Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends

Abstract

Slender body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178), we developed a PDE framework for analyzing the error introduced by the slender body approximation for closed-loop fibers with constant radius $\varepsilon$, and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to $\varepsilon |\log \varepsilon |$. Here we extend the slender body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D slender body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the slender body PDE and the slender body approximation in the free endpoint setting. The bound is a sum of the same $\varepsilon |\log \varepsilon |$ term appearing in the closed loop setting and an endpoint term proportional to $\varepsilon$, where $\varepsilon$ is now the maximum fiber radius.

Appendix

1.1 Dependence of Well-Posedness Constants on $\varepsilon$

In this appendix, we prove each of the key well-posedness inequalities stated in Section 2.1, paying close attention to how each resulting constant depends on $\varepsilon$.

First, we prove a simple centerline straightening lemma (Lemma A.1) to facilitate computations in the following sections. Using Lemma A.1, we prove the form (2.6) of the trace inequality for ${\mathcal {A}}_\varepsilon$ functions. As in the closed loop setting [23], we show that the $L^2$ trace along the fiber blows up like $\left|\log \varepsilon \right|^{1/2}$ as $\varepsilon \rightarrow 0$; however, unlike in the closed loop case, we show that an additional weight is needed at the fiber endpoints to ensure that the trace does not diverge. This results in the required decay condition (1.27) for any prescribed force $\varvec{f}$.

Next we show the existence of an extension operator for slender bodies with free, spheroidal endpoints whose symmetric gradient is bounded independent of $\varepsilon$. As far as we know, this result is new and may be more widely useful beyond the scope of this paper. This extension operator is then used to show $\varepsilon$-independence for the Korn and Sobolev inequalities in $\Omega _\varepsilon$.

Finally we verify that the pressure estimate (2.16) does not depend on $\varepsilon$. The proof here is essentially the same as in the closed loop setting, relying only on being able to write the region $\mathcal {O}$ (1.15) as a finite union of star-shaped domains with respect to an $\varepsilon$-independent ball.

1.1.1 Centerline Straightening

To facilitate the calculations in the following sections, it will be useful to define an $\varepsilon$-independent straightening operator within the neighborhood $\mathcal {O}$ (1.15) about $\Sigma _\varepsilon$, taking $\varvec{X}$ as in Section 1.1 to a straight line. We show the following lemma.

Lemma A.1

Consider a fiber centerline $\varvec{X}(\varphi )$ as in Section 1.1 and the neighborhood $\mathcal {O}$ given by (1.15). Within $\mathcal {O}$, there exists a $\mathcal {C}^1$ operator $\Psi$ taking the moving frame basis vectors $\varvec{e}_{\mathrm{t}}(\varphi )$, $\varvec{e}_\rho (\varphi ,\theta )$, $\varvec{e}_\theta (\varphi ,\theta )$ defined with respect to a Bishop frame about $\varvec{X}_\text {ext}(\varphi )$ to straight cylindrical coordinates $\varphi \varvec{e}_{\mathrm{t}} + \rho \varvec{e}_\rho + \theta \varvec{e}_\theta$ about a straight fiber centerline such that $\nabla \Psi$ and $\nabla (\Psi ^{-1})$ are both bounded independent of $\varepsilon$.

Proof

Let $\mathcal {O}_{\text {str}}$ denote the region $\mathcal {O}$ with straight centerline $\varvec{X}_{\text {ext}}$, parameterized with respect to straight cylindrical coordinates $(\varphi ,\theta ,\rho )$. Let $\Psi _0: \mathcal {O}_{\text {str}} \rightarrow \mathcal {O}$ denote the coordinate map taking the straight cylindrical coordinates $(\varphi ,\theta ,\rho ) \mapsto \varvec{X}_\text {ext}(\varphi )+ \rho \varvec{e}_{\rho }(\varphi ,\theta )$ about the curved slender body centerline. Note that the map $\Psi _0$ is $\mathcal {C}^1$ due to the Bishop frame (1.3). Furthermore, $\nabla \Psi _0$ is full rank at each point $(\varphi ,\theta ,\rho )$, which can be seen as follows. Letting ${{\overline{\varvec{e}}}}_{\mathrm{t}},{{\overline{\varvec{e}}}}_\theta ,{{\overline{\varvec{e}}}}_\rho$ denote the straight cylindrical basis vectors, for any $\varphi _0\in (1-r_{\max },1+r_{\max })$, we can choose a rotation $\varvec{R}\in SO(2)$ such that $\varvec{R}\varvec{e}_{\mathrm{t}}(\varphi _0)={{\overline{\varvec{e}}}}_{\mathrm{t}}$, $\varvec{R}\varvec{e}_\rho (\varphi _0,\theta )={{\overline{\varvec{e}}}}_\rho$, and $\varvec{R}\varvec{e}_\theta (\varphi _0,\theta )={{\overline{\varvec{e}}}}_\theta$. Then, using (1.3),

$$\begin{aligned} \nabla (\varvec{R}\Psi _0) = (1-\rho \widehat{\kappa }){{\overline{\varvec{e}}}}_{\mathrm{t}}{{\overline{\varvec{e}}}}_{\mathrm{t}}^{\mathrm{T}} + {{\overline{\varvec{e}}}}_\theta {{\overline{\varvec{e}}}}_\theta ^{\mathrm{T}} + {{\overline{\varvec{e}}}}_\rho {{\overline{\varvec{e}}}}_\rho ^{\mathrm{T}} \end{aligned}$$

has positive determinant, since $\rho <r_{\max }$.

We then define the straightening operator $\Psi$ on the $\varvec{X}_\text {ext}$ by $\Psi = \Psi _0^{-1}$. Note $\Psi _0^{-1}\in \mathcal {C}^1$ by the inverse function theorem. Also, the map $\Psi$ depends only on the shape of the fiber centerline – in particular, on the constants $c_{\Gamma }$ and $\kappa _{\max }$. $\quad \square$

Remark A.2

We can use Lemma A.1 to work locally with a straight fiber without introducing additional $\varepsilon$-dependence into resulting estimates.

Let $\mathcal {O}_2$ denote the region

$$\begin{aligned} \mathcal {O}_2&:= \bigg \{\varvec{x}\in \Omega _\varepsilon : \varvec{x}= \varvec{X}(\varphi )+\rho \varvec{e}_\rho (\varphi ,\theta ); \left|\varphi \right|\\&\quad \leqq 1, 0\leqq \theta<2\pi , \varepsilon a(\varphi )< \rho< \frac{r_{\max }}{2} \bigg \} \\&\quad \bigcup \bigg \{\varvec{x}\in \Omega _\varepsilon : \varvec{x}= \varvec{X}_\text {ext}(\varphi )+\rho \varvec{e}_\rho (\varphi ,\theta ); 1<\left|\varphi \right|< 1+\frac{r_{\max }}{2},\\&\quad 0\leqq \theta<2\pi , \varepsilon a(\varphi )< \rho < \sqrt{r_{\max }^2/4-(\left|\varphi \right|-1)^2} \bigg \}, \end{aligned}$$

that is, the region $\mathcal {O}$ with extent $r_{\max }/2$ rather than $r_{\max }$.

We use the following partition of unity to divide $\Omega _\varepsilon$ in two regions:

$$\begin{aligned} \phi _1&= {\left\{ \begin{array}{ll} 1 &{} \text {if } \varvec{x}\in \mathcal {O}_2 \\ 0 &{} \text {if } \varvec{x}\in {\mathbb {R}}^3\backslash \mathcal {O} \end{array}\right. }, \quad \phi _2 = 1-\phi _1, \end{aligned}$$

with smooth transition between $\mathcal {O}_2$ and $\mathcal {O}$ for both $\phi _1$ and $\phi _2$. Since $r_{\max }$ depends only on the constants $c_{\Gamma }$ and $\kappa _{\max }$ and not on $\varepsilon$, these cutoffs are both independent of $\varepsilon$. Therefore we can consider the region $\mathcal {O}$ (where $\phi _1$ is supported) separately from ${\mathbb {R}}^3\backslash \mathcal {O}_2$. In particular, whenever only information near the slender body surface is needed, we can use Lemma A.1 to consider only slender bodies $\Sigma _\varepsilon$ with straight centerline. This will be useful for both the trace and extension operators constructed in the following sections, both of which rely only on information very close to $\Gamma _\varepsilon$.

1.1.2 Trace Inequality

Here we prove the weighted trace inequality (Lemma 2.3) for functions in the space ${\mathcal {A}}_\varepsilon$. We show that the weight $(1+\left|\log (a) \right|)^{-1/2}$ is needed to avoid a logarithmic divergence in the trace at the fiber endpoints.

Proof of Lemma 2.3

It suffices to show that (2.3) holds within the region $\mathcal {O}$. In particular, using the cutoff described in Remark A.2, we consider only functions $\varvec{u}$ supported within a distance $r_{\max }\leqq 1$ of the fiber centerline.

Using Lemma A.1, we proceed to show that the estimate (2.10) holds in the exterior of a straight slender body, parameterized with respect to straight cylindrical coordinates as

$$\begin{aligned} \Sigma ^\varepsilon _\text {str}= \{ (\varphi ,\theta ,\rho ) : -\eta _\varepsilon<\varphi<\eta _\varepsilon , 0\leqq \theta <2\pi , \rho \leqq \varepsilon a(\varphi ) \} \end{aligned}$$

(A.1)

for $a(\varphi )$ as in Definition 1.1. Let

$$\begin{aligned} \Gamma ^\varepsilon _\text {str} = \partial \Sigma ^\varepsilon _\text {str}, \quad \Omega ^\varepsilon _\text {str} = {\mathbb {R}}^3\backslash \overline{\Sigma ^\varepsilon _\text {str}}. \end{aligned}$$

(A.2)

We define the admissible set

$$\begin{aligned} {\mathcal {A}}^\varepsilon _\text {str} = \big \{ \varvec{v}\in D^{1,2}(\Omega ^\varepsilon _\text {str}) : \varvec{v}\big |_{\Gamma ^\varepsilon _\text {str}} = \varvec{v}(\varphi ); \quad \text {supp}(\varvec{v}) \subset \{\varvec{x}(\varphi ,\theta ,\rho ) : \rho < 1\} \big \}. \end{aligned}$$

Following the same outline as in the closed loop setting [23], we show that (2.10) holds for $\varvec{u}\in \mathcal {C}^\infty _0(\overline{\Omega ^\varepsilon _\text {str}})\cap {\mathcal {A}}^\varepsilon _\text {str}$, where again the $\overline{\Omega ^\varepsilon _\text {str}}$ notation denotes functions supported up to $\Gamma ^\varepsilon _\text {str}$. The result for $\varvec{u}\in {\mathcal {A}}^\varepsilon _\text {str}$ then follows by density.

Now, for $\varvec{x}\in \Gamma ^\varepsilon _\text {str}$, we use the fundamental theorem of calculus to write $\varvec{u}(\varvec{x})=\varvec{u}(\varepsilon a(\varphi ), \theta ,\varphi )$ as

$$\begin{aligned} \varvec{u}(\varphi ,\theta ,\varepsilon a(\varphi )) = \int _{\varepsilon a(\varphi )}^1 \frac{\partial \varvec{u}}{\partial \rho } \,\mathrm{d}\rho . \end{aligned}$$

Then we have

$$\begin{aligned} \left|\varvec{u}(\varphi ,\theta ,\varepsilon a(\varphi )) \right|&\leqq \int _{\varepsilon a(\varphi )}^1 \left|\frac{\partial \varvec{u}}{\partial \rho } \right| \,\mathrm{d}\rho \leqq \int _{\varepsilon a(\varphi )}^1 \frac{1}{\sqrt{\rho }}\sqrt{\rho } \left|\frac{\partial \varvec{u}}{\partial \rho } \right| \,\mathrm{d}\rho \\&\leqq \bigg (\int _{\varepsilon a(\varphi )}^1 \frac{1}{\rho } \,\mathrm{d}\rho \bigg )^{1/2} \bigg ( \int _{\varepsilon a(\varphi )}^1 \left|\frac{\partial \varvec{u}}{\partial \rho } \right|^2 \rho \,\mathrm{d}\rho \bigg )^{1/2} \\&= \left|\log (\varepsilon a(\varphi )) \right|^{1/2} \bigg ( \int _{\varepsilon a(\varphi )}^1 \left|\frac{\partial \varvec{u}}{\partial \rho } \right|^2 \rho \,\mathrm{d}\rho \bigg )^{1/2}, \end{aligned}$$

and therefore

$$\begin{aligned} \left|\mathrm{Tr}(\varvec{u}) \right|^2 \leqq \left|\log (\varepsilon a(\varphi )) \right| \int _{\varepsilon a(\varphi )}^1 \left|\frac{\partial \varvec{u}}{\partial \rho } \right|^2 \rho \,\mathrm{d}\rho . \end{aligned}$$

(A.3)

Already we can see that the bound (A.3) diverges at the fiber endpoints $\varphi =\pm \eta _\varepsilon$. This is due to the fact that as $a(\varphi )\rightarrow 0$ at the fiber endpoints, we are trying to limit to a truly one-dimensional trace of an $H^1$ function in ${\mathbb {R}}^3$.

To correct for this, we multiply $\left|\mathrm{Tr}(\varvec{u}) \right|^2$ in (A.3) by $(1+\left|\log (a(\varphi )) \right|)^{-1}$. Then, since $\varvec{u}$ belongs to ${\mathcal {A}}_\varepsilon ^{{\mathrm{div}}\,}$, using the $\theta$-independence of $\mathrm{Tr}(\varvec{u})=\mathrm{Tr}(\varvec{u})(\varphi )$, we may write

$$\begin{aligned}&\left||\mathrm{Tr}(\varvec{u}) (1+\left|\log (a) \right|)^{-1/2} \right||_{L^2(-\eta _\varepsilon ,\eta _\varepsilon )}^2 \\&\quad = \frac{1}{2\pi } \int _{-\eta _\varepsilon }^{\eta _\varepsilon } \int _0^{2\pi } \left|\mathrm{Tr}(\varvec{u})(\varphi ) \right|^2(1+\left|\log (a(\varphi )) \right|)^{-1} \,\hbox {d}\theta \,\hbox {d}\varphi . \end{aligned}$$

Using (A.3), we then have that $\mathrm{Tr}(\varvec{u})(\varphi )$ satisfies

$$\begin{aligned}&\left||\mathrm{Tr}(\varvec{u}) (1+\left|\log (a) \right|)^{-1/2} \right||_{L^2(-\eta _\varepsilon ,\eta _\varepsilon )}^2 \\&\quad \leqq \frac{1}{2\pi }\int _{-\eta _\varepsilon }^{\eta _\varepsilon } \int _0^{2\pi } \frac{\left|\log (\varepsilon a(\varphi )) \right| }{1+\left|\log (a(\varphi )) \right|} \int _{\varepsilon a(\varphi )}^1 \left|\frac{\partial \varvec{u}}{\partial \rho } \right|^2 \rho \,\mathrm{d}\rho \,\hbox {d}\theta \,\hbox {d}\varphi \\&\quad \leqq \frac{1}{\pi } \left|\log \varepsilon \right| \left||\nabla \varvec{u} \right||_{L^2(\Omega ^\varepsilon _\text {str})}^2, \end{aligned}$$

where we have used that $a\leqq 1$. $\quad \square$

Note that, for $\varvec{u}\in {\mathcal {A}}_\varepsilon$, this one-dimensional $L^2$ trace inequality along the fiber centerline relates to a (weighted) trace inequality over the entire fiber surface $\Gamma _\varepsilon$ by

$$\begin{aligned}&\left||\mathrm{Tr}(\varvec{u})(1+\left|\log (a) \right|)^{-1/2} \right||_{L^2(\Gamma _\varepsilon )}^2 \\&\quad = \int _{-\eta _\varepsilon }^{\eta _\varepsilon }\int _0^{2\pi } \left|\mathrm{Tr}(\varvec{u})(\varphi ) \right|^2 (1+\left|\log (a(\varphi )) \right|)^{-1} \mathcal {J}_\varepsilon (\varphi ,\theta ) \,\hbox {d}\theta \,\hbox {d}\varphi \\&\quad \leqq \int _{-\eta _\varepsilon }^{\eta _\varepsilon }\int _0^{2\pi } \left|\mathrm{Tr}(\varvec{u})(\varphi ) \right|^2(1+\left|\log (a(\varphi )) \right|)^{-1}(\varepsilon a+\varepsilon ^2a^2\kappa _{\max }+\varepsilon ^2{{\bar{c}}}_a) \,\hbox {d}\theta \,\hbox {d}\varphi \\&\quad \leqq 2\pi \varepsilon (1+\varepsilon \kappa _{\max }+\varepsilon {{\bar{c}}}_a)\left||\mathrm{Tr}(\varvec{u}) (1+\left|\log (a) \right|)^{-1/2} \right||_{L^2(-\eta _\varepsilon ,\eta _\varepsilon )}^2. \end{aligned}$$

Here we have used (1.14) and (1.11) to bound $\mathcal {J}_\varepsilon$.

1.1.3 Extension Operator for Thin Fiber with Spheroidal Endpoints

In order to determine the $\varepsilon$-dependence in the Korn inequality (Lemma (2.4)), as in [23], it will be convenient to show the existence of a $D^{1,2}$ extension into the interior of $\Sigma _\varepsilon$ whose symmetric gradient is bounded independent of $\varepsilon$. Again, due to Lemma A.1, it suffices to show the existence of such an operator for a slender filament with straight centerline. We again parameterize the straight centerline with respect to cylindrical coordinates $(\varphi ,\theta ,\rho )$ as $\{ (\varphi ,0,0) : -\eta _\varepsilon<\varphi <\eta _\varepsilon \}$.

Let $\Sigma ^\varepsilon _\text {str}$ and $\Gamma ^\varepsilon _\text {str}$ be as in (A.1) and (A.2). We define the domain $\mathcal {W}^\varepsilon$, a “fattened” version of $\Sigma ^\varepsilon _\text {str}$, as

$$\begin{aligned} \begin{aligned}&\mathcal {W}^\varepsilon = \bigg \{(\varphi ,\theta ,\rho ) : -2\eta _\varepsilon +1< \varphi< 2\eta _\varepsilon -1, 0\leqq \theta<2\pi , \\&\quad 0\leqq \rho< {\left\{ \begin{array}{ll} 2\varepsilon a(\varphi ), &{} \left|\varphi \right|\leqq 1 \\ 2\varepsilon a(\frac{\varphi +1}{2}), &{} 1< \left|\varphi \right| <2\eta _\varepsilon -1 \end{array}\right. } \bigg \}, \end{aligned} \end{aligned}$$

(A.4)

and take

$$\begin{aligned} \mathcal {H}^\varepsilon =\mathcal {W}^\varepsilon \backslash \overline{\Sigma ^\varepsilon _\text {str}}. \end{aligned}$$

(A.5)

Note that on the bounded domain $\mathcal {H}^\varepsilon$, the function spaces $D^{1,2}(\mathcal {H}^\varepsilon )$ and $H^1(\mathcal {H}^\varepsilon )$ coincide; thus it suffices to construct an $H^1$ extension from $\mathcal {H}^\varepsilon$ to $\mathcal {W}^\varepsilon$ whose symmetric gradient is bounded independent of $\varepsilon$. We show the following lemma.

Lemma A.3

(Free endpoint slender body extension) Let $\mathcal {W}^\varepsilon$ and $\mathcal {H}^\varepsilon$ be as in (A.4) and (A.5), respectively. For $\varvec{u}\in H^1(\mathcal {H}^\varepsilon )$, there exists an operator $T_\varepsilon : H^1(\mathcal {H}^\varepsilon ) \rightarrow H^1(\mathcal {W}^\varepsilon )$ extending $\varvec{u}$ into the interior of the slender body such that

• 1. $T_\varepsilon \varvec{u}\big |_{\mathcal {H}^\varepsilon } = \varvec{u}$;

• 2. $\left||{\mathcal {E}}(T_\varepsilon \varvec{u}) \right||_{L^2(\mathcal {W}^\varepsilon )} \leqq C \left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\mathcal {H}^\varepsilon )}$, where the constant C depends on the constants $c_a$, ${{\bar{c}}}_a$, $a_0$, $\delta _a$, $c_{\eta ,0}$, and $c_\eta$, but is independent of $\varepsilon$.

As an immediate corollary, due to Lemma A.1, we have

Corollary A.4

(Free endpoint extension – curved centerline) Let $\Sigma _\varepsilon$ and $\Omega _\varepsilon$ be as in Section 1.1. For $\varvec{u}\in D^{1,2}(\Omega _\varepsilon )$, there exists an operator $\widetilde{T}_\varepsilon : D^{1,2}(\Omega _\varepsilon )\rightarrow D^{1,2}({\mathbb {R}}^3)$ extending $\varvec{u}$ into the interior of the slender body $\Sigma _\varepsilon$ such that

• 1. $\widetilde{T}_\varepsilon \varvec{u}\big |_{\Omega _\varepsilon } = \varvec{u}$;

• 2. $\left||{\mathcal {E}}(\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2(\mathcal {W}^\varepsilon )} \leqq C \left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\mathcal {H}^\varepsilon )}$, where C now depends on $\kappa _{\max }$ and $c_\Gamma$ in addition to $c_a$, ${{\bar{c}}}_a$, $a_0$, $\delta _a$, $c_{\eta ,0}$, and $c_\eta$, but remains independent of $\varepsilon$.

Proof of Lemma A.3

We subdivide the somewhat long and technical proof into 6 main steps.

Step 1: Reflection E into the interior of $\Sigma _\text {str}^\varepsilon$

First, for any $\varvec{x}=\varvec{x}(\varphi ,\theta ,\rho )\in \mathcal {W}^\varepsilon$, we let

$$\begin{aligned} \delta (\varphi ,\rho ) = \inf _{s\in (-1,1)} \left|\varvec{x}- (s,0,0) \right| \end{aligned}$$

denote the distance from $\varvec{x}$ to the straightened effective centerline (1.19). Note that since $\mathcal {W}^\varepsilon$ is radially symmetric for each $\varphi$, the distance $\delta (\varphi ,\rho )$ depends only on $\varphi$ and $\rho$ and not on $\theta$. In particular, we have

$$\begin{aligned} \delta (\varphi ,\rho ) = {\left\{ \begin{array}{ll} \rho , &{} \left|\varphi \right| \leqq 1 \\ \sqrt{(\left|\varphi \right|-1)^2+\rho ^2}, &{} 1< \left|\varphi \right| <\eta _\varepsilon . \end{array}\right. } \end{aligned}$$

Now, for any $(\varphi ,\theta ,\rho )\in \mathcal {W}^\varepsilon$, define $\varphi ^*\in (-\eta _\varepsilon ,\eta _\varepsilon )$ such that $(\varphi ^*,\theta , \varepsilon a(\varphi ^*))$ is the projection of $(\varphi ,\theta ,\rho )$ onto $\Gamma ^\varepsilon _{\text {str}}$ along the straight line distance $\delta (\varphi ,\rho )$ to the effective centerline. Note that for $\left|\varphi \right|\leqq 1$, $\varphi ^*=\varphi$. Toward the fiber endpoints, $1<\left|\varphi \right|<\eta _\varepsilon$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\varepsilon a(\varphi ^*)}{\varphi ^*-1}=\frac{\rho }{\varphi -1}, &{} 1<\varphi<\eta _\varepsilon \\ \frac{\varepsilon a(\varphi ^*)}{\varphi ^*+1}=\frac{\rho }{\varphi +1}, &{} -\eta _\varepsilon<\varphi < -1, \end{array}\right. } \end{aligned}$$

(A.6)

This means that the pair $(\varphi ^*,a(\varphi ^*))$ lies on the line connecting $(\varphi ,\rho )$ to the effective endpoint $(\pm 1,0)$. Note that as $\rho \rightarrow 0$, $\varphi ^*\rightarrow \pm \eta _\varepsilon$, and as $\varphi \rightarrow 1$, $\varphi ^*\rightarrow 1$. Also, for $1<\left|\varphi \right|<\eta _\varepsilon$, we have

$$\begin{aligned} \frac{\partial \varphi ^*}{\partial \varphi } = \frac{\varepsilon a(\varphi ^*)}{\rho +\varepsilon |a'(\varphi ^*)|(|\varphi | - 1)}, \quad \frac{\partial \varphi ^*}{\partial \rho } =\frac{-(|\varphi ^*|-1)}{\rho + \varepsilon |a'(\varphi ^*)|(|\varphi |- 1)}. \end{aligned}$$

(A.7)

Consider $\varvec{v}=\varvec{v}(\varphi ,\theta ,\rho )\in H^1(\mathcal {H}^\varepsilon )$ supported only at $(\varphi ,\theta ,\rho )\in \mathcal {H}^\varepsilon$ satisfying $\delta (\varphi ,\rho ) \leqq \frac{5}{3}\delta (\varphi ^*,\varepsilon a(\varphi ^*))$ (see Fig. 4). This initial constraint on the support of $\varvec{v}$ is important for the definition (A.8) and will be addressed for functions with more general support in Step 3. We define an extension $E\varvec{v}$ into the interior of $\Sigma ^\varepsilon _\text {str}$ as the reflection of $\varvec{v}$ across $\Gamma ^\varepsilon _\text {str}$ along the shortest straight line connecting $\varvec{x}$ to the effective centerline segment $[-1,1]$ (See Fig. 4). More precisely,

$$\begin{aligned} E\varvec{v}= {\left\{ \begin{array}{ll} \varvec{v}(\varvec{x}), &{} \text { if } \varvec{x}\in \mathcal {H}^\varepsilon ; \\ {\left\{ \begin{array}{ll} \varvec{v}\big (\varphi ,\theta , 2\varepsilon a(\varphi )-\rho \big ), &{} \left|\varphi \right| \leqq 1 \\ \varvec{v}\big (2\varphi ^*-\varphi -1,\theta ,2\varepsilon a(\varphi ^*)-\rho \big ), &{} 1< \varphi \leqq \eta _\varepsilon \\ \varvec{v}\big (2\varphi ^*-\varphi +1,\theta ,2\varepsilon a(\varphi ^*)-\rho \big ), &{} -\eta _\varepsilon \leqq \varphi < -1, \end{array}\right. }&\text { if } \varvec{x}= (\varphi ,\theta ,\rho )\in \Sigma ^\varepsilon _\text {str}. \end{array}\right. }\qquad \end{aligned}$$

(A.8)

Here $\varphi ^*$ is as in (A.6). Note that this extension is well-defined for each $\varvec{x}\in \Gamma _\text {str}^\varepsilon$ due to the limits on supp$(\varvec{v})$.

Fig. 4

Consider $\varvec{v}(\varphi ,\theta ,\rho )\in H^1(\mathcal {H}^\varepsilon )$ supported only at $(\varphi ,\theta ,\rho )\in \mathcal {H}^\varepsilon$ satisfying $\delta (\varphi ,\rho ) \leqq \frac{5}{3}\delta (\varphi ^*,\varepsilon a(\varphi ^*))$. We construct the basic extension operator $E: H^1(\mathcal {H^\varepsilon })\rightarrow H^1(\mathcal {W}^\varepsilon )$ by reflecting $\varvec{v}\in H^1(\mathcal {H}^\varepsilon )$ across the straight line distance to the effective centerline of the fiber ($-1\leqq \varphi \leqq 1$)

Step 2: Gradient estimates for E

We now aim to prove the following estimate on the (Euclidean) gradient of the reflection E within $\Sigma _\text {str}^\varepsilon$:

Proposition A.5

Let E be as defined in (A.8). For each $\varvec{x}=(\varphi ,\theta ,\rho )\in \Sigma _{\text {str}}^\varepsilon$, we have

$$\begin{aligned} \left|\nabla (E\varvec{v})(\varphi ,\theta ,\rho ) \right| \leqq C {\left\{ \begin{array}{ll} \left|\nabla \varvec{v}(\varphi ,\theta ,2\varepsilon a(\varphi ^*)-\rho ) \right|, &{} \left|\varphi \right|\leqq 1 \\ \left|\nabla \varvec{v}(2\varphi ^*-\varphi -1,\theta ,2\varepsilon a(\varphi ^*)-\rho ) \right|, &{} 1<\varphi \leqq \eta _\varepsilon \\ \left|\nabla \varvec{v}(2\varphi ^*-\varphi -1,\theta ,2\varepsilon a(\varphi ^*)-\rho ) \right|, &{} -\eta _\varepsilon \leqq \varphi <-1. \end{array}\right. }\nonumber \\ \end{aligned}$$

(A.9)

Proof of Proposition A.5

For $\varvec{x}\in \Sigma ^\varepsilon _{\text {str}}$ and $\left|\varphi \right|\leqq 1$, we differentiate equation 2 of (A.8) to obtain

$$\begin{aligned} \begin{aligned} \left|\nabla (E\varvec{v})(\varphi ,\theta ,\rho ) \right|&\leqq \left|\frac{1}{\rho }\frac{\partial \varvec{v}}{\partial \theta } \right| + \left|2\varepsilon a'(\varphi ) - 1 \right| \left|\frac{\partial \varvec{v}}{\partial \rho } \right| + \left|\frac{\partial \varvec{v}}{\partial \varphi } \right| \\&\leqq \left|\frac{C}{2\varepsilon a(\varphi )-\rho } \right|\left|\frac{\partial \varvec{v}}{\partial \theta } \right| + \bigg (\frac{\varepsilon {{\bar{c}}}_a}{a(1)}+1\bigg ) \left|\frac{\partial \varvec{v}}{\partial \rho } \right| + \left|\frac{\partial \varvec{v}}{\partial \varphi } \right| \\&\leqq C\left|\nabla \varvec{v}(\varphi ,\theta ,2\varepsilon a(\varphi )-\rho ) \right|, \end{aligned}\nonumber \\ \end{aligned}$$

(A.10)

where, in the second inequality, we used that supp($E\varvec{v}$) extends only to $\rho \geqq \varepsilon a(\varphi )/3$ to bound the $\partial /\partial \theta$ term, and Definition 1.1 to bound the $\partial /\partial \rho$ term. Note that all functions on the right hand side in (A.10) are evaluated at $(\varphi ,\theta ,2\varepsilon a(\varphi )-\rho )$.

Similarly, for $\varvec{x}\in \Sigma ^\varepsilon _{\text {str}}$ and $1< \varphi <\eta _\varepsilon$, differentiating equation 3 of (A.8) and using (A.7), the gradient of the extension $\nabla (E\varvec{v})$ satisfies

$$\begin{aligned} \left|\nabla (E\varvec{v})(\varphi ,\theta ,\rho ) \right|&\leqq \left|\frac{2\varepsilon a(\varphi ^*)-2(\varphi ^*-1)}{\rho +\varepsilon |a'(\varphi ^*)|(\varphi -1)} -1 \right|\left|\frac{\partial \varvec{v}}{\partial \varphi } \right| + \left|\frac{1}{\rho }\frac{\partial \varvec{v}}{\partial \theta } \right| \nonumber \\&\quad + \left|\frac{2\varepsilon ^2 a(\varphi ^*)a'(\varphi ^*) + 2\varepsilon |a'(\varphi ^*)|(\varphi ^*-1)}{\rho +\varepsilon |a'(\varphi ^*)|(\varphi -1)} -1 \right|\left|\frac{\partial \varvec{v}}{\partial \rho } \right|.\nonumber \\ \end{aligned}$$

(A.11)

Here each function on the right hand side of (A.11) is evaluated at $(2\varphi ^*-\varphi -1,\theta ,2\varepsilon a(\varphi ^*)-\rho )$. We must now bound each of the above coefficients to obtain a bound as in (A.10). Using Definition 1.1, since $\varphi ^*\geqq 1$ we have $|a'(\varphi ^*)|\geqq \frac{C}{\varepsilon }$. Thus

$$\begin{aligned} \rho +\varepsilon \left|a'(\varphi ^*) \right|(\varphi -1) \geqq C\big (\rho +(\varphi -1) \big ). \end{aligned}$$

Furthermore, using that $\varvec{v}(\varphi ,\theta ,\rho )$ is only supported at $(\varphi ,\theta ,\rho )\in \mathcal {H}^\varepsilon$ satisfying $\delta (\varphi ,\rho )<\frac{5}{3} \delta (\varphi ^*,\varepsilon a(\varphi ^*))$, we have that $\nabla (E\varvec{v})$ is only supported at $(\varphi ,\theta ,\rho )\in \Sigma _{\text {str}}^\varepsilon$ satisfying

$$\begin{aligned} \rho + (\varphi -1) \geqq \sqrt{\rho ^2+(\varphi -1)^2} \geqq \frac{1}{3}\delta (\varphi ^*,\varepsilon a(\varphi ^*)) =\frac{1}{3}\sqrt{\varepsilon ^2a^2(\varphi ^*){+}(\varphi ^*-1)^2}.\nonumber \\ \end{aligned}$$

(A.12)

Note that this implies that for any pair $(\rho ,\varphi )$,

$$\begin{aligned} \frac{\varphi ^*-1}{\rho +(\varphi -1)} \leqq \frac{\sqrt{\varepsilon ^2a^2(\varphi ^*)+(\varphi ^*-1)^2}}{\rho +(\varphi -1)} \leqq 3. \end{aligned}$$

Also, by (A.7), we have $\frac{\partial \varphi ^*}{\partial \rho }\leqq 0$, and therefore $\frac{\varphi ^*-1}{\rho +(\varphi -1)}$ is largest when $\rho =0$. Thus

$$\begin{aligned} \frac{\varphi ^*-1}{\varphi -1} \leqq 3. \end{aligned}$$

(A.13)

Then, using (A.12), we can address the first coefficient in (A.11) as

$$\begin{aligned} \left|\frac{2\varepsilon a(\varphi ^*)-2(\varphi ^*-1)}{\rho +\varepsilon |a'(\varphi ^*)|(\varphi -1)} -1 \right| \leqq C\bigg ( \frac{\varepsilon a(\varphi ^*)+(\varphi ^*-1)}{\sqrt{(\varphi ^*-1)^2+\varepsilon ^2a^2(\varphi ^*)}} +1\bigg ) \leqq C. \end{aligned}$$

Similarly, by (A.12) and (A.13), we can address the third coefficient in (A.11) as

$$\begin{aligned}&\left|\frac{2\varepsilon ^2 a(\varphi ^*)a'(\varphi ^*) + 2\varepsilon |a'(\varphi ^*)|(\varphi ^*-1)}{\rho +\varepsilon |a'(\varphi ^*)|(\varphi -1)} -1 \right| \\&\quad \leqq \frac{2\varepsilon ^2 a(\varphi ^*)|a'(\varphi ^*)|}{\sqrt{\varepsilon ^2a^2(\varphi ^*)+(\varphi ^*-1)^2}} + 2\frac{\varphi ^*-1}{\varphi -1} +1 \\&\quad \leqq \frac{2{{\bar{c}}}_a\varepsilon ^2}{\sqrt{\varepsilon ^2a^2(\varphi ^*)+(\varphi ^*-1)^2} }+ 7. \end{aligned}$$

Now, if $0<(\varphi ^*-1)\leqq (\eta _\varepsilon -1)/2$, then by the spheroidal endpoint requirement of Definition 1.1, $a(\varphi ^*) \geqq C\varepsilon$. If $(\varphi ^*-1)\geqq (\eta _\varepsilon -1)/2$, then by (1.9), $(\varphi ^*-1)\geqq C\varepsilon ^2$. Therefore

$$\begin{aligned} \left|\frac{2\varepsilon ^2 a(\varphi ^*)a'(\varphi ^*) + 2\varepsilon |a'(\varphi ^*)|(\varphi ^*-1)}{\rho +\varepsilon |a'(\varphi ^*)|(\varphi -1)} -1 \right|&\leqq \frac{2{{\bar{c}}}_a\varepsilon ^2}{\sqrt{(\varphi ^*-1)^2+\varepsilon ^2a^2(\varphi ^*)} }+ 7 \\&\leqq C\frac{\varepsilon ^2}{\varepsilon ^2}+7 \leqq C. \end{aligned}$$

Finally, using (A.6) and (A.13), we can address the middle $\frac{1}{\rho }$ coefficient of (A.11). We have

$$\begin{aligned} \left|2\varepsilon a(\varphi ^*)-\rho \right| = \left|2\frac{\varphi ^*-1}{\varphi -1}-1 \right|\rho \leqq 7\rho , \end{aligned}$$

and therefore

$$\begin{aligned} \frac{1}{\rho } \leqq \frac{C}{2\varepsilon a(\varphi ^*)-\rho }. \end{aligned}$$

Altogether, for $\varvec{x}\in \Sigma ^\varepsilon _{\text {str}}$ and $1< \varphi <\eta _\varepsilon$, we have that $\nabla (E\varvec{v})$ satisfies

$$\begin{aligned} \begin{aligned} \left|\nabla (E\varvec{v})(\varphi ,\theta ,\rho ) \right|&\leqq C\bigg (\left|\frac{\partial \varvec{v}}{\partial \varphi } \right| + \left|\frac{1}{2\varepsilon a(\varphi ^*)-\rho } \right|\left|\frac{\partial \varvec{v}}{\partial \theta } \right| + \left|\frac{\partial \varvec{v}}{\partial \rho } \right| \bigg ) \\&\leqq C\left|\nabla \varvec{v}(2\varphi ^*-\varphi -1,\theta ,2\varepsilon a(\varphi ^*)-\rho ) \right|. \end{aligned} \end{aligned}$$

(A.14)

A similar bound holds for $-\eta _\varepsilon< \varphi < -1$. $\quad \square$

Step 3: Cutoff and definition of T

To make use of the extension E for $H^1(\mathcal {H}^\varepsilon )$ functions with arbitrary support, we must define a cutoff function $\psi \in \mathcal {C}^1(\mathcal {H}^\varepsilon )$. In particular, we take

$$\begin{aligned}&\psi (\varphi ,\rho ) \nonumber \\&\quad = {\left\{ \begin{array}{ll} {\left\{ \begin{array}{ll} 1 &{} \text { if } \delta (\varphi ,\varepsilon a(\varphi )) \leqq \rho \leqq \frac{4\delta (\varphi ,\varepsilon a(\varphi ))}{3} \\ 0 &{} \text { if } \rho> \frac{5\delta (\varphi ,\varepsilon a(\varphi ))}{3} \end{array}\right. } &{} \text { if } \left|\varphi \right| \leqq 1 \\ {\left\{ \begin{array}{ll} 1 &{} \text { if } \delta (\varphi ^*,\varepsilon a(\varphi ^*)) \leqq \sqrt{(\left|\varphi \right|-1)^2+\rho ^2} \leqq \frac{4\delta (\varphi ^*,\varepsilon a(\varphi ^*))}{3} \nonumber \\ 0 &{} \text { if } \sqrt{(\left|\varphi \right|-1)^2+\rho ^2} > \frac{5\delta (\varphi ^*,\varepsilon a(\varphi ^*))}{3} \end{array}\right. }&\text { if } 1\leqq \left|\varphi \right| \leqq \eta _\varepsilon \end{array}\right. }\\ \end{aligned}$$

(A.15)

with smooth transition between 0 and 1, and $\mathcal {C}^1$ transition from $\left|\varphi \right| \leqq 1$ to the endpoint sections. Here $\varphi ^*$ is as in (A.6).

For $\left|\varphi \right|<1$, we require that the decay rate $\partial \psi /\partial \rho$ is such that $\nabla \psi$ satisfies

$$\begin{aligned} \left|\nabla \psi (\varphi ,\rho ) \right|\leqq & {} \left|\frac{\partial \psi }{\partial \rho } \right|+\left|\frac{\partial \psi }{\partial \varphi } \right| \leqq C\bigg (\frac{1}{\varepsilon a(\varphi )} + \varepsilon |a'(\varphi )|\bigg ) \leqq \frac{1}{\varepsilon a(\varphi )}C\big (1 + \varepsilon ^2{{\bar{c}}}_a \big ) \nonumber \\\leqq & {} \frac{C}{\varepsilon a(\varphi )} \end{aligned}$$

(A.16)

for $\varepsilon$ sufficiently small. Similarly, for $1\leqq \left|\varphi \right|<\eta _\varepsilon$, we require that $\nabla \psi$ satisfies

$$\begin{aligned} \begin{aligned} \left|\nabla \psi (\varphi ,\rho ) \right|&\leqq \frac{C}{\delta (\varphi ^*,\varepsilon a(\varphi ^*))} \leqq C {\left\{ \begin{array}{ll} \frac{1}{\varepsilon (1-\varepsilon ^2c_a)\sqrt{\eta _\varepsilon ^2 - (\eta _\varepsilon +1)^2/4}}, &{} 1 \leqq \varphi ^*< \frac{\eta _\varepsilon +1}{2} \\ \frac{2}{\eta _\varepsilon -1}, &{} \frac{\eta _\varepsilon +1}{2}\leqq \varphi ^*<\eta _\varepsilon \end{array}\right. } \\&\leqq C {\left\{ \begin{array}{ll} \frac{1}{(\eta _\varepsilon -1)\sqrt{c_{\eta ,0}}(1-\varepsilon ^2c_a)}, &{} 1 \leqq \varphi ^*< \frac{\eta _\varepsilon +1}{2} \\ \frac{2}{\eta _\varepsilon -1}, &{} \frac{\eta _\varepsilon +1}{2} \leqq \varphi ^*<\eta _\varepsilon \end{array}\right. }\\&\leqq \frac{C}{\eta _\varepsilon -1}, \end{aligned}\nonumber \\ \end{aligned}$$

(A.17)

where we have used Definition 1.1 and (1.9) to rewrite the bound.

We then define our preliminary extension operator $T: H^1(\mathcal {H}^\varepsilon ) \rightarrow H^1(\mathcal {W}^\varepsilon )$ by

$$\begin{aligned} T\varvec{u}= E(\psi \varvec{u}) + (1-\psi )\varvec{u}. \end{aligned}$$

(A.18)

Note that the $L^2$ norm of the extension satisfies

$$\begin{aligned} \left||T\varvec{u} \right||_{L^2(\mathcal {W}^\varepsilon )} \leqq C \left||\varvec{u} \right||_{L^2(\mathcal {H}^\varepsilon )}, \end{aligned}$$

(A.19)

where the constant C is independent of $\varepsilon$.

Furthermore, using (A.9), (A.16), and (A.17), the symmetric gradient ${\mathcal {E}}(T\varvec{u})$ satisfies

$$\begin{aligned} \begin{aligned} \left|{\mathcal {E}}(T\varvec{u}) \right|&\leqq 2\left|\nabla (E(\psi \varvec{u})) \right| + \left|{\mathcal {E}}(\varvec{u}) \right| + 2\left|\nabla \psi \right|\left|\varvec{u} \right| \\&\leqq C\big (\left|\nabla (\psi \varvec{u}) \right|+\left|{\mathcal {E}}(\varvec{u}) \right| \big )+ C {\left\{ \begin{array}{ll} \frac{1}{\varepsilon a(\varphi )}\left|\varvec{u} \right|, &{} \left|\varphi \right|<1 \\ \frac{1}{\eta _\varepsilon -1}\left|\varvec{u} \right|, &{} 1\leqq \left|\varphi \right|<\eta _\varepsilon \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(A.20)

for C independent of $\varepsilon$.

Step 4: Auxiliary Korn-type lemmas

Now, we would like to be able to adapt this extension operator T to satisfy Lemma A.3. However, the $L^2(\mathcal {W}^\varepsilon )$ bound for ${\mathcal {E}}(T\varvec{u})$ is problematic (see the last term of (A.20)). To address this issue, we follow a similar construction to [22], Chapter 3, which develops an extension operator for a thin, infinite cylinder with gradient bounds independent of $\varepsilon$, as well as [23], Section 3.2, which adapts these arguments for the symmetric gradient. The new difficulty here is the change in the radius function $a(\varphi )$ from O(1) to 0 along the filament, as well as the introduction of endpoints.

The key to removing the $\varepsilon$-dependence in the estimate (A.20) will involve dividing the fiber into many small segments with length roughly equal to radius on each segment. We will then construct an extension operator that makes use of the Korn-Poincaré inequality (Lemma A.10) along with homogeneous rescaling (Corollary A.11) to get rid of the problematic last term of (A.20). This construction will involve working with a series of domains that are very similar to each other but not exactly identical. The main goal of this step is therefore to show that the Korn-Poincaré inequality (Lemma A.10) holds with a uniform constant on domains that are similar but not exactly the same. To arrive at this result, we must first prove a series of Korn-type inequalities with a uniform constant over slightly deformed domains.

To this end, for a bounded, Lipschitz domain ${\mathcal {D}}\subset {\mathbb {R}}^3$ and neighborhood $\mathcal {N}$ of ${\mathcal {D}}$, we will work with $\mathcal {C}^2$ maps $\Psi : \mathcal {N}\rightarrow {\mathbb {R}}^3$ satisfying

$$\begin{aligned} \Psi : \mathcal {N}\rightarrow \Psi (\mathcal {N}) \text { is a diffeomorphism}; \quad \left||\Psi \right||_{\mathcal {C}^2(\mathcal {N})} + \left||\Psi ^{-1} \right||_{\mathcal {C}^2(\Psi (\mathcal {N}))} \leqq M\nonumber \\ \end{aligned}$$

(A.21)

for some $M>0$. Note that $\Psi ({\mathcal {D}})$ is also a bounded, Lipschitz domain.

The reason we consider Lipschitz domains rather than smoother domains is that we will be using the following results in a truncated cylinder domain; in particular, ${\mathcal {D}}$ must be allowed to have corners. However, the mappings $\Psi$ that we consider will be $\mathcal {C}^2$ in the sense that second derivatives of $\Psi$ are uniformly continuous up to $\partial {\mathcal {D}}$.

We proceed to summarize the necessary tools to show Lemma A.10. We begin by recalling the following important result from elasticity theory, which does not depend on the domain. The proof may be found in [9].

Lemma A.6

(Rigid motion) Let ${\mathcal {D}}\subset {\mathbb {R}}^3$ be any domain. If $\varvec{u}: {\mathcal {D}}\rightarrow {\mathbb {R}}^3$ with $\nabla \varvec{u}\in L^2({\mathcal {D}})$ satisfies $\nabla \varvec{u}+(\nabla \varvec{u})^{\mathrm{T}}=0$, then $\varvec{u}$ is a rigid body motion: $\varvec{u}(\varvec{x}) = \varvec{A}\varvec{x}+{\varvec{b}}$ for some constant, antisymmetric $\varvec{A}\in {\mathbb {R}}^{3\times 3}$ and constant ${\varvec{b}}\in {\mathbb {R}}^3$.

Now, for a bounded, Lipschitz domain ${\mathcal {D}}$, we let ${\mathcal {R}}$ denote the space of rigid motions:

$$\begin{aligned} {\mathcal {R}} = \{ \varvec{v}\in H^1({\mathcal {D}}) : \varvec{v}= \varvec{A}\varvec{x}+ \varvec{b} \text { for some } \varvec{A}= -\varvec{A}^{\mathrm{T}} \in {\mathbb {R}}^{3\times 3} \text { and } \varvec{b}\in {\mathbb {R}}^3\}. \end{aligned}$$

For any $\varvec{u}\in H^1({\mathcal {D}})$, we define the $L^2$ projection $P_{\mathcal {R}}\varvec{u}$ onto the space ${\mathcal {R}}$:

$$\begin{aligned} P_{{\mathcal {R}}}\varvec{u}= \varvec{v}\in {\mathcal {R}} \text { such that } \Vert \varvec{u}-\varvec{v}\Vert _{L^2({\mathcal {D}})} \leqq \Vert \varvec{u}- \varvec{w}\Vert _{L^2({\mathcal {D}})}\quad \text {for all } \varvec{w}\in {\mathcal {R}}. \end{aligned}$$

By Lemma A.6, for any $\varvec{v}\in {\mathcal {R}}$, we have ${\mathcal {E}}(\varvec{v})=0$.

The first varying-domain result that we will need is a uniform Nečas inequality for slightly deformed domains. Here we verify just the uniformity of the constant for small domain deformations; the proof of the inequality itself can be found in [3, 9].

Lemma A.7

(Varying domain Nečas inequality). Let ${\mathcal {D}}\subset {\mathbb {R}}^3$ be a bounded, Lipschitz domain and consider any diffeomorphism $\Psi$ as in (A.21) for fixed $M>0$. Then for any $\varvec{w}\in H^{-1}(\Psi ({\mathcal {D}}))$ with $\nabla \varvec{w}\in H^{-1}(\Psi ({\mathcal {D}}))$, we in fact have $\varvec{w}\in L^2(\Psi ({\mathcal {D}}))$ and $\varvec{w}$ satisfies

$$\begin{aligned} \left||\varvec{w} \right||_{L^2(\Psi ({\mathcal {D}}))} \leqq C(M) \big ( \left||\varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} + \left||\nabla \varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} \big ), \end{aligned}$$

(A.22)

where the constant C(M) is uniform for any $\Psi$ satisfying (A.21).

Proof

Let $\widetilde{\varvec{w}}=\varvec{w}\circ \Psi ^{-1}\in L^2({\mathcal {D}})$, and let $\widetilde{C}$ denote the constant for which (A.22) holds for $\widetilde{\varvec{w}}$. Note that for any $\varvec{\phi }\in H^1_0({\mathcal {D}})$ with $\left||\nabla \varvec{\phi } \right||_{L^2({\mathcal {D}})}=1$, we have

$$\begin{aligned} \left| \int _{{\mathcal {D}}}\varvec{\phi } \cdot \widetilde{\varvec{w}} \right|&= \bigg | \int _{\Psi ({\mathcal {D}})}(\varvec{\phi }\circ \Psi ) \cdot \varvec{w}\left|\det \nabla \Psi \right| \bigg | = \bigg | \int _{\Psi ({\mathcal {D}})}\overline{\varvec{\phi }} \cdot \varvec{w}\bigg | \\&= \Vert \nabla \overline{\varvec{\phi }}\Vert _{L^2(\Psi ({\mathcal {D}}))} \bigg | \int _{\Psi ({\mathcal {D}})}\frac{\overline{\varvec{\phi }}\cdot \varvec{w}}{\Vert \nabla \overline{\varvec{\phi }}\Vert _{L^2(\Psi ({\mathcal {D}}))}} \bigg |, \end{aligned}$$

where we have defined $\overline{\varvec{\phi }} := (\varvec{\phi }\circ \Psi ) \left|\det \nabla \Psi \right|$. Taking the supremum over all such $\varvec{\phi }$, we obtain

$$\begin{aligned} \left||\widetilde{\varvec{w}} \right||_{H^{-1}({\mathcal {D}})} \leqq C(M) \left||\varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))}. \end{aligned}$$

Likewise,

$$\begin{aligned} \left||\nabla \widetilde{\varvec{w}} \right||_{H^{-1}({\mathcal {D}})} \leqq C(M) \left||\nabla \varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \int _{\Psi ({\mathcal {D}})} \left|\varvec{w} \right|^2 = \int _{{\mathcal {D}}} \left|\widetilde{\varvec{w}} \right|^2\left|\det \nabla \Psi ^{-1} \right| \leqq M\int _{{\mathcal {D}}} \left|\widetilde{\varvec{w}} \right|^2. \end{aligned}$$

Altogether,

$$\begin{aligned} \left||\varvec{w} \right||_{L^2(\Psi ({\mathcal {D}}))}&\leqq M\left||\widetilde{\varvec{w}} \right||_{L^2({\mathcal {D}})} \leqq M\widetilde{C}\big (\left||\widetilde{\varvec{w}} \right||_{H^{-1}({\mathcal {D}})} + \left||\nabla \widetilde{\varvec{w}} \right||_{H^{-1}({\mathcal {D}})} \big ) \\&\leqq M \widetilde{C}\big (C(M)\left||\varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} + C(M)\left||\nabla \varvec{w} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} \big ). \end{aligned}$$

$\square$

Using Lemma A.7, we immediately obtain the following result:

Lemma A.8

(Varying domain Korn-type inequality, 1) Let ${\mathcal {D}}\subset {\mathbb {R}}^3$ be a bounded, Lipschitz domain and let $\Psi$ be as in (A.21) for fixed $M>0$. Then $\varvec{v}\in H^1(\Psi ({\mathcal {D}}))$ satisfies

$$\begin{aligned} \left||\varvec{v} \right||_{H^1(\Psi ({\mathcal {D}}))} \leqq C(M) \big ( \left||{\mathcal {E}}(\varvec{v}) \right||_{L^2(\Psi ({\mathcal {D}}))} + \left||\varvec{v} \right||_{L^2(\Psi ({\mathcal {D}}))} \big ), \end{aligned}$$

(A.23)

where the constant C(M) is uniform for any $\Psi$ satisfying (A.21).

Proof

Define the set $K = \{\varvec{v}\in L^2(\Psi ({\mathcal {D}})) : {\mathcal {E}}(\varvec{v})\in L^2(\Psi ({\mathcal {D}}))\}$. Then K is a Hilbert space with norm

$$\begin{aligned} \left||\varvec{v} \right||_K^2 = \left||{\mathcal {E}}(\varvec{v}) \right||_{L^2(\Psi ({\mathcal {D}}))}^2 + \left||\varvec{v} \right||_{L^2(\Psi ({\mathcal {D}}))}^2. \end{aligned}$$

Letting $\varvec{x}=(x_1,x_2,x_3)$, $\varvec{v}=(v_1,v_2,v_3)$, and ${\mathcal {E}}_{ik}(\varvec{v}) = \frac{1}{2}\big (\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}\big )$, we have that for $\varvec{v}\in K$, $\frac{\partial {\mathcal {E}}_{ik}(\varvec{v})}{\partial x_j} \in H^{-1}(\Psi ({\mathcal {D}}))$ for each $1\leqq i,j,k\leqq 3$. Note that

$$\begin{aligned} \frac{\partial ^2 v_k}{\partial x_i\partial x_j} = \frac{\partial {\mathcal {E}}_{jk}(\varvec{v})}{\partial x_i} + \frac{\partial {\mathcal {E}}_{ik}(\varvec{v})}{\partial x_j} - \frac{\partial {\mathcal {E}}_{ij}(\varvec{v})}{\partial x_k}, \end{aligned}$$

and therefore $\left||\nabla ^2\varvec{v} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} \leqq 3 \left||\nabla \big ({\mathcal {E}}(\varvec{v})\big ) \right||_{H^{-1}(\Psi ({\mathcal {D}}))}$. Then, by Lemma A.7,

$$\begin{aligned} \left||\nabla \varvec{v} \right||_{L^2(\Psi ({\mathcal {D}}))}&\leqq C(M) \big ( \left||\nabla \varvec{v} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} + \left||\nabla ^2\varvec{v} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} \big ) \\&\leqq C(M) \big ( \left||\nabla \varvec{v} \right||_{H^{-1}(\Psi ({\mathcal {D}}))} + \left||\nabla \big ({\mathcal {E}}(\varvec{v})\big ) \right||_{H^{-1}(\Psi ({\mathcal {D}}))} \big ) \\&\leqq C(M) \big ( \left||\varvec{v} \right||_{L^2(\Psi ({\mathcal {D}}))} + \left||{\mathcal {E}}(\varvec{v}) \right||_{L^2(\Psi ({\mathcal {D}}))} \big ). \end{aligned}$$

$\square$

We now use Lemma A.8 to show that the following pair of varying-domain Korn-type inequalities hold with a uniform constant:

Corollary A.9

(Varying domain Korn-type inequalities, 2) Consider a bounded, Lipschitz domain $\mathcal {D}\subset {\mathbb {R}}^3$ along with any diffeomorphism $\Psi$ satisfying (A.21) for fixed $M>0$. Suppose $\varvec{v}\in H^1(\Psi ({\mathcal {D}}))$ satisfies at least one of the following conditions:

• 1. $\varvec{v}\perp {\mathcal {R}}$, the space of rigid motions on $\Psi ({\mathcal {D}})$, or

• 2. $\varvec{v}$ vanishes on an open set of $\partial (\Psi ({\mathcal {D}}))$ containing four non-coplanar points.

Then there exists a constant $C(M)>0$ such that $\varvec{v}$ satisfies

$$\begin{aligned} \Vert \nabla \varvec{v}\Vert _{L^2(\Psi ({\mathcal {D}}))} \leqq C(M)\Vert {\mathcal {E}}(\varvec{v})\Vert _{L^2(\Psi ({\mathcal {D}}))}. \end{aligned}$$

(A.24)

Proof

We first prove case 1. Assume that (A.24) does not hold. Then there is a sequence of $\mathcal {C}^2$-diffeomorphisms $\Psi _k$ and functions $\varvec{v}_k\in H^1(\Psi _k({\mathcal {D}}))$ with $\varvec{v}_k\perp {\mathcal {R}}$ and $\left||\varvec{v}_k \right||_{L^2(\Psi _k({\mathcal {D}}))} =1$ satisfying

$$\begin{aligned} \left||\nabla \varvec{v}_k \right||_{L^2(\Psi _k({\mathcal {D}}))} > k \left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))}. \end{aligned}$$

Let $\widetilde{\varvec{v}}_k=\varvec{v}_k\circ \Psi _k^{-1} \in H^1({\mathcal {D}})$. Then

$$\begin{aligned} 1=\int _{\Psi ({\mathcal {D}})}\left|\varvec{v}_k \right|^2 = \int _{{\mathcal {D}}}\left|\widetilde{\varvec{v}}_k \right|^2\left|\det \nabla \Psi _k^{-1} \right| \leqq M\int _{{\mathcal {D}}}\left|\widetilde{\varvec{v}}_k \right|^2, \end{aligned}$$

so $\left||\widetilde{\varvec{v}}_k \right||_{L^2({\mathcal {D}})} \geqq \frac{1}{\sqrt{M}} >0$.

Using Lemma A.8, we also have

$$\begin{aligned} \left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))}< & {} \frac{1}{k}\left||\nabla \varvec{v}_k \right||_{L^2(\Psi _k({\mathcal {D}}))} \leqq \frac{1}{k}\left||\varvec{v}_k \right||_{H^1(\Psi _k({\mathcal {D}}))} \\\leqq & {} \frac{C(M)}{k}\big ( \left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))} + 1 \big ). \end{aligned}$$

Taking k sufficiently large, we have

$$\begin{aligned} \bigg (1- \frac{C(M)}{k}\bigg )\left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))}< \frac{C(M)}{k} \rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$. Then, by definition of $\widetilde{\varvec{v}}_k$ and Lemma A.8, for k sufficiently large we have

$$\begin{aligned} \left||\widetilde{\varvec{v}}_k \right||_{H^1({\mathcal {D}})} \leqq C(M) \left||\varvec{v}_k \right||_{H^1(\Psi _k({\mathcal {D}}))} \leqq C(M)\bigg (\frac{C(M)}{k-C(M)} +1 \bigg ). \end{aligned}$$

Passing to a subsequence (which we continue to denote by $\widetilde{\varvec{v}}_k$) and using Rellich compactness, $\widetilde{\varvec{v}}_k\rightarrow \widetilde{\varvec{v}}_\infty$ in $L^2({\mathcal {D}})$. Note also that by (A.21), $\Psi _k\rightarrow \Psi _\infty$ in $\mathcal {C}^1$.

Now, choose any rigid motion $\varvec{A}\varvec{x}+\varvec{b} \in {\mathcal {R}}$ on $\Psi _\infty ({\mathcal {D}})$, where $\varvec{A}$ is a constant, antisymmetric matrix and $\varvec{b}$ is a constant vector. Note that this is also a well-defined rigid motion on $\Psi _k({\mathcal {D}})$ for any k. Then we have

$$\begin{aligned} \begin{aligned} 0&= \int _{\Psi _k({\mathcal {D}})} \varvec{v}_k\cdot (\varvec{A}\varvec{x}+\varvec{b}) = \int _{{\mathcal {D}}}\widetilde{\varvec{v}}_k \cdot (\varvec{A}\Psi ^{-1}_k(\varvec{x})+\varvec{b})\left|\det \nabla \Psi ^{-1}_k \right| \\&\quad \rightarrow \int _{{\mathcal {D}}}\widetilde{\varvec{v}}_\infty \cdot (\varvec{A}\Psi ^{-1}_\infty (\varvec{x})+\varvec{b}) \left|\det \nabla \Psi ^{-1}_\infty \right| = \int _{\Psi _\infty ({\mathcal {D}})}\varvec{v}_\infty \cdot (\varvec{A}\varvec{x}+\varvec{b}). \end{aligned} \end{aligned}$$

(A.25)

Thus $\varvec{v}_\infty \perp {\mathcal {R}}$ on $\Psi _\infty ({\mathcal {D}})$. However, since $\liminf _k\left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))}\geqq \left||{\mathcal {E}}(\varvec{v}_\infty ) \right||_{L^2(\Psi _\infty ({\mathcal {D}}))}$, we have ${\mathcal {E}}(\varvec{v}_\infty )=0$, and by Lemma A.6, $\varvec{v}_\infty \in {\mathcal {R}}$. Thus $\varvec{v}_\infty =0$, so $\varvec{v}_k\rightarrow 0$, contradicting $\left||\varvec{v}_k \right||_{L^2(\Psi ({\mathcal {D}}))}\geqq \frac{1}{\sqrt{M}}>0$.

To show case (2.) of Corollary A.9, we instead take $\varvec{v}_k\in H^1(\Psi _k({\mathcal {D}}))$ vanishing on an open set of $\partial (\Psi ({\mathcal {D}}))$ containing four non-coplanar points. The proof proceeds as above, except instead of arriving at a contradiction via (A.25), we use that each $\widetilde{\varvec{v}}_k$ vanishes on an open set of $\partial {\mathcal {D}}$ and thus $\widetilde{\varvec{v}}_\infty$ does as well. Then $\varvec{v}_\infty$ also vanishes on an open set of $\partial (\Psi _\infty ({\mathcal {D}}))$, but ${\mathcal {E}}(\varvec{v}_\infty )=0$, and therefore $\varvec{v}_\infty =0$. $\quad \square$

Finally, using Corollary A.9, we can show the following:

Lemma A.10

(Varying domain Korn-Poincaré inequality) Let $\mathcal {D}\subset {\mathbb {R}}^3$ be a bounded, Lipschitz domain and consider any diffeomorphism $\Psi$ satisfying (A.21) for fixed $M>0$. Then there exists a constant $C(M)>0$ such that any $\varvec{v}\in H^1(\Psi ({\mathcal {D}}))$ with $\varvec{v}\perp {\mathcal {R}}$, the space of rigid motions on $\Psi ({\mathcal {D}})$, satisfies

$$\begin{aligned} \left||\varvec{v} \right||_{L^2(\Psi ({\mathcal {D}}))} \leqq C(M) \left||{\mathcal {E}}(\varvec{v}) \right||_{L^2(\Psi ({\mathcal {D}}))}. \end{aligned}$$

(A.26)

Proof

Suppose that Lemma A.10 does not hold. Then there exist a sequence of diffeomorphisms $\Psi _k$ and functions $\varvec{v}_k\in H^1(\Psi _k({\mathcal {D}}))$ with $\varvec{v}_k\perp {\mathcal {R}}$ and $\left||\varvec{v}_k \right||_{L^2(\Psi _k({\mathcal {D}}))} =1$ such that

$$\begin{aligned} 1 = \left||\varvec{v}_k \right||_{L^2(\Psi _k({\mathcal {D}}))} > k\left||{\mathcal {E}}(\varvec{v}_k) \right||_{L^2(\Psi _k({\mathcal {D}}))}. \end{aligned}$$

As in the proof of Corollary A.9, let $\widetilde{\varvec{v}}_k = \varvec{v}_k\circ \Psi _k^{-1}\in H^1({\mathcal {D}})$. Then $\left||\widetilde{\varvec{v}}_k \right||_{L^2({\mathcal {D}})} \geqq \frac{1}{\sqrt{M}} >0$. Furthermore, using Corollary A.9, we have

$$\begin{aligned} \int _{{\mathcal {D}}} \left|\nabla \widetilde{\varvec{v}}_k \right|^2&\leqq \int _{\Psi _k({\mathcal {D}})} \left|\nabla \varvec{v}_k \right|^2\left|\nabla \Psi _k^{-1} \right|^2\left|\det \nabla \Psi _k \right| \leqq M^3\int _{\Psi _k({\mathcal {D}})} \left|\nabla \varvec{v}_k \right|^2 \\&\leqq C(M) \int _{\Psi _k({\mathcal {D}})} \left|{\mathcal {E}}(\varvec{v}_k) \right|^2\leqq \frac{1}{k^2} \rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$. Passing to a subsequence (which we continue to denote $\widetilde{\varvec{v}}_k$) and using Rellich compactness, we have that $\widetilde{\varvec{v}}_k\rightarrow \widetilde{\varvec{v}}_\infty$ in $L^2({\mathcal {D}})$. Also, since $\liminf _k\left||\nabla \widetilde{\varvec{v}}_k \right||_{L^2({\mathcal {D}})} \leqq \left||\nabla \widetilde{\varvec{v}}_\infty \right||_{L^2({\mathcal {D}})}$, we have $\nabla \widetilde{\varvec{v}}_\infty =0$, so $\widetilde{\varvec{v}}_\infty =\varvec{b}$ for some constant vector $\varvec{b}$.

However, using that $\Psi _k\rightarrow \Psi _\infty$ in $\mathcal {C}^1$, by (A.21), we have $\varvec{v}_\infty = \widetilde{\varvec{v}}_\infty \circ \Psi _\infty = \varvec{b}$, and $\varvec{v}_\infty \perp {\mathcal {R}}$, so $\varvec{b}=0$. Thus $\widetilde{\varvec{v}}_\infty =0$, which contradicts $\left||\widetilde{\varvec{v}}_k \right||_{L^2({\mathcal {D}})} \geqq \frac{1}{\sqrt{M}} >0$. $\quad \square$

Lemma A.10 under homogeneous rescaling then gives rise to the following corollary:

Corollary A.11

(Korn-Poincaré rescaling) Let $\mathcal {D}\subset {\mathbb {R}}^3$ be a bounded, Lipschitz domain and $\Psi$ a diffeomorphism as in (A.21) for fixed $M>0$. Consider $\varvec{v}\in H^1(\Psi (\mathcal {D}))$ with $\varvec{v}\perp {\mathcal {R}}$, the space of rigid motions on $\Psi ({\mathcal {D}})$. Under homogeneous rescaling of the domain $\Psi (\mathcal {D})\rightarrow \lambda \Psi (\mathcal {D})$, $\lambda \in {\mathbb {R}}_+$, we have

$$\begin{aligned} \left||\varvec{v} \right||_{L^2(\lambda \Psi (\mathcal {D}))} \leqq \lambda C(M)\left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\lambda \Psi (\mathcal {D}))}, \end{aligned}$$

(A.27)

where the constant C(M) is uniform for all $\Psi$ satisfying (A.21).

Step 5: Decomposition of domain $\mathcal {W}^\varepsilon$

Now, we would like to make use of the scaling in Corollary A.11 to design an extension operator that removes the $\varepsilon$-dependence coming from the lower part of the norm in (A.20). To do so, we consider the slender body as the union of many tiny segments, each sufficiently small to be able to make use of the scaling in Corollary A.11. We consider the positive half of the fiber ($\varphi \geqq 0$) here; the negative half ($\varphi \leqq 0$) follows by the same arguments.

Recalling the constant $\delta _a$ given by Definition 1.1, we partition the segment [0, 1] in the following way: we define a collection of points $\{p_n\}$ by

$$\begin{aligned} \begin{aligned} p_1 = 1;\quad p_{n+1} = {\left\{ \begin{array}{ll} p_n-\varepsilon a(p_n) &{} \text { if } p_n > 1-\delta _a \\ p_n-\varepsilon &{} \text { if } 0\leqq p_n \leqq 1-\delta _a. \end{array}\right. } \end{aligned} \end{aligned}$$

(A.28)

Note that we continue iterating this procedure until we reach N such that $p_{N+1}<0$. It will also be necessary to define $p_{N+2}=p_{N+1}-\varepsilon$. We also let $N_\delta$ denote

$$\begin{aligned} N_\delta := \min \{ n\geqq 0 : p_n \leqq 1-\delta _a \}. \end{aligned}$$

(A.29)

Using the collection of points $\{p_n\}$, we define an open cover $\{Q_n\}$ of the interval $(0,\eta _\varepsilon )$ as follows: for $\varphi \in (p_{N+2},2\eta _\varepsilon -1)$, define

$$\begin{aligned} \begin{aligned} Q_0 = \big \{\varphi&: p_1<\varphi<2\eta _\varepsilon -1 \big \}, \quad Q_1 = \big \{\varphi : p_2<\varphi<(\eta _\varepsilon +1)/2 \big \}, \\ \quad Q_n&= \big \{\varphi : p_{n+1}<\varphi <p_{n-1} \big \}, \quad n\geqq 2. \end{aligned}\nonumber \\ \end{aligned}$$

(A.30)

Using the cover $\{Q_n\}$, we subdivide the domain $\mathcal {W}^\varepsilon$ as (see Fig. 5)

$$\begin{aligned} \mathcal {W}_n^\varepsilon = \big \{ \varvec{x}=\varvec{x}(\varphi ,\theta ,\rho ) \in \mathcal {W}^\varepsilon : \varphi \in Q_n \big \}. \end{aligned}$$

(A.31)

Fig. 5

We subdivide the domain $\mathcal {W}^\varepsilon$ using the points $\{p_n\}$ defined via the process in (A.28)

We also define

$$\begin{aligned} \mathcal {H}_n^\varepsilon :=\mathcal {W}_n^\varepsilon \cap \mathcal {H}^\varepsilon . \end{aligned}$$

(A.32)

Now, using Definition 1.1 – in particular, the endpoint monotonicity requirement – along with (A.28), for each $n\in [2,N_\delta ]$ we can write

$$\begin{aligned} a(p_n)&= a(p_{n-1})- \varepsilon {\overline{a}}(p_n,p_{n-1}) \quad \text { for some function } \left|{\overline{a}} \right| \leqq {{\bar{c}}}_a, \end{aligned}$$

where ${{\bar{c}}}_a$ and $\delta _a$ are as defined in Section 1.1. Then the ratios $a(p_n)/a(p_{n-1})$ and $a(p_{n-1})/a(p_n)$ satisfy

$$\begin{aligned} 0 < \frac{a(p_n)}{a(p_{n-1})}, \frac{a(p_{n-1})}{a(p_n)} \leqq 1+ \varepsilon {{\bar{c}}}_a/a(1), \quad 2\leqq n\le N_\delta . \end{aligned}$$

(A.33)

Using the spheroidal endpoint requirement of Definition 1.1, we have that these ratios are bounded independent of $\varepsilon$.

Furthermore, using (1.9) and Definition 1.1, we have that the ratios $(\eta _\varepsilon -1)/(\varepsilon a(1))$ and $\varepsilon a(1)/(\eta _\varepsilon -1)$ satisfy

$$\begin{aligned} 0<\frac{\eta _\varepsilon -1}{\varepsilon a(1)} \leqq \frac{c_{\eta }}{\sqrt{2c_{\eta ,0}}- \varepsilon c_a}, \quad 0<\frac{\varepsilon a(1)}{\eta _\varepsilon -1} \leqq \frac{\sqrt{2c_\eta }+\varepsilon c_a}{c_{\eta ,0}}, \end{aligned}$$

(A.34)

and are thus also bounded independent of $\varepsilon$.

Therefore we can can define the sets $\mathcal {W}_n^*$ as follows:

$$\begin{aligned} \mathcal {W}_n^\varepsilon = {\left\{ \begin{array}{ll} \varepsilon a(p_1)\mathcal {W}_0^*, &{}n=0 \\ \varepsilon a(p_n)\mathcal {W}_n^*, &{} 1\leqq n\le N_\delta \\ \varepsilon \mathcal {W}_n^*, &{} N_\delta +1 \leqq n\le N+1, \end{array}\right. } \end{aligned}$$

(A.35)

where each $\mathcal {W}_n^*$ is contained in a cylinder of radius

$$\begin{aligned} {\left\{ \begin{array}{ll} 2, &{}n=0 \\ 2\frac{a(p_{n+1})}{a(p_n)}, &{} 1\leqq n \leqq N_\delta \\ 2, &{} N_\delta +1 \leqq n\le N+1 \end{array}\right. } \end{aligned}$$

and height

$$\begin{aligned} {\left\{ \begin{array}{ll} 2\frac{\eta _\varepsilon -1}{a(1)}, &{} n=0 \\ 1+\frac{\eta _\varepsilon -1}{2a(1)}, &{} n=1 \\ 1+ \frac{a(p_{n-1})}{a(p_n)}, &{} 2\leqq n\le N_\delta \\ 2, &{} N_\delta +1 \leqq n \leqq N+1. \end{array}\right. } \end{aligned}$$

See Fig. 6 for a depiction of $\mathcal {W}_n^*$ for $2\leqq n\le N_\delta$. In particular, due to (A.33) and (A.34), the volume of each slice $\mathcal {W}_n^*$ is bounded independent of $\varepsilon$. By the same arguments, we can also write $\mathcal {H}_n^\varepsilon$ (A.32) as

$$\begin{aligned} \mathcal {H}_n^\varepsilon = {\left\{ \begin{array}{ll} \varepsilon a(p_1)\mathcal {H}_0^*, &{} n=0 \\ \varepsilon a(p_n)\mathcal {H}_n^*, &{} 1\leqq n \leqq N_\delta \\ \varepsilon \mathcal {H}_n^*, &{} N_\delta +1\leqq n\le N+1, \end{array}\right. }\nonumber \\ \end{aligned}$$

(A.36)

where the volume of each $\mathcal {H}_n^*$ is bounded independent of $\varepsilon$.

Fig. 6

For $2\leqq n \leqq N_\delta$, we may think of the domain $\mathcal {W}_n^\varepsilon$ as $\mathcal {W}_n^\varepsilon =\varepsilon a(p_n)\mathcal {W}_n^*$, a rescaled version of the domain $\mathcal {W}_n^*$, the volume of which is bounded independent of $\varepsilon$. A similar result holds on the endpoint segments $n=0,1$ and on the segments $N_\delta +1\leqq n\le N+1$

Remark A.12

The domains $\mathcal {H}^*_n$, $1\leqq n\le N_\delta$, are each slightly different in size and shape; however, each can be deformed via a diffeomorphism satisfying (A.21) for M small to a cylinder of height 2 and radius 2 with a cylindrical hole of radius 1. For any $1\leqq n\le N_\delta$, we may parameterize $\mathcal {H}^*_n$ with respect to cylindrical coordinates as

$$\begin{aligned}&\mathcal {H}^*_n = \bigg \{ (\varphi ,\theta ,\rho ) : 0\leqq \varphi \leqq 1+\frac{a(p_{n-1})}{a(p_n)}, 0\leqq \theta <2\pi , \frac{a(p_{n+1}-\varphi )}{a(p_n)} \leqq \rho \\&\quad \qquad \leqq 2\frac{a(p_{n+1}-\varphi )}{a(p_n)} \bigg \}. \end{aligned}$$

Then, using (A.33), the transformation

$$\begin{aligned} \Psi _n: (\varphi ,\theta ,\rho ) \mapsto \bigg (\frac{a(p_n)}{a(p_n)+a(p_{n-1})}\varphi ,\theta ,\frac{a(p_n)}{a(p_{n+1}-\varphi ) }\rho \bigg ) \end{aligned}$$

gives the desired diffeomorphism. Note that then, by Lemma A.10, the Korn-Poincaré inequality over each of the domains $\mathcal {H}^*_n$, $1\leqq n\le N+1$, has a uniform constant.

The endpoint segment ($n=0$) must be treated separately from the above construction. Instead, using (A.34), we may deform $\mathcal {H}^*_0$ to a hemisphere of radius 2. Since the endpoint shape implicitly depends on $\varepsilon$ via Definition 1.1, this will provide us with a uniform Korn-Poincaré inequality (Lemma A.10) on the endpoint segment for any value of $\varepsilon$.

We may then take the maximum of the two different Korn-Poncaré constants to serve as a uniform Korn-Poincaré constant for all segments of the fiber.

Now, for $\varvec{u}$ defined in $\mathcal {H}^\varepsilon$, let $\varvec{u}_n$ denote the restriction $\varvec{u}\big |_{\mathcal {H}_n^\varepsilon }$. On each $\mathcal {H}^\varepsilon _n$, we consider the projection $P_{{\mathcal {R}}}\varvec{u}_n$ onto rigid motions, and write $P_{{\mathcal {R}}}\varvec{u}_n = \varvec{A}_n\varvec{x}+\varvec{b}_n$ for some $\varvec{A}_n=-\varvec{A}_n^{\mathrm{T}}\in {\mathbb {R}}^{3\times 3}$ and $\varvec{b}_n\in {\mathbb {R}}^3$. We omit the dependence of the projection $P_{{\mathcal {R}}}$ on n and $\varepsilon$. We take

$$\begin{aligned} {\overline{P}}_{{\mathcal {R}}}\varvec{u}_n = \varvec{A}_n\varvec{x}+\varvec{b}_n, \quad \varvec{x}\in \Sigma _n^\varepsilon \end{aligned}$$

to be the extension of $P_{{\mathcal {R}}}\varvec{u}_n$ to all of $\mathcal {W}_n^\varepsilon$. Note that on each $\mathcal {W}_n^\varepsilon$, we have

$$\begin{aligned} \left||{\overline{P}}_{{\mathcal {R}}}\varvec{u}_n \right||_{L^2(\mathcal {W}_n^\varepsilon )} \leqq C\left||P_{{\mathcal {R}}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}. \end{aligned}$$

(A.37)

Let $\{\Phi _n \}$ be a partition of unity subordinate to the cover $\{Q_n\}$. We require each $\Phi _n$ to satisfy

$$\begin{aligned} \left|\nabla \Phi _n \right|\leqq C {\left\{ \begin{array}{ll} \frac{1}{\varepsilon a(1)}, &{} n=0,1 \\ \frac{1}{\varepsilon a(p_{n-1})}, &{} 2\leqq n\le N_\delta \\ \frac{1}{\varepsilon }, &{} N_\delta +1\leqq N+1 \end{array}\right. } \end{aligned}$$

(A.38)

where the constant C is independent of $\varepsilon$. Note that due to (A.34), the bound for $n=0,1$ is equivalent to requiring that $\left|\nabla \Phi _n \right|\leqq C/(\eta _\varepsilon -1)$, $n=0,1$.

Step 6: Definition of extension operator $T_\varepsilon$

We now define the extension operator $T_\varepsilon : H^1(\mathcal {H}^\varepsilon ) \rightarrow H^1(\mathcal {W}^\varepsilon )$ that satisfies Lemma A.3. For $\varvec{x}\in \Sigma ^\varepsilon _\text {str}$, we take

$$\begin{aligned} \begin{aligned} T_\varepsilon \varvec{u}(\varvec{x})&= \varvec{T}_1(\varvec{x})+\varvec{T}_2(\varvec{x}); \\ \varvec{T}_1(\varvec{x})&:= \sum _{n=0}^{N+1} \Phi _n(\varphi ) \bigg ( {\overline{P}}_{{\mathcal {R}}}\varvec{u}_n\bigg )(\varvec{x}) \\ \varvec{T}_2(\varvec{x})&:= \sum _{n=0}^{N+1} \Phi _n(\varphi ) \bigg (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\bigg )(\varvec{x}), \end{aligned} \end{aligned}$$

(A.39)

where the operator T is as in (A.18). Note that $T_\varepsilon \varvec{u}\big |_{\mathcal {H}^\varepsilon } = \varvec{u}$.

It remains to estimate the symmetric gradient ${\mathcal {E}}(T_\varepsilon \varvec{u})$ of the extension (A.39). We begin by estimating $\left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\mathcal {W}^\varepsilon )}$, which is simpler.

Define a finer partition $\{\widetilde{Q}_n\}$ of the interval $(0,\eta _\varepsilon )$:

$$\begin{aligned} \widetilde{Q}_0 = \big \{\varphi : p_1<\varphi<\eta _\varepsilon \big \}, \quad \widetilde{Q}_n = \big \{\varphi : p_{n+1} <\varphi \leqq p_n \big \}, \quad n\geqq 2.\nonumber \\ \end{aligned}$$

(A.40)

For each $n\geqq 0$, define the slices $\widetilde{\mathcal {W}}_n^\varepsilon$ of the fiber as follows:

$$\begin{aligned} \widetilde{\mathcal {W}}_n^\varepsilon = \big \{ \varvec{x}= \varvec{x}(\varphi ,\theta ,\rho )\in \mathcal {W}^\varepsilon : \varphi \in \widetilde{Q}_n \big \}. \end{aligned}$$

(A.41)

Accordingly, we define $\widetilde{\mathcal {H}}_n^\varepsilon = \widetilde{\mathcal {W}}_n^\varepsilon \cap \mathcal {H}^\varepsilon$.

Note that $\widetilde{\mathcal {W}}_n^\varepsilon \subset \mathcal {W}_n^\varepsilon$ and $\widetilde{\mathcal {W}}_n^\varepsilon \subset \mathcal {W}_{n+1}^\varepsilon$ for each n, and $\Phi _n(\varphi )+\Phi _{n+1}(\varphi )=1$ on $\widetilde{\mathcal {W}}_n^\varepsilon$. Then on each $\widetilde{\mathcal {W}}_n^\varepsilon$, we may write

$$\begin{aligned} \varvec{T}_1(\varvec{x}) = {\overline{P}}_{\mathcal {R}}\varvec{u}_n + \Phi _{n+1}(\varphi ) ({\overline{P}}_{\mathcal {R}}\varvec{u}_{n+1}-{\overline{P}}_{\mathcal {R}}\varvec{u}_n). \end{aligned}$$

Using the bounds (A.37) and (A.38), for $2\leqq n\le N_\delta$ we have

$$\begin{aligned} \left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\widetilde{\mathcal {W}}_n^\varepsilon )}&\leqq \frac{C}{\varepsilon a(p_{n-1})}\left||{\overline{P}}_{\mathcal {R}}\varvec{u}_{n+1}-{\overline{P}}_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\widetilde{\mathcal {W}}_n^\varepsilon )}\\&\leqq \frac{C}{\varepsilon a(p_{n-1})}\left||P_{\mathcal {R}}\varvec{u}_{n+1}- P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\widetilde{\mathcal {H}}_n^\varepsilon )} \\&\leqq \frac{C}{\varepsilon a(p_{n-1})}\bigg (\left||\varvec{u}_{n+1}-P_{\mathcal {R}}\varvec{u}_{n+1} \right||_{L^2(\mathcal {H}_{n+1}^\varepsilon )}\\&\quad +\left||\varvec{u}_n -P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )} \bigg ), \end{aligned}$$

since $\widetilde{\mathcal {H}}_n^\varepsilon \subset \mathcal {H}_n^\varepsilon$ and $\widetilde{\mathcal {H}}_n^\varepsilon \subset \mathcal {H}_{n+1}^\varepsilon$. Then, using Remark A.12, by (A.36) and Lemma A.11 we have

$$\begin{aligned}&\left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\widetilde{\mathcal {W}}_n^\varepsilon )} \\&\quad \leqq C\frac{a(p_n)}{a(p_{n-1})} \bigg (\left||{\mathcal {E}}(\varvec{u}_{n+1}) \right||_{L^2(\mathcal {H}_{n+1}^\varepsilon )}+\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} \bigg ), \quad 2\leqq n\le N_\delta . \end{aligned}$$

Following the same arguments on the endpoint segments $n=0,1$ and on segments $n\geqq N_\delta +1$, we obtain

$$\begin{aligned}&\left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\widetilde{\mathcal {W}}_n^\varepsilon )} \\&\quad \leqq C \bigg (\left||{\mathcal {E}}(\varvec{u}_{n+1}) \right||_{L^2(\mathcal {H}_{n+1}^\varepsilon )}+\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} \bigg ),&n=0,1 \text { or } n\geqq N_\delta +1. \end{aligned}$$

Squaring and summing over n, using orthogonality from the nearly-disjoint supports of each $\varvec{u}_n$ along with (A.33), we thus have

$$\begin{aligned} \left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\mathcal {W}^\varepsilon )}^2\leqq C\sum _{n=0}^{N+1} \left||{\mathcal {E}}(\varvec{T}_1) \right||_{L^2(\mathcal {H}_n^\varepsilon )}^2 \leqq C\left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\mathcal {H}^\varepsilon )}^2 \end{aligned}$$

(A.42)

for C independent of $\varepsilon$.

Next we estimate $\varvec{T}_2$. Using (A.38), we have

$$\begin{aligned} \begin{aligned}&\left||{\mathcal {E}}(\varvec{T}_2) \right||_{L^2(\mathcal {W}^\varepsilon )}^2 \\&\quad \leqq \sum _{n=0}^1 \bigg (\frac{C}{\varepsilon a(1)}\left||T(\varvec{u}_n- P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2 + \left||{\mathcal {E}}\big (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\big ) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2 \bigg ) \\&\qquad + \sum _{n=2}^{N_\delta } \bigg (\frac{C}{\varepsilon a(p_{n-1})}\left||T(\varvec{u}_n- P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2 + \left||{\mathcal {E}}\big (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\big ) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2 \bigg ) \\&\qquad + \sum _{n=N_\delta +1}^{N+1} \bigg (\frac{C}{\varepsilon }\left||T(\varvec{u}_n- P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2 + \left||{\mathcal {E}}\big (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\big ) \right||_{L^2(\mathcal {W}_n^\varepsilon )}^2\bigg ). \end{aligned}\nonumber \\ \end{aligned}$$

(A.43)

Now, using (A.19) along with Remark A.12, the scaling (A.36), and Corollary A.11, we have

$$\begin{aligned} \begin{aligned} \left||T(\varvec{u}_n- P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {W}_n^\varepsilon )}&\leqq C\left||\varvec{u}_n- P_{{\mathcal {R}}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )} \\&\leqq C {\left\{ \begin{array}{ll} \varepsilon a(p_1)\left||{\mathcal {E}}(\varvec{u}_0) \right||_{L^2(\mathcal {H}_0^\varepsilon )}, &{} n=0 \\ \varepsilon a(p_n)\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} 1\leqq n\le N_\delta \\ \varepsilon \left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} N_\delta +1\leqq n \leqq N+1. \end{array}\right. } \end{aligned} \end{aligned}$$

(A.44)

Furthermore, noting that ${\mathcal {E}}(\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n) = \varvec{u}_n$ and using (A.20), we have

$$\begin{aligned} \left||{\mathcal {E}}\big (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\big ) \right||_{L^2(\mathcal {W}_n^\varepsilon )}&\leqq C \big (\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )}+\left||\nabla (\psi \varvec{u}_n-\psi P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} \big ) \\&\quad + C {\left\{ \begin{array}{ll} \frac{1}{\eta _\varepsilon -1} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} n=0,1 \\ \frac{1}{\varepsilon a(p_{n-1})} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} 2\leqq n \leqq N_\delta \\ \frac{1}{\varepsilon a_0} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} N_\delta +1 \leqq n, \end{array}\right. } \end{aligned}$$

where $a_0$ is as in Definition 1.1.

Now, due to the cutoff function $\psi$, we have that $\psi \varvec{u}_n-\psi P_{{\mathcal {R}}}\varvec{u}_n$ vanishes for $\delta (\varphi ,\rho )> \frac{5}{3}\delta (\varphi ^*,\varepsilon a(\varphi ^*))$, and thus by Corollary A.9, case (2.), we obtain

$$\begin{aligned}&\left||\nabla (\psi \varvec{u}_n-\psi P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} \leqq C\left||{\mathcal {E}}(\psi \varvec{u}_n-\psi P_{{\mathcal {R}}}\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} \\&\quad \leqq C\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )}+ C {\left\{ \begin{array}{ll} \frac{1}{\eta _\varepsilon -1} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} n=0,1 \\ \frac{1}{\varepsilon a(p_{n-1})} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} 2\leqq n \leqq N_\delta \\ \frac{1}{\varepsilon a_0} \left||\varvec{u}_n-P_{\mathcal {R}}\varvec{u}_n \right||_{L^2(\mathcal {H}_n^\varepsilon )}, &{} N_\delta +1 \leqq n. \end{array}\right. } \end{aligned}$$

Then, using Remark A.12, the scaling (A.36), and Corollary A.11, we have

$$\begin{aligned}&\left||{\mathcal {E}}\big (T(\varvec{u}_n-P_{{\mathcal {R}}}\varvec{u}_n)\big ) \right||_{L^2(\mathcal {W}_n^\varepsilon )} \nonumber \\&\quad \leqq C\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}_n^\varepsilon )} {\left\{ \begin{array}{ll} 1+ \frac{\varepsilon a(1)}{\eta _\varepsilon -1}, &{} n=0,1 \\ 1+ \frac{a(p_n)}{a(p_{n-1})}, &{} 2\leqq n \leqq N_\delta \\ 1+ \frac{1}{a_0}, &{} N_\delta +1 \leqq n. \end{array}\right. } \end{aligned}$$

(A.45)

Using the estimates (A.44) and (A.45) in (A.43), and again using orthogonality from the nearly-disjoint supports of each $\varvec{u}_n$, we have

$$\begin{aligned} \left||{\mathcal {E}}(\varvec{T}_2) \right||_{L^2(\mathcal {W}^\varepsilon )}^2 \leqq C\sum _{n=0}^{N-1}\left||{\mathcal {E}}(\varvec{u}_n) \right||_{L^2(\mathcal {H}^\varepsilon _n)}^2 \leqq C\left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\mathcal {H}^\varepsilon )}^2, \end{aligned}$$

(A.46)

where, by the ratio bounds (A.33) and (A.34), C is independent of $\varepsilon$. Combining (A.42) and (A.46), we obtain the $\varepsilon$-independent estimate (2.) of Lemma A.3. $\quad \square$

1.1.4 Korn and Sobolev Inequalities

Using the extension operator guaranteed by Corollary A.4, we can easily show that the Korn and Sobolev inequalities (Lemmas 2.4 and 2.2, respectively) hold in $\Omega _\varepsilon$ with constants that are independent of $\varepsilon$.

Proof of Lemma 2.4

Recall that for any $\varvec{v}\in D^{1,2}({\mathbb {R}}^3)$, the Korn inequality over all of ${\mathbb {R}}^3$ holds with constant $c_K=\sqrt{2}$:

$$\begin{aligned} \int _{{\mathbb {R}}^3} |{\mathcal {E}}(\varvec{v})|^2 \hbox {d}\varvec{x}&= \int _{{\mathbb {R}}^3} \bigg (2|\nabla \varvec{v}|^2 + 2\nabla \varvec{v}:(\nabla \varvec{v})^{\mathrm{T}}\bigg ) \hbox {d}\varvec{x}\\&= \int _{{\mathbb {R}}^3} 2|\nabla \varvec{v}|^2 \hbox {d}\varvec{x}+ 2\int _{{\mathbb {R}}^3} |{\mathrm{div}}\, \varvec{v}|^2 \hbox {d}\varvec{x}\geqq \int _{{\mathbb {R}}^3} 2|\nabla \varvec{v}|^2 \hbox {d}\varvec{x}. \end{aligned}$$

Here we have used integration by parts twice on the second integral term.

Then, using the extension operator $\widetilde{T}_\varepsilon$ from Corollary A.4, we have

$$\begin{aligned} \left||\nabla \varvec{u} \right||_{L^2(\Omega _\varepsilon )}&\leqq \left||\nabla (\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2({\mathbb {R}}^3)} \leqq \sqrt{2}\left||{\mathcal {E}}(\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2({\mathbb {R}}^3)} \leqq C\left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\Omega _\varepsilon )}. \end{aligned}$$

$\square$

Similarly, we can use the extension operator $\widetilde{T}_\varepsilon$ to show that the Sobolev inequality (2.2) holds in $\Omega _\varepsilon$ with a constant independent of $\varepsilon$.

Proof of Lemma 2.2

First note that, using the Korn inequality on ${\mathbb {R}}^3$, the extension $\widetilde{T}_\varepsilon$ from Corollary A.4 satisfies

$$\begin{aligned} \left||\nabla (\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2({\mathbb {R}}^3)} \leqq \sqrt{2}\left||{\mathcal {E}}(\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2({\mathbb {R}}^3)} \leqq C\left||{\mathcal {E}}(\varvec{u}) \right||_{L^2(\Omega _\varepsilon )} \leqq C\left||\nabla \varvec{u} \right||_{L^2(\Omega _\varepsilon )}. \end{aligned}$$

We then have

$$\begin{aligned} \left||\varvec{u} \right||_{L^6(\Omega _\varepsilon )} \leqq \left||\widetilde{T}_\varepsilon \varvec{u} \right||_{L^6({\mathbb {R}}^3)} \leqq c_R\left||\nabla (\widetilde{T}_\varepsilon \varvec{u}) \right||_{L^2({\mathbb {R}}^3)} \leqq C\left||\nabla \varvec{u} \right||_{L^2(\Omega _\varepsilon )}, \end{aligned}$$

where $c_R$ is the Sobolev constant over all of ${\mathbb {R}}^3$. Since $c_R$ is a universal constant independent of the slender body geometry, we do not note its dependence in the statement of Lemma 2.2 or elsewhere where the Sobolev inequality is used. $\quad \square$

1.1.5 Pressure Estimate

In this section we prove Lemma 2.5 detailing the $\varepsilon$-independence of the pressure constant $c_P$. The proof is essentially the same as in the closed loop setting [23], but we recap the arguments here to make note of slight differences due to the free endpoint geometry.

Proof of Lemma 2.5

Consider the region $\mathcal {O}$ (1.15) of the slender body. Recall that by [10], Theorem III.3.1, given $P\in L^2(\mathcal {O})$, there exists $\varvec{B}\in H^1_0(\mathcal {O})$ satisfying

$$\begin{aligned} {\mathrm{div}}\, \varvec{B}=P \text { in }\mathcal {O}, \quad \left||\nabla \varvec{B} \right||_{L^2(\mathcal {O})} \leqq c_B\left||P \right||_{L^2(\mathcal {O})}, \end{aligned}$$

(A.47)

where the constant $c_B$ has a useful explicit representation provided that $\mathcal {O}$ can be written as a finite union of star-shaped domains.

We verify that the region $\mathcal {O}$ can indeed be written as a finite union of domains star-shaped with respect to balls of uniform radius. Note that by construction, since $\varepsilon a \leqq \varepsilon \leqq \frac{r_{\max }}{4}$ and the fiber is non-self-intersecting (1.2), the region $\mathcal {O}$ satisfies a uniform interior sphere condition with radius $r_{\max }/2$. In particular, we can consider $\mathcal {O}$ as the infinite union of balls of radius $r_{\max }/2$.

As in [23], we can follow the construction in Lemma 2, Chapter 1.1.9 of [21] to show that this uniform sphere condition implies that

$$\begin{aligned} \mathcal {O} = \bigcup _{k=1}^N \mathcal {O}_k, \end{aligned}$$

where each $\mathcal {O}_k$ is star-shaped with respect to balls of radius $r_{\max }/2$ and N depends only on $\kappa _{\max }$ and $c_\Gamma$. Then, according to [10], equation III.3.27, the constant $c_B$ in (A.47) satisfies

$$\begin{aligned} c_B \leqq c_0 \bigg (\frac{\delta ({\mathcal {O}})}{r_{\max }} \bigg )^3\bigg (1+ \frac{\delta ({\mathcal {O}})}{r_{\max }} \bigg ), \end{aligned}$$

where $\delta ({\mathcal {O}})$ is the diameter of ${\mathcal {O}}$ and $c_0$ depends on the diameters of each ${\mathcal {O}}_k$, which are bounded independent of $\varepsilon$. Using this form of $c_B$, the $\varepsilon$-independence of the constant $c_P$ follows by exactly the same construction as in the closed loop case (see [23], Appendix A.2.5), which we do not repeat here. $\quad \square$