Lower Bounds of Optimal Exponentials of Thickness in Geometry Rigidity Inequality for Shells

TL;DR

This paper establishes lower bounds for the exponential factors related to the geometry rigidity inequality in shells, revealing how these bounds depend on the shell's curvature type.

Contribution

It provides the first explicit lower bounds for the optimal exponentials in the geometry rigidity inequality for different shell geometries.

Findings

Lower bound of 4/3 for hyperbolic shells

Lower bound of 3/2 for parabolic shells

Lower bound of 1 for elliptic shells

Abstract

The optimal exponentials of the thickness in the geometry rigidity inequality of shells represent the geometry rigidity of the shells. We obtain that the lower bounds of the optimal exponentials are $4/3,$ $3/2,$ and $1,$ for the hyperbolic shell, the parabolic shell, and the elliptic shell, respectively, through the construction of the Ans\"{a}tze.

Equations

$$\Omega =\{p+t\vec{n}(p)|p\in S,-h/2<t<h/2\}.$$

$$\kappa (p)=0,|\Pi (p)|>0\text{mbox}forp\in \overline{S},$$

$$\kappa (p)>0\text{mbox}forp\in \overline{S},$$

$$\kappa (p)<0\text{mbox}forp\in \overline{S}.$$

$$\Vert \nabla u-Q{\Vert }_{L^2(\Omega )}^2\le \frac{C}{h^{\mu }}{\int }_{\Omega }{\mathrm{dist}}^2(\nabla u(z),\mathrm{SO}(3))dz.$$

$$\mu (\Omega )=\inf \{\mu |\mu >0\text{mbox}suchthat(\text{ref}1.1)holds\}.$$

$$\mu (\Omega )=2$$

$$\mu (\Omega )\le 2$$

$$\mu (\Omega )\ge 4/3.$$

$$\mu (\Omega )\ge 1.$$

$$\gamma (0,p)=p,|\dot{\gamma }(t,p)|=1,{\nabla }_{\dot{\gamma }(t,p)}\vec{n}=0\text{mbox}fort\in IR.$$

$$\gamma (t_{\pm },p_0)\in \partial S,({\nabla }_{{\tau }_{\pm }}\vec{n})(\gamma (t_{\pm },p_0))\ne 0,\gamma (t,p_0)\in S\text{mbox}fort\in (t_{-},t_{+}),$$

$$\mu (\Omega )\ge 3/2.$$

$$S=\{\mathbf{r}(z,\theta )|(z,\theta )\in [1,1+l]\times [0,{\theta }_0]\},$$

$${\nabla }_{\partial z}\vec{n}={\kappa }_z\partial z,{\nabla }_{\partial \theta }\vec{n}={\kappa }_{\theta }\partial \theta \text{mbox}forp\in S$$

$$\det\Big{(}e_{1},e_{2},\vec{n}(p)\Big{)}=1.$$

$$Q\alpha =⟨\alpha ,e_2⟩e_1-⟨\alpha ,e_1⟩e_2\text{mbox}forall\alpha \in T_{p}M.$$

$$\det\Big{(}\hat{e}_{1},\hat{e}_{2},\vec{n}(p)\Big{)}=1.$$

$${\hat{e}}_i=\sum _{j=1}^2{\alpha }_{ij}{e}_j\text{mbox}fori=1,2.$$

$$1=\mathcal{E}({\hat{e}}_1,{\hat{e}}_2)={\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}{\alpha }_{21}.$$

$$⟨\alpha ,{\hat{e}}_2⟩{\hat{e}}_1-⟨\alpha ,{\hat{e}}_1⟩{\hat{e}}_2=⟨\alpha ,e_2⟩e_1-⟨\alpha ,e_1⟩e_2.$$

$$Q^T=-Q,Q^2=-\mathrm{Id}.$$

$$\Pi (\partial x_1,\partial x_1)=\Pi (\partial x_2,\partial x_2)=0\text{mbox}forp\in \mathcal{B}(p_0,3\delta ),$$

$$\det\Big{(}\partial x_{1},\partial x_{2},\vec{n}(p)\Big{)}>0{\quad\mbox{for}\quad}p\in{\,\cal B\,}(p_{0},3\delta).$$

$$\phi (p)=1\text{mbox}forp\in \mathcal{B}(p_0,2\delta ).$$

$$f(p)=\phi (p)x_1(p)\text{mbox}forp\in S,$$

$$f(p)=0\text{mbox}forp\in S/\mathcal{B}(p_0,3\delta ).$$

$$\Pi (QDf,QDf)(p)=0\text{mbox}forp\in \mathcal{B}(p_0,2\delta ),$$

$$E_1=a_1\partial x_1,E_2=b_1\partial x_1+b_2\partial x_2,$$

$$a_1=\frac{1}{\sqrt{g_{11}}},b_1=-\frac{g_{12}}{\sqrt{g_{11}\det G}},b_2=\sqrt{\frac{g_{11}}{\det G}},$$

Full text

Lower Bounds of Optimal Exponentials of Thickness in Geometry Rigidity Inequality for Shells

Peng-Fei YAO

Key Laboratory of Systems and Control

Institute of Systems Science, Academy of Mathematics and Systems Science

Chinese Academy of Sciences, Beijing 100190, P. R. China

School of Mathematical Sciences

University of Chinese Academy of Sciences, Beijing 100049, China

e-mail: pfyao@iss.ac.cn

This work is supported by the National Science Foundation of China, grants no. 61473126 and no. 61573342, and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011.

**Abstract    The optimal exponentials of the thickness in the geometry rigidity inequality of shells represent the geometry rigidity of the shells. We obtain that the lower bounds of the optimal exponentials are $4/3,$ $3/2,$ and $1,$ for the hyperbolic shell, the parabolic shell, and the elliptic shell, respectively, through the construction of the Ansätze.

Keywords   geometry rigidity inequality, shell, nonlinear elasticity, Riemannian geometry

Mathematics Subject Classifications (2010)   74K20(primary), 74B20(secondary). **

1 Introduction and Main Results

The geometry rigidity inequality, namely the Friesecke-James-Muller estimate [10, 11], plays a central role in models in nonlinear elasticity. In their basic form, these estimates assert that for a deformation $u\in H^{1}(\Omega,I\!\!R^{n})$ the distance of $\nabla u$ to a suitably chosen proper rotation $Q\in{\,\rm SO\,}(n)$ is dominated in $L^{2}$ by the distance function of $\nabla u$ to ${\,\rm SO\,}(n).$ The proof [10] is based on the fact that the nonlinear estimate can be related to the linear one since the tangent space to the smooth manifold ${\,\rm SO\,}(n)$ at the identity matrix is given by the linear space of all skew-symmetric matrices. In fact, geometric rigidity results are the cornerstone of rigorous derivations of two dimensional plate and shell theories from three-dimensional models in the framework of nonlinear elasticity theory. The $L^{2}$ version by Friesecke et al. [10] generalized previous work [20, 21, 22, 38, 39] and allowed for the first time the derivation of limiting theories as the thickness of the three-dimensional structure tends to zero without a priori assumptions on the deformations in various scaling regimes [9, 10, 11, 19, 27, 28, 29, 46] and many others.

It is known that the rigidity of a shell is closely related to the optimal constant of thickness in the geometric rigidity estimate [6, 7, 11, 14] and the optimal constant is crucial to shell theories being derived from $3$-dimensional elasticity by $\Gamma$-convergence like [9, 10, 19, 27, 28, 29, 46]. As a linear version of the geometric rigidity estimate, the the optimal constants of thickness in Korn’s inequalities have been calculated subject to the Gaussian curvature of the middle surface of a shell under the assumption that the middle surface is given by a single principal curvature coordinate [13, 15, 18]. This assumption that the middle surface being a single principal curvature coordinate is generalized in [47, 48]. In Korn’s inequalities the optimal constant for the plate calculated in [11] scales like $h^{2},$ for cylindrical shells in [13], $h^{3/2},$ for the positive curvature in [18], $h,$ and for the negative curvature in [18], $h^{4/3},$ respectively. It is expected that the analogous nonlinear estimates will have the same scaling of the constant in terms of the shell thickness.

Here we calculate some lower bounds of the optimal exponentials of the thickness in the nonlinear geometry rigidity inequality subject to the curvature of the middle surface.

Let $M\subset I\!\!R^{3}$ be a ${\,\rm C\,}^{3}$ surface with the induce metric $g$ and a normal field $\vec{n}.$ Let $S\subset M$ be an open, simply connected, bounded set with a regular boundary $\partial S.$ We consider a shell with thickness $h>0$

[TABLE]

Let $\kappa$ be the Gaussian curvature of $M.$ We say that $\Omega$ is parabolic if

[TABLE]

where $\Pi=\nabla\vec{n}$ is the second fundamental form of $M.$ If

[TABLE]

then $\Omega$ is said to be elliptic. In addition, $\Omega$ is said to be hyperbolic if

[TABLE]

For $A\in I\!\!R^{3\times 3},$ we denote the Euclidean norm by $|A|=\sqrt{{{\,\rm tr\,}AA^{T}}}.$ The distance from $A$ to ${\,\rm SO\,}(3)$ is denoted ${\,\rm dist\,}(A,{\,\rm SO\,}(3)).$ Let $\mu>0$ be such that estimate (1.3) below holds true. There is a constant $C>0,$ independent of $h>0,$ such that for every $u\in H^{1}(\Omega,I\!\!R^{3})$ there exists a constant rotation $Q\in{\,\rm SO\,}(3),$ such that

[TABLE]

Set

[TABLE]

It follows from [11] that

[TABLE]

if $\Omega$ is a plate, and from [27] that

[TABLE]

for a shell, respectively.

We have the following.

**     Theorem 1.1**

If $\Omega$ is hyperbolic, then

[TABLE]

In the case of $\Omega$ being elliptic,

[TABLE]

Next, we consider the case of the parabolic. We need

**     Proposition 1.1**

Let $M\subset I\!\!R^{3}$ be a parabolic surface without boundary. Then for given $p\in M,$ there exists a unique regular geodesic $\gamma(t,p)$ on $M$ such that

[TABLE]

Moreover, $\gamma(t,p)$ is a straight line in $I\!\!R^{3}.$

**     Theorem 1.2**

Let $M\subset I\!\!R^{3}$ be a parabolic surface and let $S\subset M$ be a bounded open set. We suppose that there is a point $p_{0}\in S$ such that the following assumption holds: Let $t_{-}<0<t_{+}$ be such that

[TABLE]

where $\gamma$ is given in $(\ref{pra})$ and $\tau_{\pm}\in T_{\gamma(t_{\pm},p_{0})}\partial S$ with $|\tau_{\pm}|=1.$ Then

[TABLE]

**     Remark 1.1**

It is conjectured that all the equal signs in $(\ref{e2}),$ $(\ref{e1}),$ and $(\ref{e3})$ hold.

The estimates (1.4), (1.5), and (1.8) will be obtain by constructing the Ansätze. The main observation is that such Ansätzes may come from the improvement in the ones for the Korn inequality. In the case of the Korn inequality the Ansätzes are constructed in [13], [15], and [18], which are based on the main assumption that the middle surface is given by a single principal curvature coordinate, i.e.,

[TABLE]

where the properties

[TABLE]

hold. In the case of the parabolic or hyperbolic shell, a principal coordinate only exists locally ([48]). There is even no such a local existence for the elliptic shell.

Here we will construct the Ansätze for the korn inequality without assumption (1.9) and then improve them to obtain the ones for the geometric rigidity estimate.

2 Proof of the Main Results

Let $\nabla$ and $D$ denote the connection of $I\!\!R^{3}$ in the Euclidean metric and the one of $M$ in the reduced metric, respectively. We have to treat the relationship between $\nabla$ and $D$ carefully.

We need a linear operator $Q$ as follows. Let $M$ be oriented and ${\cal E}$ be the volume element of $M$ with the positive orientation. Let $p\in M$ be given and let $e_{1},$ $e_{2}$ be an orthonormal basis of $T_{p}M$ with positive orientation, that is,

[TABLE]

We define $Q:$ $T_{p}M\rightarrow T_{p}M$ by

[TABLE]

$Q$ is well defined in the following sense: Let $\hat{e}_{1},$ $\hat{e}_{2}$ be a different orthonormal basis of $T_{p}M$ with positive orientation,

[TABLE]

Let

[TABLE]

Then

[TABLE]

Using the above formula, a simple computation yields

[TABLE]

Clearly, $Q:$ $T_{p}M\rightarrow T_{p}M$ is an isometry and

[TABLE]

The operator $Q$ plays an important role in the case of the hyperbolic surface [46].

Let $S$ be hyperbolic and $p_{0}\in S$ be given. Let $\psi(p)=(x_{1},x_{2}):$ ${\,\cal B\,}(p_{0},3\delta)\rightarrow I\!\!R^{2}$ be an asymptotic coordinate system with $\psi(p_{0})=0$

[TABLE]

where ${\,\cal B\,}(p_{0},3\delta)\subset S$ is the geodesic ball centered at $p_{0}$ with radius $3\delta$ where $\delta>0$ is small. It is further assumed that $\psi(p)=x$ is positively orientated, i.e.,

[TABLE]

Let $\varphi\in{\,\rm C\,}^{1}_{0}({\,\cal B\,}(p_{0},3\delta))$ be given such that

[TABLE]

We define

[TABLE]

where $x_{1}$ is the first component of the $\psi(p)=(x_{1},x_{2})$ for $p\in{\,\cal B\,}(p_{0},3\delta).$ Then

[TABLE]

**     Lemma 2.1**

Let $S$ be hyperbolic and $f$ be given in $(\ref{f}).$ Then

[TABLE]

where $D$ is the connection of $S$ in the induced metric $g.$

Proof   Consider the asymptotic coordinate system (2.2). Set

[TABLE]

where

[TABLE]

where

[TABLE]

Then $E_{1},$ $E_{2}$ forms an orthonormal frame with positive orientation on ${\,\cal B\,}(p_{0},\delta)$ since it follows from (2.3) that

[TABLE]

for $p\in{\,\cal B\,}(p_{0},\delta).$ Using (2.1) and (2.6), we have

[TABLE]

From (2.4),

[TABLE]

We thus obtain

[TABLE]

where \Big{(}g^{ij}\Big{)}=\Big{(}g_{ij}\Big{)}^{-1}, and

[TABLE]

(2.5) follows from (2.2) and (2.8). $\Box$

**     Lemma 2.2**

Let $S$ be hyperbolic. Then the shape operator $\nabla\vec{n}:$ $T_{p}S\rightarrow T_{p}S$ is reversible. Let $\kappa$ be the Gaussian curvature and let $f$ be given in $(\ref{f}).$ Set

[TABLE]

Then

[TABLE]

Proof   Let $p\in{\,\cal B\,}(p_{0},\delta)$ be given. Let $e_{1},$ $e_{2}$ be an orthonormal basis of $T_{p}S$ with positive orientation such that

[TABLE]

Then

[TABLE]

Moreover, it follows from Lemma 2.1 and (2.11) that

[TABLE]

Using (2.11) and (2.13), we have

[TABLE]

It follows from (2.11)-(2.14) that

[TABLE]

[TABLE]

[TABLE]

Thus (2.10) follows from the above formulas. $\Box$

Let $(M,g)$ be a Riemanniann manifold. Let $T$ be a 2-order tensor field on $(M,g)$ and let $X$ be a vector field on $(M,g).$ We define the inner multiplication of $T$ with $X$ to be another vector field, denoted by ${\,\rm i\,}(X)T,$ given by

[TABLE]

For any $y\in H^{1}(\Omega,I\!\!R^{3}),$ we decompose $y$ into

[TABLE]

where $w={\langle}y,\vec{n}{\rangle}$ and $W(\cdot,t)$ is a vector field on $S$ for $|t|<h/2.$ It follows from (2.15) that

[TABLE]

[TABLE]

where $\nabla$ and $D$ are the covariant differentials of the dot metric in $I\!\!R^{3}$ and of the induced metric in $S,$ respectively, and $W_{t}=\partial_{t}W$ and $w_{t}=\partial_{t}w.$

By defining $\nabla\vec{n}\vec{n}=0,$ we introduce an 2-order tensor $P(y)$ on $I\!\!R^{3}_{p}$ by

[TABLE]

We have

**     Lemma 2.3**

([47]) Let $y\in H^{1}(\Omega,I\!\!R^{3})$ be given in $(\ref{2.1}).$ Then

[TABLE]

[TABLE]

where

[TABLE]

Moreover, the following estimates hold

[TABLE]

[TABLE]

for $h>0$ small, where $|t|\leq h/2.$

For $A,$ $B\in I\!\!R^{3\times 3},$ let

[TABLE]

where $I$ is the $3\times 3$ unit matrix. Then

[TABLE]

**     Lemma 2.4**

[TABLE]

where $A=I+B$ for $B\in I\!\!R^{3\times 3}.$

Proof   (2.24) follows from the identities

[TABLE]

$\Box$

Proof of Theorem 1.1   (a)   Let the middle surface $S$ be hyperbolic. Let $\delta>0$ be given in (2.2). Let $\hat{\varphi}\in{\,\rm C\,}_{0}^{2}({\,\cal B\,}(p_{0},3\delta))$ be such that

[TABLE]

Let $f$ and $Z,$ $v$ be given in (2.4) and (2.9), respectively. First, we look for the ansatze for the Korn inequality in the form

[TABLE]

where

[TABLE]

We have

[TABLE]

[TABLE]

and

[TABLE]

It follows from (2.29) and(2.10) that

[TABLE]

where $\Upsilon(y)$ is given in (2.21). In addition, from (2.27) and (2.28), we have

[TABLE]

Let $\psi(p)=x$ be the asymptotic coordinate system with $\psi(p_{0})=0,$ given in (2.2). Let $\delta_{0}>0$ be given small such that

[TABLE]

From (2.28) and (2.7), we obtain

[TABLE]

where

[TABLE]

A similar argument as above yields

[TABLE]

Noting that $|t|\leq h/2,$ from (2.30) and (2.32), we have

[TABLE]

Using the formulas (2.19), (2.29), and (2.31), we obtain

[TABLE]

which gives

[TABLE]

Moreover, it follows from (2.19) and (2.33) that

[TABLE]

that is,

[TABLE]

Let $X(y)$ be given in (2.21). It follows from (2.26) and (2.31) that

[TABLE]

In addition, by an argument as for (2.32), we have

[TABLE]

From (2.20), (2.34), (2.36), and (2.37), we obtain

[TABLE]

and, by (2.36) and (2.38),

[TABLE]

respectively.

Now we consider the ansatze for the geometry rigidity inequality, given by

[TABLE]

where $y$ is given in (2.25). Then

[TABLE]

Let $\Phi(B)$ be given in (2.22). From (2.35) and (2.39), we have

[TABLE]

and from (2.40)

[TABLE]

respectively.

We set $A=\nabla u$ and $B=h^{\tau}\nabla y$ in Lemma 2.4 and use (2.23), (2.24), and (2.42) to obtain

[TABLE]

Similarly, it follows from (2.23), (2.24), (2.35) and (2.43) that

[TABLE]

Finally, using (2.44), (2.45), and (2.36), we obtain

[TABLE]

The estimate (1.4) follows.

(b)   Let $S$ be elliptic. We look for the ansatz in the form

[TABLE]

where

[TABLE]

is the ansatz, given in the proof [47, Theorem 1.4] for the optimal constant of the Korn inequality. From [47], we have

[TABLE]

We have

[TABLE]

It follows from (2.48) that

[TABLE]

Thus the proof is complete by Lemma 2.4. $\Box$

Proof of Proposition 1.1   Let $p\in M$ be given and let $e\in T_{p}M$ be such that

[TABLE]

Consider the geodesic

[TABLE]

on $M$ in the induced metric $g,$ where $\exp_{q}:$ $T_{p}M\rightarrow M$ is the exponential map. We will show that $\gamma(t,p)$ satisfies (1.6).

From [47, Lemma 2.7], there are a neighbourhood $N$ of $p$ and a vector field $X$ such that

[TABLE]

Set

[TABLE]

where $Q$ is given in (2.1). Then

[TABLE]

where $\lambda\not=0$ is the nonzero principal curvature.

We have

[TABLE]

Thus

[TABLE]

which yields, by $\lambda\not=0,$

[TABLE]

We obtain

[TABLE]

[TABLE]

Consider the flow by $X:$

[TABLE]

Set

[TABLE]

The formula (2.51) shows that

[TABLE]

Next, we prove that

[TABLE]

Let $q\in N$ be given. From (2.54) and (2.50), there is a largest number $\epsilon_{0}>0$ such that (2.55) hold for all $t\in[0,\varepsilon_{0}).$ Let $\varepsilon_{0}<\infty.$ Clearly

[TABLE]

By [47, Lemma 2.7] again, there is a vector field $Z$ on a neighbourhood of $\gamma(\varepsilon_{0},q)$ such that

[TABLE]

From the uniqueness of a geodesic, (2.55) would hold true for all $t\in[0,\varepsilon_{1})$ for some $\varepsilon_{1}>\varepsilon_{0}.$ This contradiction shows that (2.55) hold for all $t\in[0,\infty).$ A similar argument shows that (2.55) also hold for all $t\in(-\infty,0].$

We extend the vector field $X$ from $N$ to $\hat{N},$ still denoted by $X,$ by

[TABLE]

where

[TABLE]

Moreover, we extend $Y$ from $N$ to $\hat{N}$ by parallelling translation $Y(q)$ to $Y(\gamma(t,q))$ along the geodesic $\gamma(t,q)$ in the induced metric of $M.$ Then $X$ and $Y$ forms an orthonormal frame on $\hat{N}$ with

[TABLE]

Let

[TABLE]

Then

[TABLE]

Thus the formulas $\gamma^{\prime}(t,q)=X(\gamma^{\prime}(t,q))$ imply that

[TABLE]

Thus $\gamma(t,q)$ is a straight line in $I\!\!R^{3}$ for given $q\in N.$ The proof is complete. $\Box$

**     Lemma 2.5**

Let $p_{0}\in S$ be such that $(\ref{ga7})$ holds. For any $a>0,$ there exists a principal coordinate system $\psi^{-1}:$ $\in(-a,a)\times(-\varepsilon,\varepsilon)\rightarrow M$ such that

[TABLE]

where $\varepsilon>0$ is a number small.

Proof   Let $\gamma(t,q),$ $X,$ and $Y$ be given in the proof of Proposition 1.1. Define

[TABLE]

Then

[TABLE]

Let $\beta(s,q)$ be the flow by the vector $\eta Y,$ i.e., for each $q\in\hat{N},$ there is $\varepsilon(q)>0$ such that

[TABLE]

Since the interval $[-a,a]$ is compact, there is a constant $\varepsilon>0$ small such that

[TABLE]

From [26, p.233, Theorem 9.44], (2.58) implies that

[TABLE]

We define $\psi^{-1}:$ $(-a,a)\times(-\varepsilon,\varepsilon)\rightarrow M$ by

[TABLE]

From Proposition 1.1, there is a $\varepsilon>0$ such that $\psi(q)=x$ defines a coordinate satisfying (2.56). Furthermore, (2.59) implies that $\partial x_{1}=X$ and $\partial x_{2}=\eta Y.$ $\Box$

We make some further preparations for the proof of Theorem 1.2. Let assumption (1.7) hold. Let $a>\max\{|t_{-}|,\,|t_{+}|\}$ be given. Let $\psi(q)=(x_{1},x_{2})$ be the principal coordinate given in Lemma 2.5. Let $X$ and $\eta Y$ be the vector field given in Lemma 2.5 such that

[TABLE]

Then

[TABLE]

where $\varrho={\langle}D_{Y}X,Y{\rangle}.$ For $x_{2}\in(-\varepsilon,\varepsilon),$ let $t_{+}(x_{2})>0>t_{-}(x_{2})$ be such that

[TABLE]

Set

[TABLE]

Then $S_{0}\subset S.$

We will construct the ansatz with its values supported on $S_{0}.$ Let the functions $\eta$ and $\varrho$ be given in (2.57) and (2.61), respectively, on $S_{0}.$ Set

[TABLE]

We further define

[TABLE]

Then

[TABLE]

Consider the ansatz

[TABLE]

where

[TABLE]

It follows from (2.61) and (2.60) that

[TABLE]

[TABLE]

Using the above formulas and from Lemma 2.3, we obtain

[TABLE]

where the following formulas have been used

[TABLE]

Noting (2.63), by a similar computation, we have

[TABLE]

[TABLE]

Proof of Theorem 1.2   Let $\varphi_{0}\in{\,\rm C\,}^{4}_{0}(-\varepsilon,\varepsilon)$ be given such that

[TABLE]

Set

[TABLE]

in (2.62).

[TABLE]

Then it follows from (2.62) that

[TABLE]

[TABLE]

Noting $|t|\leq h/2$ and from (2.65)-(2.67) and (2.20), we obtain

[TABLE]

[TABLE]

Moreover, a similar argument as in (2.32) yields

[TABLE]

Using the above formulas, we have

[TABLE]

[TABLE]

Consider the ansatz

[TABLE]

where $y$ is given in (2.64). Then

[TABLE]

It follows from (2.68) and (2.69) that

[TABLE]

[TABLE]

Since $\det\nabla u=\det(I+h^{\tau}\nabla y)>0$ when $h>0$ is small enough, from Lemma 2.4 and the above estimates, we obtain

[TABLE]

The proof is complete. $\Box$

Conflict of interest statement

There is no conflict of interests.

Ethical approval: This article does not contain any studies with human participants or animals performed by the author.