 4.2

(Griso’s decomposition, [14]) Let $\omega \subset {\mathbb {R}}^2$ with Lipschitz boundary and ${\varvec{\psi }}\in H^1(\omega \times I,{\mathbb {R}}^3)$, then for arbitrary $h>0$ the following identity holds

$$\begin{aligned} {\varvec{\psi }} = \hat{{\varvec{\psi }}}(x') + {\varvec{r}}(x')\wedge x_3{\varvec{e}}_3 + \bar{{\varvec{\psi }}}(x) = \left\{ \begin{array}{l} \hat{\psi }_1(x') + r_2(x')x_3 + \bar{\psi }_1(x)\\ \hat{\psi }_2(x') - r_1(x')x_3 + \bar{\psi }_2(x)\\ \hat{\psi }_3(x') + \bar{\psi }_3(x) \end{array} \right. , \end{aligned}$$

(61)

where

$$\begin{aligned} \hat{{\varvec{\psi }}}(x') = \int _I {\varvec{\psi }}(x',x_3){\mathrm {d}}x_3,\quad {\varvec{r}}(x') = \frac{3}{2}\int _Ix_3{\varvec{e}}_3\wedge {\varvec{\psi }}(x',x_3){\mathrm {d}}x_3. \end{aligned}$$

(62)

Moreover, the following inequality holds

$$\begin{aligned} \Vert {\text {sym}}\,\nabla _h(\hat{{\varvec{\psi }}} + {\varvec{r}}\wedge x_3{\varvec{e}}_3)\Vert _{L^2}^2 + \Vert \nabla _h\bar{{\varvec{\psi }}}\Vert _{L^2}^2 + \frac{1}{h^2}\Vert \bar{{\varvec{\psi }}}^h\Vert _{L^2}^2 \le C\Vert {\text {sym}}\,\nabla _h{\varvec{\psi }}\Vert _{L^2}^2, \end{aligned}$$

(63)

with constant $C>0$ depending only on $\omega$.

The following corollary is the direct consequence of Theorem 4.2 [relation (63), i.e. (65)], Poincare and Korn inequalities.

Corollary 4.2

(Korn’s inequalities for thin domains) Let $\omega \subset {\mathbb {R}}^2$ and $\varGamma \subset \partial \varOmega$ of positive measure, then the following inequalities hold

$$\begin{aligned} \Vert (\psi _1,\psi _2, h\psi _3)\Vert ^2_{H^1}&\le C_T\left( \Vert (\psi _1,\psi _2, h\psi _3)\Vert _{L^2}^2 + \Vert {\text {sym}}\,\nabla _h {\varvec{\psi }}\Vert _{L^2}^2\right) ,\\&\quad \forall {\varvec{\psi }}\in H^1(\omega \times I,{\mathbb {R}}^3);\,\\ \Vert (\psi _1,\psi _2, h\psi _3)\Vert _{H^1}^2&\le C_T^{\varGamma }(\Vert |(\psi _1,\psi _2, h\psi _3)\Vert ^2_{L^2(\varGamma )}+\left\| {\text {sym}}\,\nabla _h {\varvec{\psi }}\Vert _{L^2}^2\right) , \\&\quad \forall {\varvec{\psi }}\in H^1(\omega \times I,{\mathbb {R}}^3), \end{aligned}$$

where positive constants $C_T$ and $C_T^{\varGamma }$ depend only on $\omega$ and $\varGamma$.

The following lemma tells us additional information on the weak limit of sequence that has bounded symmetrized scaled gradients.

Lemma 4.3

Let $\omega \subset {\mathbb {R}}^2$ be a bounded set with Lipschitz boundary and $(h_n)_n$ monotonically decreasing to zero sequence of positive reals. Let $({\varvec{\psi }}^{h_n})_{n}\subset H^1(\omega \times I,{\mathbb {R}}^3)$ which for all $n\in {\mathbb {N}}$ equals zero on $\varGamma _d\times I\subset \partial \omega \times I$ of strictly positive surface measure, and

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert {\text {sym}}\,\nabla _{h_n}{\varvec{\psi }}^{h_n}\Vert _{L^2} < \infty , \end{aligned}$$

then there exists a subsequence (still denoted by $(h_n)_n$) for which it holds

$$\begin{aligned} {\text {sym}}\,\nabla _{h_n}{\varvec{\psi }}^{h_n} = \imath (-x_3\nabla '^2 v + {\text {sym}}\,\nabla ' {\varvec{w}}) + {\text {sym}}\,\nabla _{h_n}\bar{{\varvec{\psi }}}^{h_n}, \end{aligned}$$

for some $v\in H_{\varGamma _d}^2(\omega ), {\varvec{w}}\in H_{\varGamma _d}^1(\omega ,{\mathbb {R}}^2)$ and a sequence $(\bar{{\varvec{\psi }}}^{h_n})_{n}\subset H^1(\omega \times I,{\mathbb {R}}^3)$ satisfies $\bar{{\varvec{\psi }}}^{h_n} = 0$ on $\varGamma _d\times I$ and $(\bar{\psi }_1^{h_n},{\bar{\psi }}_2^{h_n},h_n{\bar{\psi }}_3^{h_n})\rightarrow 0$ in the $L^2$-norm. Furthermore,

$$\begin{aligned} \Vert v\Vert _{L^2}^2 + \Vert {\varvec{w}}\Vert _{L^2}^2 \le \limsup _{n\rightarrow \infty }\left\| \left( \psi _1^{h_n},\psi _2^{h_n},h_n\psi _3^{h_n}\right) \right\| _{L^2}^2. \end{aligned}$$

(64)

Proof

Applying the Griso’s decomopsition (Theorem 4.2) on each ${{\varvec{\psi }}}^{h_n}$, we find

$$\begin{aligned} {{\varvec{\psi }}}^{h_n}(x) = \hat{{\varvec{\psi }}}^{h_n}(x') + {{\varvec{r}}}^{h_n}(x')\wedge x_3{{\varvec{e}}}_3 + \bar{{\varvec{u}}}^{h_n}(x), \end{aligned}$$

with sequences $\hat{{\varvec{\psi }}}^{h_n}(x') = \int _I {\varvec{\psi }}^{h_n}(x',x_3){\mathrm {d}}x_3, {\varvec{r}}^{h_n}(x')= \frac{3}{2}\int _Ix_3{\varvec{e}}_3\wedge {\varvec{\psi }}^{h_n}(x',x_3){\mathrm {d}}x_3$ and $\bar{{\varvec{\psi }}}^{h_n}$ satisfying the following a priori estimate

$$\begin{aligned}&\left\| {\text {sym}}\,\nabla '\left( \hat{\psi }_1^{h_n},{\hat{\psi }}_2^{h_n}\right) \right\| _{L^2}^2 + \frac{1}{12}\left\| {\text {sym}}\,\nabla '\left( r_2^{h_n},-r_1^{h_n}\right) \right\| _{L^2}^2 \nonumber \\&\quad + \frac{1}{2h_n^2}\left\| \partial _1\left( h_n\hat{\psi }_3^{h_n}\right) + r_2^{h_n}\right\| _{L^2}^2 \nonumber \\&\quad + \frac{1}{2h_n^2}\left\| \partial _2\left( h_n\hat{\psi }_3^{h_n}\right) - r_1^{h_n}\right\| _{L^2}^2\, + \Vert \nabla _{h_n}\bar{{\varvec{\psi }}}^{h_n}\Vert _{L^2}^2 \nonumber \\&\quad + \frac{1}{h_n^2}\Vert \bar{{\varvec{\psi }}}^{h_n}\Vert _{L^2}^2 \le C\Vert {\text {sym}}\,\nabla _{h_n}{\varvec{\psi }}^{h_n}\Vert _{L^2}^2 \le C. \end{aligned}$$

(65)

Together with the Korn’s inequality with boundary condition (cf. Theorem 4.1), the above estimate implies (on a subsequence)

$$\begin{aligned} (\hat{\psi }_1^{h_n},{\hat{\psi }}_2^{h_n}) \rightharpoonup {{\varvec{w}}}\quad \text {weakly in }H^1(\omega ,{\mathbb {R}}^2)\quad \text {and} \quad {{\varvec{r}}}^{h_n} \rightharpoonup {{\varvec{r}}}\quad \text {weakly in }H^1(\omega ,{\mathbb {R}}^2). \end{aligned}$$

Furthermore, using the triangle inequality and estimate (65)

$$\begin{aligned} \partial _1(h_n\hat{{\varvec{\psi }}}^{h_n}_3) \rightarrow -r_2\quad \text {and}\quad \partial _2(h_n\hat{{\varvec{\psi }}}^{h_n}_3) \rightarrow r_1 \quad \text {strongly in }L^2(\omega ). \end{aligned}$$

By the compactness of the trace operator, ${{\varvec{r}}}\in H^1_{\varGamma _d}(\omega ,{\mathbb {R}}^2)$, thus, by the Korn’s inequality, there exists $v\in H^2_{\varGamma _d}(\omega )$ such that

$$\begin{aligned} h_n\hat{{\varvec{\psi }}}^{h_n}_3 \rightarrow v \quad \text { strongly in }H^1(\omega ),\quad \text {and}\quad r_1 = \partial _2 v,\quad r_2 = -\partial _1v. \end{aligned}$$

Defining

$$\begin{aligned} {\bar{{\varvec{\psi }}}}^{h_n}(x) = {\varvec{\psi }}^{h_n}(x) - \left( \begin{array}{c}{\varvec{w}}(x') \\ \dfrac{v(x')}{h_n}\end{array}\right) + x_3\left( \begin{array}{c} \nabla 'v(x') \\ 0 \end{array} \right) , \end{aligned}$$

it is easy to check $\left( {\bar{\psi }}_1^{h_n},{\bar{\psi }}_2^{h_n},h_n{\bar{\psi }}_3^{h_n}\right) \rightarrow 0$ in the $L^2$-norm, which finishes the proof. $\square$

The following lemma is given in [26, Proposition 3.3]. It is a consequence of Theorem 4.2. It tells us how we can further decompose the sequence of deformations that has bounded symmetrized scaled gradient.

Lemma 4.4

Let $A \subset {\mathbb {R}}^2$ with $C^{1,1}$ boundary. Denote by $\{A_i\}_{i=1,\dots ,k}$ the connected components of A. Let $({{\varvec{\psi }}}^{h_n})_n \subset H^1(A \times I,{\mathbb {R}}^3)$ be such that

$$\begin{aligned}&\left( \psi _1^{h_n},\psi _2^{h_n},h_n\psi _3^{{h_n}}\right) \rightarrow 0, \text { strongly in } L^2,\quad \forall n \in {\mathbb {N}},\ \forall i=1, \dots ,k ,\ \int _{A_i} \psi _3^{h_n}=0, \\&\limsup _{n \rightarrow \infty } \Vert {\text {sym}}\,\nabla _{h_n} {\varvec{\psi }}^{h_n}\Vert _{L^2} \le M<\infty . \end{aligned}$$

Then there exist $(\varphi ^{h_n})_{n \in {\mathbb {N}}} \subset H^2(A), (\tilde{{{\varvec{\psi }}}}^{h_n})_{n \in {\mathbb {N}}} \subset H^1(A \times I ,{\mathbb {R}}^3)$ such that

$$\begin{aligned} {\text {sym}}\,\nabla _{h_n}{{\varvec{\psi }}}^{h_n}=-x_3\iota ({\text {sym}}\,\nabla '^2 \varphi ^{h_n})+{\text {sym}}\,\nabla _{h_n} \tilde{{\varvec{\psi }}}^{h_n}+o^{h_n}, \end{aligned}$$

where $o^{h_n} \in L^2(A \times I,{\mathbb {R}}^{3 \times 3})$ is such that $o^{h_n} \rightarrow 0$, strongly in $L^2$, and the following properties hold

$$\begin{aligned}&\lim _{n \rightarrow \infty } \left( \Vert \varphi ^{h_n}\Vert _{H^1}+\Vert \tilde{{{\varvec{\psi }}}}^{h_n}\Vert _{L^2} \right) =0,\\&\limsup _{n \rightarrow \infty } \left( \Vert \varphi ^{h_n} \Vert _{H^2}+\Vert \nabla _{h_n} \tilde{{{\varvec{\psi }}}}^{h_n}\Vert _{L^2} \right) \le C(A) M. \end{aligned}$$

The following lemma is given in [26, Lemma 3.6]. Although the claim can be put in more general form (since the argument is simple truncation), we state it only in the form we need here.

Lemma 4.5

Let $p\ge 1$ and $A\subset {\mathbb {R}}^2$ be an open, bounded set. Let $({{\varvec{\psi }}}^{h_n})_{n \in {\mathbb {N}}} \subset W^{1,p}(A \times I,{\mathbb {R}}^3)$ and $(\varphi ^{h_n})_{n \in {\mathbb {N}}} \subset W^{2,p}(A)$. Suppose that $\left( |\nabla '^2 \varphi ^{h_n}|^p\right) _{n\in {\mathbb {N}}}$ and $\left( | \nabla _{h_n} {{\varvec{\psi }}}^{h_n} |^p\right) _{n \in {\mathbb {N}}}$ are equi-integrable and

$$\begin{aligned} \lim _{n \rightarrow \infty } \left( \Vert \varphi ^{h_n} \Vert _{W^{1,p}}+\Vert {\varvec{\psi }}^{h_n} \Vert _{L^p} \right) =0. \end{aligned}$$

Then there exist sequences $(\tilde{\varphi }^{h_n})_{n \in {\mathbb {N}}} \subset W^{2,p}(A), (\tilde{\psi }^{h_n})_{n\in {\mathbb {N}}}\subset W^{1,p}(A \times I,{\mathbb {R}}^3)$ and a sequence of sets $(A_n)_{n\in {\mathbb {N}}}$ such that for each $n \in {\mathbb {N}}, A_n \ll A_{n+1} \ll A$ and $\cup _{n \in {\mathbb {N}}} A_n=A$ and

• (a) $\tilde{\varphi }^{h_n}=0, \nabla ' \tilde{\varphi }^{h_n}=0$ in a neighborhood of $\partial A$, $\tilde{{{\varvec{\psi }}}}^{h_n}=0$ in a neighborhood of $\partial A \times I$;

• (b) $\tilde{{{\varvec{\psi }}}}^{h_n}={{\varvec{\psi }}}^{h_n} \text { on } A_n \times I,\ \tilde{\varphi }^{h_n}=\varphi ^{h_n} \text { on } A_n$;

• (c) $\Vert \tilde{\varphi }^{h_n}-\varphi ^{h_n}\Vert _{W^{2,p}}\rightarrow 0,\ \Vert \tilde{{{\varvec{\psi }}}}^{h_n}-{{\varvec{\psi }}}^{h_n}\Vert _{W^{1,p}}\rightarrow 0, \Vert \nabla _{h_n} \tilde{{{\varvec{\psi }}}}^{h_n}-\nabla _{h_n}{{\varvec{\psi }}}^{h_n}\Vert _{L^p} \rightarrow 0$, as $n \rightarrow \infty$.

The following two lemmas are given in [13] and [6]. The proof of the second claim in the first lemma is given in [26, Proposition A.5] as an adaptation of the result given in [13].

Lemma 4.6

Let $p>1$ and $A \subset {\mathbb {R}}^n$ be an open bounded set.

• (a) Let $( w^n)_{n \in {\mathbb {N}}}$ be a bounded sequence in $W^{1,p}(A)$. Then there exist a subsequence $( w^{n(k)})_{k \in {\mathbb {N}}}$ and a sequence $( z^k)_{k \in {\mathbb {N}}} \subset W^{1,p}(A)$ such that

  $$\begin{aligned} |\{ z^k \ne w^{n(k)} \}| \rightarrow 0 , \end{aligned}$$

  as $k \rightarrow \infty$ and $\big (|\nabla z_k|^p \big )_{k \in {\mathbb {N}}}$ is equi-integrable. Each $z^k$ may be chosen to be Lipschitz function. If $w^n \rightharpoonup w$ weakly in $W^{1,p}$, then $z^k \rightharpoonup w$ weakly in $W^{1,p}$.

• (b) Let $( w^n)_{n \in {\mathbb {N}}}$ be a bounded sequence in $W^{2,p}(A)$. Then there exist a subsequence $( w^{n(k)})_{k \in {\mathbb {N}}}$ and a sequence $( z^k)_{k \in {\mathbb {N}}} \subset W^{2,p}(A)$ such that

  $$\begin{aligned} |\{ z^k \ne w^{n(k)} \}| \rightarrow 0 , \end{aligned}$$

  as $k \rightarrow \infty$ and $\big (|\nabla ^2 z^k|^p \big )_{k \in {\mathbb {N}}}$ is equi-integrable. Each $z^k$ may be chosen such that $z^k \in W^{2,\infty }(S)$. If $w^n \rightharpoonup w$ weakly in $W^{2,p}$, then $z^k \rightharpoonup w$ weakly in $W^{2,p}$.

Lemma 4.7

Let $p>1$ and $A \subset {\mathbb {R}}^2$ be an open bounded set with Lipschitz boundary. Let $(h_n)_{n\in {\mathbb {N}}}$ be a sequence of positive numbers converging to 0 and let $({{\varvec{w}}}^{h_n})_{n \in {\mathbb {N}}} \subset W^{1,p} (A \times I, {\mathbb {R}}^3)$ be a bounded sequence satisfying:

$$\begin{aligned} \limsup _{n \in {\mathbb {N}}} \Vert \nabla _{h_n} {{\varvec{w}}}^{h_n}\Vert _{L^p}<+\infty . \end{aligned}$$

Then there exists a subsequence $({{\varvec{w}}}^{h_{n(k)}})_{k \in {\mathbb {N}}}$ and a sequence $({{\varvec{z}}}^{h_{n(k)}})_{k \in {\mathbb {N}}}$ such that

$$\begin{aligned} |{{\varvec{z}}}^{h_{n(k)}} \ne {{\varvec{w}}}^{h_{n(k)}} | \rightarrow 0, \end{aligned}$$

as $k \rightarrow \infty$ and $|\nabla _{h_{n(k)}} {{\varvec{z}}}^{h_{n(k)}}|^p$ is equi-integable. If ${{\varvec{w}}}^{h_{n}}\rightharpoonup {{\varvec{w}}} \in W^{1,p}(A,{\mathbb {R}}^3)$ weakly in $W^{1,p}$, then ${{\varvec{z}}}^{h_{n(k)}}\rightharpoonup {{\varvec{w}}}$ in $W^{1,p}$. Each ${{\varvec{z}}}^{h_{n(k)}}$ may be chosen such that $|\nabla _{h_{n(k)}}{{\varvec{z}}}^{h_{n(k)}}|$ is bounded.

The following lemma is given in [26, Lemma 3.4]. For the sake of completeness, we will give also the proof.

Lemma 4.8

(continuity in ${{\varvec{M}}}$) Under the uniform coerciveness and boundedness assumption (2), there exists a constant $C>0$ depending only on $\alpha$ and $\beta$, such that for every sequence $(h_n)_{n}$ monotonically decreasing to zero and $A \subset \omega$ open set with Lipschitz boundary it holds:

$$\begin{aligned} \left| \mathcal {K}_{(h_n)}^\pm ({{\varvec{M}}}_1,A)-\mathcal {K}_{(h_n)}^\pm ({{\varvec{M}}}_2,A)\right|\le & {} C \Vert {{\varvec{M}}}_1-{{\varvec{M}}}_2\Vert _{L^2}\left( \Vert {{\varvec{M}}}_1\Vert _{L^2}+\Vert {{\varvec{M}}}_2\Vert _{L^2}\right) ,\nonumber \\&\ \forall {{\varvec{M}}}_1, {{\varvec{M}}}_2 \in \mathcal S(\omega ), \end{aligned}$$

(66)

Proof

Due to relation (13), it is enough to assume $A=\omega$. For fixed ${{\varvec{M}}}_1,\, {{\varvec{M}}}_2 \in \mathcal S(\omega )$ and $r,\,h_n>0$ take arbitrary ${{\varvec{\psi }}}_{{{\varvec{M}}}_\alpha }^{r,h_n}\in H^1(\varOmega ,{\mathbb {R}}^3)$, which for $\alpha =1,2$ satisfy:

$$\begin{aligned}&\int _\varOmega Q^{h_n}\left( x,\iota ({{\varvec{M}}}_\alpha )+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_\alpha }^{r,h_n}\right) {\mathrm {d}}x \nonumber \\&\quad \le \inf _{\begin{array}{c} {{\varvec{\psi }}} \in H^1(\varOmega ,{\mathbb {R}}^3)\\ \Vert (\psi _1,\psi _2,h_n\psi _3)\Vert _{L^2} \le r \end{array}} \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_\alpha )+\nabla _{h_n} {{\varvec{\psi }}}\right) {\mathrm {d}}x + h_n. \nonumber \\&\Vert (\psi _{\alpha ,1}^{r,h_n},\psi _{\alpha ,2}^{r,h_n},h_n\psi _{\alpha ,3}^{r,h_n})\Vert _{L^2} \le r\, \end{aligned}$$

(67)

We want to prove that for every $r>0$ we have

$$\begin{aligned}&\left| \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1 )+\nabla _{h_n}{{\varvec{\psi }}}_{{{\varvec{M}}}_1 }^{r,h_n}\right) {\mathrm {d}}x - \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_2 )+\nabla _{h_n} {\varvec{\psi }}_{{{\varvec{M}}}_2}^{r,h_n}\right) {\mathrm {d}}x \right| \nonumber \\&\quad \le C \Vert {{\varvec{M}}}_1-{{\varvec{M}}}_2\Vert _{L^2}\left( \Vert {{\varvec{M}}}_1\Vert _{L^2}+\Vert {{\varvec{M}}}_2\Vert _{L^2}\right) + h_n. \end{aligned}$$

(68)

From that, (66) can be easily obtained by using (14) for a family of balls of radius r.

Let us prove (68). From (67) and (2), by testing with zero function, we can assume for $\alpha =1,2$

$$\begin{aligned} \alpha \Vert {{\varvec{M}}}_\alpha + {\text {sym}}\,\nabla _{h_n} {{\varvec{\psi }}}^{r,h_n}_{{{\varvec{M}}}_\alpha }\Vert ^2_{L^2} \le \int _{A \times I} Q^{h_n}\left( x,\iota ({{\varvec{M}}}_\alpha )+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_\alpha }^{r,h_n}\right) {\mathrm {d}}x \le \beta \Vert {{\varvec{M}}}_{\alpha }\Vert ^2_{L^2}. \end{aligned}$$

From this we have for $\alpha =1,2$

$$\begin{aligned} \Vert {\text {sym}}\,\nabla _{h_n} {{\varvec{\psi }}}^{r,h_n}_{{{\varvec{M}}}_\alpha }\Vert ^2_{L^2} \le C(\alpha ,\beta ) \Vert {{\varvec{M}}}_{\alpha }\Vert ^2_{L^2}. \end{aligned}$$

(69)

Without any loss of generality we can also assume that

$$\begin{aligned} \int _{\varOmega } Q^{h_n} \left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {\varvec{\psi }}_{{{\varvec{M}}}_1}^{r,h_n}\right) {\mathrm {d}}x \ge \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_2)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_2}^{r,h}\right) {\mathrm {d}}x. \end{aligned}$$

(70)

We have

$$\begin{aligned}&\left| \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_1}^{r,h_n}\right) {\mathrm {d}}x - \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_2)+\nabla _{h_n} {\varvec{\psi }}_{{{\varvec{M}}}_2}^{r,h_n}\right) {\mathrm {d}}x\right| \\&= \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_1}^{r,h_n}\right) {\mathrm {d}}x - \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_2)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_2}^{r,h_n}\right) {\mathrm {d}}x \\&= \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_1}^{r,{h_n}}\right) {\mathrm {d}}x - \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {\varvec{\psi }}_{2}^{r,h_n}\right) {\mathrm {d}}x \\&+ \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_1)+\nabla _{h_n} {{\varvec{\psi }}}_{{{\varvec{M}}}_2}^{r,h_n}\right) {\mathrm {d}}x - \int _{\varOmega } Q^{h_n}\left( x,\iota ({{\varvec{M}}}_2)+\nabla _{h_n} {\varvec{\psi }}_{{{\varvec{M}}}_2}^{r,h_n}\right) {\mathrm {d}}x \\&\le h_n + C(\alpha ,\beta ) \Vert {{\varvec{M}}}_1-{{\varvec{M}}}_2\Vert _{L^2}\left( \Vert {{\varvec{M}}}_1\Vert _{L^2}+\Vert {{\varvec{M}}}_2\Vert _{L^2}\right) , \end{aligned}$$

where we used (70), (3) and (69) respectively. This concludes the proof. $\square$