# Lower regularity assumption for an Euler-Lagrange equation on the contact
line of the phase dependent Helfrich energy.

Sascha Eichmann
Mathematisch-Naturwissenschaftliche Fakultät,
Eberhard Karls Universität Tübingen,
Auf der Morgenstelle 10,
D-72076 Tübingen, Germany
E-mail<EMAIL_ADDRESS>
Phone: +49/7071/2976886

###### Abstract

We examine the phase dependent Helfrich energy and show an Euler-Lagrange
equation on the phase seperation line. This result has already been observed
by e.g. Jülicher-Lipowski and later Elliot-Stinner. Here we are able to lower
the regularity assumption for this result down to $C^{1,1}$ for the seperation
line.
In the proof we employ a carefully choosen test function utilising the signed
distance function.

Keywords: Canham-Helfrich energy, Euler-Lagrange equation, phase dependency
MSC.: 49K10, 49Q10, 35B30, 92B99

##  1 Introduction

The Canham-Helfrich energy (or short Helfrich energy) is defined for a two
dimensional smooth, orientable surface $S\subset\mathbb{R}^{3}$ with mean
curvature $H$ by

$W_{H_{0}}(S)=\int_{S}(H-H_{0})^{2}\,dA.$ (1.1)

Here $H_{0}\in\mathbb{R}$ is called spontaneous curvature. This energy is used
in e.g. modelling the shape of lipid bilayers, see e.g. [18] or red blood
cells, see [6]. In more layman’s terms lipid bilayers form the boundary of
depots in biological cells. These kind of depots are called vesicles. The
thickness of these boundaries is usually small compared to the whole vesicle,
hence modelling the shape of it by a two-dimensional surface is feasible (see
figure 1).

Figure 1: Sketch of a vesicle with homogeneous lipid bilayer.

The parameter $H_{0}$ represents an asymmetry of the lipid bilayer. I.e. it
has been observed, that it has a prefered curvature, which is dependent on the
material the bilayer consists of (see e.g. [27] and the references therein).

Since the bilayer itself does not consist of a homogeneous material, it may
happen that the spontaneous curvature differs throughout the bilayer. Then it
has been observed that two phases of the bilayer form, each having their own
prefered spontaneous curvature. These different phases usually do not mingle,
but rather form a sharp contact line (see e.g. [3] for some experimental
results or [22] for some numerical simulations). The Helfrich energy has been
adapted to model this kind of behaviour in [23] and [24].

Let us describe this model now (cf. figure 2).

Figure 2: Sketch of a surface with domain/phase seperation.

Let $S\subset\mathbb{R}^{3}$ be an oriented compact two dimensional surface
with scalar mean curvature $H$. Furthermore we describe the two different
lipid bilayers by a function $\phi:S\rightarrow\\{0,1\\}$, i.e. $x\in S$
belongs to the bilayer of type $i\in\\{0,1\\}$, iff $\phi(x)=i$. We call this
type phase or domain. The spontaneous curvature is now a function depending on
the phase, i.e. $H_{0}:\\{0,1\\}\rightarrow\mathbb{R}$. Since the phases have
been seperated, it is natural to assume, that their contact is minimal. Hence
the length of the contact line should be minimised as well (see e.g. [23] and
the references therein). For that let $J\subset S$ be the jump set of $\phi$
and let it be a one-dimensional curve (in the Hausdorff sense, see e.g. [13,
§2.1] and [33, §2] for additional informations). Then the domain dependend
Helfrich energy (or phase dependent Helfrich energy) is defined by

$W_{H_{0},\sigma}(S,\phi)=\int_{S}(H(x)-H_{0}(\phi(x)))^{2}\,dA+\sigma\mathcal{H}^{1}(J).$
(1.2)

We call the term $\sigma\mathcal{H}^{1}(J)$ line tension. Here
$\sigma\in\mathbb{R}$ is a parameter describing the contribution of the length
of $J$.

The aim of this article is to calculate an Euler-Lagrange equation on $J$. The
theorem for $S$ being in $C^{1}$ is as follows:

######  Theorem 1.1.

Let $S\subset R^{3}$ be an oriented surface with mean curvature $H$ and phase
seperation $\phi:S\rightarrow\\{0,1\\}$. We assume $S$ to be a $C^{1}$
surface. Furthermore $S|_{\\{\phi=i\\}}$ is supposed to be in $C^{4}$, such
that the derivatives up to third order can be extended to the contact line $J$
(i.e. the jump set of $\phi$) for $i=0,1$. Now we additionally assume
$(S,\phi)$ to be critical for $W_{H_{0},\sigma}$, i.e. for any smooth
vectorfield $V\in\mathbb{C}^{\infty}_{c}(\mathbb{R}^{3},\mathbb{R}^{3})$ and
their associated flow
$\Phi:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}$ we have

$0=\frac{d}{dt}W_{H_{0},\sigma}(\Phi(S,t),\phi(\Phi(\cdot,t))).$

Finally we assume $J$ to be of class $C^{1,1}$. We call $H^{i}$ the extension
of the scalar mean curvature of $S|_{\\{\phi=i\\}}$ to $J$. This extension
satisfies

$H^{0}(x)-H_{0}(0)=H^{1}(x)-H_{0}(1)$ (1.3)

for all $x\in J$.

Unfortunately there are some experimental results in [3], which indicate, that
$S$ is not necessarily $C^{1}$, but only $C^{0}$. In this case (1.3) becomes:

$H^{i}(x)-H_{0}(i)=0\mbox{ for all }x\in J\mbox{ and }i=0,1.$ (1.4)

A precise statements for $S$ being a graph is given in Theorem 1.3.

The equalities (1.3) and (1.4) have first been observed in [23] and [24] for
$S$ and $J$ being axially symmetric. Unfortunately as seen in the experiments
conducted in [3] and the numeric in [22] this symmetry cannot be expected in
general. Equations (1.3) and (1.4) have been extended to the more general case
of surfaces in [12, Problem 2.16] (see also [35, Eq. (4.12)] for a more
background information). There $J$ is usually assumed to be smooth. Here we
will lower this requirement to $C^{1,1}$.
The proof involves finding a suitable test function for the first variation.
In this argument the signed distance function (and its regularity) of $J$ will
be paramount. There we will follow an argument given by Foote in [15].

Existence of minimisers of the phase dependend Helfrich energy were first
shown in [7] for axially symmetric surfaces. Later in [5] this was extended to
the more general class of curvature varifolds. Unfortunately regularity for
this varifold minimiser and the corresponding contact line is still mainly
open. Our Theorem 1.1 shows that equality (1.3) (rsp. (1.4)) has to be
incorporated into such a proof, since it is a necessary condition for
regularity. How to do this and if (and how) one can show (1.3) for such a
minimizer, are open questions for future research.

A phase field approach of the phase dependend Helfrich energy for axially
symmetric surfaces and the corresponding analysis can be found in [20] and
[21]. A parametric finite element approach to the corresponding flow of
axially symmetric surfaces has been done recently in [16].

Further numerical studies for more general surfaces for the corresponding flow
have been conducted in e.g. [1] with a phase field apprach and [2] in a sharp
interface setting. In [2, §3] a weaker version of the flow was formulated,
which does not need as much regularity. Furthermore an analogue formulation
for flows of (1.3) has been given in [2, Eq. (1.4a)].

Without the phase dependency the available literatur is quite vast and
developing quite rapidly. A calculation of the Euler-Lagrange equation without
phase dependencies has already been done in [30]. There are also several
existence results, e.g. in the axially symmetric case [8], [9], for graphs
[10] and for compact immersions [29], [11]. Further some other modifications
to the Helfrich energy have been analysed, e.g. adding an elastic energy for
the boundary in [31]. Also some stability results in [26] and [4] are
available.

If we additionally consider $H_{0}=0$ our energy becomes the famous Willmore
energy. We refer to the survey articles [25], [17], [19], the seminal paper of
Willmore [34] and the proof of the Willmore conjecture [28] in this case.

###  1.1 Strategy of the proof

Since equation (1.3) is of local nature, we can assume without loss of
generality $S$ to be a graph of a function $u:\Omega\rightarrow\mathbb{R}$
with $\Omega\subset\mathbb{R}^{2}$ a smooth domain. Let us further refine the
phase seperation $\phi$ in this case: Let $A_{i}\subset\Omega$ be the maximal
open set on which $\phi(\operatorname{graph}u|_{A_{i}})=i$. For the proof let
$E$ be the contact line in the parameter region $\Omega$, i.e.
$E:=\Omega\setminus(A_{0}\cup A_{1})$ (or the projection of $E$ onto
$\Omega$). Then $E$ is a $C^{1,1}$ curve by our assumptions on $J$ in Theorem
1.1. In this case we call $A_{0},A_{1}$ a phase seperation of $\Omega$ with
contact line $E$. The phase dependend Helfrich energy for graphs is

$\displaystyle\begin{split}&W_{H_{0},\sigma}(u,(A_{0},A_{1}))\\\
:=&\int_{\Omega}(H_{u}(x)-H_{0}(\phi(u(x),x))^{2}\sqrt{1+|\nabla
u(x)|^{2}}\,dx+\sigma\mathcal{H}^{1}(\operatorname{graph}u|_{E}).\end{split}$
(1.5)

We reformulate Theorem 1.1 for graphs:

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{2}$ be an open bounded set. Furthermore let
$A_{0},A_{1}\subset\Omega$ be a phase seperation of $\Omega$ with contact line
$E$. Now let $u\in C^{1}(\Omega)$ and $u|_{A_{i}}\in C^{4}(A_{i})\cap
C^{3}(\overline{A_{i}})$ be a critical point of $u\mapsto
W_{H_{0},\sigma}(u,(A_{0},A_{1}))$, i.e. for all $\varphi\in
C^{\infty}_{0}(\Omega)$ we have

$0=\frac{d}{dt}W_{H_{0},\sigma}(u+t\varphi,(A_{0},A_{1}))|_{t=0}.$

Furthermore we assume $E$ to be of class $C^{1,1}$.
Then the derivatives of $u$ up to the order of $3$ can be extended
continuously to $E$. We call this extension $u_{A_{i}}$ and the corresponding
extension of the mean curvature of $\operatorname{graph}u$ is $H_{u_{A_{i}}}$.
Then for every $x\in E$ we have

$H_{u_{A_{0}}}(x)-H_{0}(0)=H_{u_{A_{1}}}(x)-H_{0}(1).$

The formulation of (1.4) as a Theorem for graphs is as follows:

######  Theorem 1.3.

Let $\Omega\subset\mathbb{R}^{2}$ be open and bounded. Furthermore let
$A_{0},A_{1}\subset\Omega$ be a phase seperation of $\Omega$ with contact line
$E$. Now let $u\in C^{0}(\Omega)$ and $u|_{A_{i}}\in C^{4}(A_{i})\cap
C^{3}(\overline{A_{i}})$ be a critical point of $u\mapsto
W_{H_{0},\sigma}(u,(A_{0},A_{1}))$, i.e. for all

$\displaystyle\varphi\in$ $\displaystyle\\{\psi:\Omega\rightarrow\mathbb{R}|\
\psi|_{A_{i}}\in C^{\infty}(A_{i})\mbox{ s.t. }D^{k}\psi$ $\displaystyle\mbox{
is uniformely continuous for }k=1,2,3\mbox{ and
}\operatorname{supp}\psi\subset\Omega\\}$

we have

$0=\frac{d}{dt}W_{H_{0},\sigma}(u+t\varphi,(A_{0},A_{1}))|_{t=0}.$

Furthermore we assume $E$ to be of class $C^{1,1}$.
Then the derivatives of $u$ up to the order of $3$ can be extended
continuously to $E$. We call this extension $u_{A_{i}}$ and the corresponding
extension of the mean curvature of $\operatorname{graph}u$ is $H_{u_{A_{i}}}$.
Then for every $x\in E$ and $i=0,1$ we have

$H_{u_{A_{i}}}(x)-H_{0}(i)=0.$

The proof of Theorem 1.2 requires two ingredients: First we calculate an
Euler-Lagrange equation in section 2, which will give us a condition on the
boundary with test functions in $C^{1}$. The other step is showing the
existence of a suitable test function itself, which will boil down to showing
the signed distance function for $E$ is $C^{1}$ close to and on $E$. We will
demonstrate this in section 3. Section 4 is dedicated to bringing all
arguments together to finish the proof. There we also explain the changes
needed to obtain Theorem 1.3

######  Remark 1.4.

Since $E$ is of Class $C^{1,1}$ the corresponding phase seperation
$A_{0},A_{1}$ of $\Omega$ satisfies

$\mathcal{L}^{2}(\Omega\setminus(A_{0}\cup A_{1}))=0\mbox{ and }A_{0}\cap
A_{1}=\emptyset.$

This will allow us to seperate the area integral in (1.5) in two distinct
parts.

##  2 A necessary condition for criticality

Under the assumptions of Theorem 1.2, we will show the following Euler-
Lagrange equation on $E$ (see e.g. [12]) for a function $\varphi\in
C^{1}_{0}(\Omega)$, which is zero on $E$.

$\displaystyle\begin{split}&0=\int_{E}\left(\langle\nabla\varphi,n_{E}\rangle-\sum_{i,j=1}^{2}\frac{(n_{E})_{i}\partial_{j}\varphi\partial_{i}u\partial_{j}u}{1+|\nabla
u|^{2}}\right)\\\
&\cdot\left(H_{u_{A_{0}}}-H_{0}(0)-(H_{u_{A_{1}}}-H_{0}(1))\right)\,d\mathcal{H}^{1}.\end{split}$
(2.1)

Here $n_{E}$ is a normal of $E$. In case the $A_{i}$ are only Caccioppoli it
is the corresponding measure theoretic normal (see [13, p. 169]). By Remark
1.4 we have $\mathcal{L}^{2}(\Omega\setminus(A_{0}\cup A_{1}))=0$ and
$A_{0}\cap A_{1}=\emptyset$. Hence

$\displaystyle W_{H_{0},\sigma}(u,(A_{0},A_{1}))=$
$\displaystyle\sigma\mathcal{H}^{1}(\operatorname{graph}u|_{E})$
$\displaystyle+\sum_{i=1}^{2}\int_{A_{i}}\left(H_{u}-H_{0}(i)\right)^{2}\sqrt{1+|\nabla
u|^{2}}\,dx$

and therefore we only need to calculate the derivative of the bulk term on
$A_{i}$. First we assume $\varphi$ to be smooth. Afterwards we get our result
by approximation. Let us start with the mean curvature of the variation (see
e.g. [10, Eq. (8)] for a formula for the mean curvature of graphs):

$\displaystyle H_{u+t\varphi}=$
$\displaystyle\sum_{i=1}^{2}\partial_{i}\left(\frac{\partial_{i}(u+t\varphi)}{\sqrt{1+\sum_{j}(\partial_{j}u+t\partial_{j}\varphi)^{2}}}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\frac{\partial_{i}^{2}(u+t\varphi)}{\sqrt{1+|\nabla(u+t\varphi)|^{2}}}$
$\displaystyle-\sum_{i=1}^{2}\frac{\partial_{i}(u+t\varphi)\sum_{j=1}^{2}(\partial_{j}u+t\partial_{j}\varphi)(\partial_{i}\partial_{j}u+t\partial_{i}\partial_{j}\varphi)}{\left(1+\sum_{j=1}^{2}(\partial_{j}u+t\partial_{j}\varphi)^{2}\right)^{\frac{3}{2}}}$

Hence

$\displaystyle\frac{d}{dt}H_{u+t\varphi}|_{t=0}=$
$\displaystyle\sum_{i}\frac{\partial_{i}^{2}\varphi}{\sqrt{1+|\nabla
u|^{2}}}-\sum_{i,j}\frac{\partial_{i}^{2}u\partial_{j}\varphi\partial_{j}u}{(1+|\nabla
u|^{2})^{\frac{3}{2}}}$
$\displaystyle-\sum_{i,j}\frac{\partial_{i}\varphi\partial_{j}u\partial_{i}\partial_{j}u+\partial_{i}u\partial_{j}\varphi\partial_{i}\partial_{j}u+\partial_{i}u\partial_{j}u\partial_{i}\partial_{j}\varphi}{(1+|\nabla
u|^{2})^{\frac{3}{2}}}$
$\displaystyle+\sum_{i,j,k}\frac{3\partial_{i}u\partial_{j}u\partial_{i}\partial_{j}u\partial_{k}\varphi\partial_{k}u}{(1+|\nabla
u|^{2})^{\frac{5}{2}}}.$

Since we assume boundedness for $u$ and the derivatives, the dominated
convergence theorem yields

$\displaystyle\frac{d}{dt}\int_{A_{i}}|H_{u+t\varphi}-H_{0}(i)|^{2}\sqrt{1+|\nabla(u+t\varphi)|^{2}}\,dx|_{t=0}$
$\displaystyle=$
$\displaystyle\int_{A_{i}}2(H_{u}-H_{0}(i))\frac{d}{dt}(H_{u}+t\varphi)|_{t=0}\sqrt{1+|\nabla
u|^{2}}$ $\displaystyle+|H_{u}-H_{0}(i)|^{2}\frac{\langle\nabla
u,\nabla\varphi\rangle}{\sqrt{1+|\nabla u|^{2}}}\,dx$

Now using partial integration twice, we obtain as the bulk term the usual
Helfrich equation, which is zero by only utilising test functions from
$C^{\infty}_{0}(A_{i})$, and the rest of the boundary terms. Almost all of
these boundary terms are zero, because $\varphi=0$ on $E$. Only the ones
remain, where originally the second derivatives of $\varphi$ were present:

$\displaystyle\frac{d}{dt}\int_{A_{i}}|H_{u+t\varphi}-H_{0}(i)|^{2}\sqrt{1+|\nabla(u+t\varphi)|^{2}}\,dx|_{t=0}$
$\displaystyle=$ $\displaystyle
2\int_{E}\left(\langle\nabla\varphi,n_{E}\rangle-\sum_{i,j}\frac{\partial_{j}\varphi\partial_{i}u\partial_{j}u(n_{E})_{i}}{1+|\nabla
u|^{2}}\right)(H_{u_{A_{i}}}-H_{0}(i))\,d\mathcal{H}^{1}\lfloor E.$

Since we have this term twice with opposite signs (we use the same normal
$n_{E}$ for both $A_{0}$ and $A_{1}$ in the partial integration), we obtain
(2.1).

##  3 The signed distance function

In this section let $c:[a,b]\rightarrow\mathbb{R}^{2}$ be an injective,
regular curve with $c\in C^{1,1}$ and being parametrised by arclength. We will
show, that there exists a neighbourhood of $c$, such that the signed distance
function is in $C^{1}$. We will follow the presentation of [15]. First we
start by showing that the projection is well defined:

######  Lemma 3.1.

For all $x\in c((a,b))$ exists a $\delta>0$ such that the Projection

$P:B_{\frac{\delta}{2}}(x)\rightarrow c((a,b))\cap B_{\delta}(x)$

is well defined, i.e. for all $y\in B_{\frac{\delta}{2}}(x)$ the nearest point
on the curve $c$ is unique.

###### Proof.

We proceed by contradiction and assume we find a sequence $y_{n}\in
B_{\frac{1}{2n}}(x)$ and $t_{n}^{1}<t_{n}^{2}\in(a,b)$, such that

$\operatorname{dist}(y_{n},c((a,b))\cap
B_{\frac{1}{n}}(x))=|c(t_{n}^{1})-y_{n}|=|c(t^{2}_{n})-y_{n}|$

For $i=1,2$ we have

$\displaystyle\begin{split}0=&\partial_{t}|c(t)-y_{n}|^{2}|_{t=t_{n}^{i}}\\\
=&2\langle\dot{c}(t_{n}^{i}),c(t_{n}^{i})\rangle-2\langle\dot{c}(t_{n}^{i}),y_{n}\rangle\end{split}$
(3.1)

and therefore $\dot{c}(t_{n}^{i})$ is orthogonal to $c(t_{n}^{i})-y_{n}$. Let
$\nu:(a,b)\rightarrow\mathbb{S}^{1}$ be a choosen lipschitz continuous unit
normal of $c$ with Lipschitz constant $L>0$. Then we find $\lambda_{n}^{i}$,
satisfying

$c(t_{n}^{i})-y_{n}=\lambda_{n}^{i}\nu(t_{n}^{i})$

and therefore

$\lambda_{n}^{i}=\pm\operatorname{dist}(y_{n},c((a,b))\cap
B_{\frac{1}{n}}(x)).$

If we choose $n$ big enough, the curve $c$ seperates $B_{\frac{1}{n}}(x)$ into
two connected regions (see Figure 3), since $c\in C^{1,1}$.

Figure 3: Sketch of nonuniqueness situation with normals.

Hence we can assume without loss of generality

$\lambda_{n}^{i}=\operatorname{dist}(y_{n},c((a,b))\cap B_{\frac{1}{n}}(x)).$

Hence $0<\lambda_{n}^{i}\leq\frac{1}{n}$, independent of $i$ (i.e.
$\lambda_{n}=\lambda_{n}^{1}=\lambda_{n}^{2}$) and furthermore

$c(t_{n}^{i})-y_{n}=\lambda_{n}\nu(t_{n}^{i}).$

Subtracting these two equations yields

$c(t_{n}^{1})-c(t_{n}^{2})=\lambda_{n}\nu(t_{n}^{1})-\lambda_{n}\nu(t_{n}^{2}).$

By the mean value theorem we find
$\xi_{n,1},\xi_{n,2}\in(t_{n}^{1},t_{n}^{2})$ with

$\left|\left(\begin{array}[]{c}\dot{c}^{1}(\xi_{n,1})\\\
\dot{c}^{2}(\xi_{n,2})\end{array}\right)\right||t_{n}^{1}-t_{n}^{2}|=\lambda_{n}|\nu(t_{n}^{1})-\nu(t_{n}^{2})|\leq
L\lambda_{n}|t_{n}^{1}-t_{n}^{2}|.$

Since $c$ is parametrised by arclength and $|t_{n}^{1}-t_{n}^{2}|\rightarrow
0$, we have for $n\rightarrow\infty$

$\left|\left(\begin{array}[]{c}\dot{c}^{1}(\xi_{n,1})\\\
\dot{c}^{2}(\xi_{n,2})\end{array}\right)\right|\rightarrow 1.$

Hence there is a constant $\varepsilon>0$ and
$n_{0}=n_{0}(\varepsilon)\in\mathbb{N}$ such that for every $n\geq n_{0}$ we
have

$\left|\left(\begin{array}[]{c}\dot{c}^{1}(\xi_{n,1})\\\
\dot{c}^{2}(\xi_{n,2})\end{array}\right)\right|\geq 1-\varepsilon.$

All in all we get

$1-\varepsilon\leq\frac{1}{n}L$

for every $n\geq n_{0}$, which is a contradiction. ∎

######  Remark 3.2.

Sets on which the projection is locally unique are called sets with positive
reach. This notion goes back to [14] and has been characterised very well
recently in [32] (see also the references therein). In our case the arguments
are a lot simpler, hence we provided them for the sake of completeness in
Lemma 3.1.
The characterisation in [32] is essentially a $C^{1,1}$ regularity assumption
for the set in question. In this sense our method is optimal, though we do not
know, whether the regularity assumption of $E$ in our theorems is optimal as
well.

The next lemma shows, that the projection $P$ is continuous. The proof given
here is by Federer [14, 4.8(4)]

######  Lemma 3.3 (see [14]).

Let $x\in c((a,b))$ and $\delta>0$ as in Lemma 3.1. Then the projection $P$ on
the curve $c$

$P:B_{\frac{\delta}{2}}(x)\rightarrow c((a,b))\cap B_{\delta}(x)$

is continuous.

###### Proof.

Let us assume the opposite. Thereby we find a sequence $y_{i}\in
B_{\frac{\delta}{2}}(x)$ with $y_{i}\rightarrow y\in B_{\frac{\delta}{2}}(x)$
and an $\varepsilon>0$ such that

$|P(y)-P(y_{i})|\geq\varepsilon$

for every $i\in\mathbb{N}$. Since $P(y_{i})\in B_{\delta}(x)$ the sequence
$P(y_{i})$ is bounded. After choosing a subsequence and relabeling, we get an
$a\in\overline{B_{\delta}(x)}\cap c([a,b])$ with $P(y_{i})\rightarrow a$.
Furthermore the distance function is Lipschitz and therefore we obtain

$\frac{\delta}{2}>\operatorname{dist}(y,c((a,b)))=\lim_{i\rightarrow\infty}\operatorname{dist}(y_{i},c((a,b)))=\lim_{i\rightarrow\infty}|P(y_{i})-y_{i}|=|a-y|.$

Hence $a\in B_{\delta}(x)$. By Lemma 3.1 the projection is unique and we get

$a=P(y).$

This yields by the convergence of $P(y_{i})\rightarrow a$

$|P(y)-P(y_{i})|=|a-P(y_{i})|\rightarrow 0$

and therefore the desired contradiction. ∎

Let us now define a signed distance function:

######  Definition 3.4.

Let $x\in c((a,b))$ and $\delta>0$ as in the preceeding lemmas. Then for every
$y\in B_{\frac{\delta}{2}}(x)$ the projection is unique and there exists a
unique $\lambda_{y}\in\mathbb{R}$ (cf. (3.1)), such that

$P(y)-y=\lambda_{y}\nu(P(y)).$

Here $\nu$ is an a priori choosen Lipschitz continuous unit normal of $c$.
Then we define a signed distance function
$d_{\pm}:B_{\frac{\delta}{2}}(x)\rightarrow\mathbb{R}$ by

$d_{\pm}(y)=\left\\{\begin{array}[]{cc}\operatorname{dist}(y,c((a,b))),&\mbox{
if }\lambda_{y}\geq 0\\\ -\operatorname{dist}(y,c((a,b))),&\mbox{ if
}\lambda_{y}<0.\end{array}\right.$

######  Remark 3.5.

The sign of $d_{\pm}$ switches, if the other unit normal is choosen.

Now we can show the central result of this section. For the readers benefit we
provide the proof already given in [15, Thm. 2] here as well.

######  Theorem 3.6 (see [15] Thm. 2).

Let $\nu$ be a lipschitz continuous unit normal of $c$. Then for every $x\in
c((a,b))$ there exists a $\delta>0$, such that the signed distance function
choosen with respect to $\nu$ satisfies $d_{\pm}\in
C^{1}(B_{\frac{\delta}{2}}(x))$. Furthermore the derivative satisfies

$\nabla d_{\pm}(y)=\left\\{\begin{array}[]{cc}\frac{y-P(y)}{|y-P(y)|},&\mbox{
if }\lambda_{y}>0\\\ -\frac{y-P(y)}{|y-P(y)|},&\mbox{ if }\lambda_{y}<0\\\
\nu(y),&\mbox{ if }y\in c((a,b)).\end{array}\right.$

Here $\lambda_{y}$ is defined as in Definition 3.4.

###### Proof.

We show the theorem first for $y\in B_{\frac{\delta}{2}}(x)\setminus
c((a,b))$. Let $d:=(d_{\pm})^{2}$ and now we proceed by contradiction and
assume there exists a $v\in\mathbb{R}^{2}$ such that

$\liminf_{t\searrow 0}\frac{d(y+tv)-d(y)}{t}<2\langle y-P(y),v\rangle$ (3.2)

or

$\limsup_{t\searrow 0}\frac{d(y+tv)-d(y)}{t}>2\langle y-P(y),v\rangle.$ (3.3)

Let us work through case (3.2) first: Hence we find an $\varepsilon>0$, a
$t>0$ arbitrarily small with

$d(y+tv)<d(y)+2\langle y-P(y),tv\rangle-t\varepsilon$

Hence

$\displaystyle|y-P(y+tv)|^{2}=|y+tv-P(y+tv)-tv|^{2}$ $\displaystyle=$
$\displaystyle d(y+tv)-2\langle y+tv-P(y+tv),tv\rangle+t^{2}|v|^{2}$
$\displaystyle<$ $\displaystyle d(y)+2\langle
P(y+tv)-P(y),tv\rangle-t^{2}|v|^{2}-t\varepsilon$

Since $P$ is continuous we have $2\langle P(y+tv)-P(y),tv\rangle=o(|t|)$ and
hence this term can be absorbed for small $t$ and we get

$|y-P(y+tv)|^{2}<d(y)-t^{2}|v|^{2}-t\frac{\varepsilon}{2}<d(y)=|y-P(y)|^{2}$

and therefore $P(y+tv)$ has a strictly smaller distance to $y$ than $P(y)$,
which is a contradiction.
Now we assume (3.3): As in the preceeding case, there is an $\varepsilon>0$
and $t>0$ arbitrarily small, such that

$d(y+tv)>d(y)+2\langle y-P(y),tv\rangle+t\varepsilon.$

Hence

$\displaystyle|y+tv-P(y)|^{2}=$
$\displaystyle|y-P(y)|^{2}+t^{2}|v|^{2}+2\langle y-P(y),tv\rangle$
$\displaystyle<$ $\displaystyle d(y+tv)-t\varepsilon+t^{2}|v|^{2}.$

By choosing $t$ small enough, we obtain

$|y+tv-P(y)|^{2}<d(y+tv).$

Therefore $P(y)$ is a strictly better projection than $P(y+tv)$, i.e. a
contradiction. Hence for all $y\in B_{\frac{\delta}{2}}(x)\setminus c((a,b))$
we have

$\nabla d(y)=2(y-P(y)).$

Since $d_{\pm}(y)=\operatorname{sign}(\lambda_{y})\sqrt{d(y)}$ we obtain by
chain rule for these $y$:

$\nabla d_{\pm}(y)=\operatorname{sign}(\lambda_{y})\frac{\nabla
d(y)}{2\sqrt{d(y)}}=\lambda_{y}\frac{y-P(y)}{|y-P(y)|}.$

Let us now examine $y\in c((a,b))$: Let $y_{i}\in
B_{\frac{\delta}{2}}(x)\setminus c((a,b))$ with $y_{i}\rightarrow y$. Then for
each $y_{i}$ exists a neighbourhood, such that $\lambda_{(\cdot)}$ does not
change sign. Then by $|\lambda_{y_{i}}|=|y_{i}-P(y_{i})|$ we get

$\displaystyle\nabla d_{\pm}(y_{i})=$
$\displaystyle\operatorname{sign}(\lambda_{y_{i}})\frac{y_{i}-P(y_{i})}{|y_{i}-P(y_{i})|}=\operatorname{sign}(\lambda_{y_{i}})\frac{\lambda_{y_{i}}}{|y_{i}-P(y_{i})|}\nu(P(y_{i}))$
$\displaystyle=$
$\displaystyle\operatorname{sign}(\lambda_{y_{i}})^{2}\nu(P(y_{i}))\rightarrow\nu(y),$

since $P$ is continuous and $P(y)=y$. Hence $d_{\pm}$ is continuously
differentiable. ∎

##  4 Finishing the proofs

In the first part of this section we show Theorem 1.2. After that we highlight
the changes we need to make to obtain a proof of Theorem 1.3.

###### Proof.

Let $\varphi\in C^{\infty}_{0}(\Omega)$ be arbitrary. Since it has compact
support and $E$ is relatively compact in $\Omega$ (because $E$ is a $C^{1,1}$
curve) the intersection $\operatorname{spt}\varphi\cap E$ is compact as well.
Using a partition of zero and Theorem 3.6 we find $\sigma_{i}\in
C^{\infty}_{0}(B_{\delta_{i}}(x_{i}))$ ($i=1,\ldots,N$) with
$x_{i}\in\operatorname{spt}\varphi\cap E$, $\delta_{i}>0$, such that the
signed distance function $d_{\pm}$ with respect to $E$ and the choosen normal
$n_{E}$ satisfies $d_{\pm}\in C^{1}(B_{\delta_{i}}(x_{i}))$. Furthermore
$\sum_{i}\sigma_{i}=1$. Then we can define

$\Phi:=\varphi\sum_{i}d_{\pm}\sigma_{i}\in C^{1}_{0}(\Omega),$ (4.1)

where we implicitly continue $d_{\pm}$ by zero to the whole of $\Omega$.
Furthermore $\Phi$ can be plugged into (2.1), because $d_{\pm}$ is zero on
$E$. We also obtain for any $y\in E$:

$\nabla\Phi(y)=\varphi(y)n_{E}(y)\sum_{i}\sigma_{i}(y)=\varphi(y)n_{E}(y).$

Hence (2.1) yields

$0=\int_{E}\varphi\left(1-\sum_{ij}\frac{(n_{E})_{i}(n_{E})_{j}\partial_{i}u\partial_{j}u}{1+|\nabla
u|^{2}}\right)(H_{u_{A_{0}}}-H_{0}(0)-(H_{u_{A_{1}}}-H_{0}(1))\,d\mathcal{H}^{1}$

Let us now analyse
$1-\sum_{ij}\frac{(n_{E})_{i}(n_{E})_{j}\partial_{i}u\partial_{j}u}{1+|\nabla
u|^{2}}$:

$\displaystyle\sum_{ij}(n_{E})_{i}(n_{E})_{j}\partial_{i}u\partial_{j}u$
$\displaystyle=$
$\displaystyle((n_{E})_{1})^{2}(\partial_{1}u)^{2}+2(n_{E})_{1}(n_{E})_{2}\partial_{1}u\partial_{2}u+((n_{E})_{2})^{2}(\partial_{2}u)^{2}$
$\displaystyle=$
$\displaystyle((n_{E})_{1}\partial_{1}u+(n_{E})_{2}\partial_{2}u)^{2}=(\langle
n_{E},\nabla u\rangle)^{2}\leq|n_{E}|^{2}|\nabla u|^{2}=|\nabla u|^{2}$

and hence

$1-\sum_{ij}\frac{(n_{E})_{i}(n_{E})_{j}\partial_{i}u\partial u_{j}}{1+|\nabla
u|^{2}}>0.$

Since $\varphi$ itself is arbitrary, the fundamental lemma of variational
calculus yields

$H_{u_{A_{0}}}-H_{0}(0)-(H_{u_{A_{1}}}-H_{0}(1))=0\quad\mathcal{H}^{1}\lfloor
E\mbox{-a.e.}$

Since the mean curvatures on $E$ are continuous continuations, we obtain the
desired conclusion. ∎

######  Remark 4.1.

Adding $\lambda_{1}\operatorname{area}(S)+\lambda_{2}\operatorname{vol}(S)$ to
the phase dependend Helfrich energy (1.2) to account for additional conditions
like prescribing the area and/or enclosed volume of $S$, does not change the
result of Theorem 1.1. Since these terms do not produce terms with derivatives
of the test function in the first variation, the proof of Theorem 1.1 remains
valid.

Now we explain the changes to the proof to obtain Theorem 1.3

###### Proof.

Since $u\in C^{0}(\Omega)$ and $C^{4}(A_{i})\cap C^{3}(\overline{A_{i}})$, we
can choose from a greater set of testfunctions $\psi$. We employ a function,
which is zero on $A_{1}$ and $E$, i.e. we take $\psi\in C^{0}_{0}(\Omega)$
with $\psi\in C^{1}(A_{i})$, such that the derivative can be extended to $E$.
Then the same calculations to obtain (2.1) yield

$\int_{E}\left(\langle\nabla\psi,n_{E}\rangle-\sum_{i,j=1}^{2}\frac{(n_{E})_{i}\partial_{j}\varphi\partial_{i}u_{0}\partial_{j}u_{0}}{1+|\nabla
u_{0}|^{2}}\right)\left(H_{u_{A_{0}}}-H_{0}(0)\right)\,d\mathcal{H}^{1}=0.$
(4.2)

Here $\nabla u_{0}$ refers to the continuation of $\nabla u|_{A_{0}}$ to $E$.
Instead of using the signed distance function to construct a suitable $\psi$,
we employ the following function instead

$d_{0}(y):=\left\\{\begin{array}[]{cc}\operatorname{dist}(y,E),&y\in A_{0}\cup
E\\\ 0,&\mbox{else.}\end{array}\right.$

This $d_{0}$ is continuous and the same techniques employed in Lemma 3.6 show
$d_{0}$ to be $C^{1}$ close to $E$ with continued derivative

$\nabla d_{0}(y)=\pm n_{E}(y)\mbox{ for }y\in E.$

The sign depends on whether $n_{E}$ points inward or outward of $A_{0}$. As in
(4.1) we can now construct a suitable test function $\psi$ with an arbitrary
$\varphi\in C^{\infty}_{0}(\Omega)$. This we can plug in (4.2) and obtain

$0=\int_{E}\varphi\left(\pm
1\mp\sum_{ij}\frac{(n_{E})_{i}(n_{E})_{j}\partial_{i}u_{0}\partial_{j}u_{0}}{1+|\nabla
u_{0}|^{2}}\right)(H_{u_{A_{0}}}-H_{0}(0))\,d\mathcal{H}^{1}.$

As in the proof of Theorem 1.2 the prefactor is never zero and then the
fundamental lemma of variational calculus yields (1.4). The same arguments can
be employed to show the result for the continuation of $u|_{A_{1}}$ on $E$. ∎

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