Lower semicontinuity for the Helfrich problem

We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^3$$\end{document}.


Introduction
This article deals with minimising the Helfrich energy for closed oriented smooth connected two-dimensional immersions f ∶ Σ → ℝ 3 , which is defined as Here H is the scalar mean curvature, i.e. the sum of the principal curvatures with respect to a choosen unit normal of f. We call g the area measure on Σ induced by f and the euclidean metric of ℝ 3 . Furthermore, H 0 ∈ ℝ is called spontaneous curvature. This energy was introduced by Helfrich [20] and Canham [3] to model the shape of blood cells. Hence it is The author thanks Prof. Reiner Schätzle for discussing the Helfrich energy and providing insight into geometric measure theory.
called the Canham-Helfrich or short Helfrich energy. We recover the Willmore energy by setting H 0 = 0 and multiplying by 1 4 , i.e. The Willmore energy goes back to Thomsen [40]. He denoted critical points of the Willmore energy as conformal minimal surfaces. Willmore later revived the mathematical discussion in [41]. Please note, that for the Willmore energy an orientation is not needed, contrary to the Helfrich energy. Hence we need a fixated normal. We assume Σ to have a continuous orientation , given as a 2-form. Then we set f ∶= * (df ( )) ∈ B 1 (0) ⊂ ℝ 3 as the unit normal of f. Here * denotes the Hodge- * -Operator. Furthermore, we like to prescribe the area and enclosed volume of f. Therefore we set Please note that, if f is an embedding and f the outer normal, Vol(f) would be the volume of the set enclosed by f. In the general case, Vol(f) may become negativ dependend on the orientation. Minimisers of such a problem satisfy the following Euler-Lagrange equation (see, e.g. [31,Eq. (31)]) Here, A ∈ ℝ and V ∈ ℝ correspond to Lagrange multipliers for the prescribed area and enclosed volume, respectively. This differential equation is highly nonlinear since the Laplace-Beltrami Δ f depends on the unknown immersion f. Furthermore, it is of fourth order, hence standard techniques like the maximum principle are not applicable. Nevertheless a lot of important results concerning such problems have been achieved: Existence of closed Willmore surfaces of arbitrary genus has been shown in the papers [39] and [2]. If the ambient space becomes a general 3-manifold, the problem has been examined in, e.g. [30].
Prescribing additional conditions and showing existence has also been very successful, i.e. the isoperimetric ratio in [23] and [36], the area in [25], boundary conditions in [6,32,35] and [9]. Finally, the Willmore conjecture was shown to be true in [28]. Further research and references for Willmore problems are summarized in the surveys [21] rsp. [19]. The class of axisymmetric surfaces is especially important for modelling purposes; see, e.g. [11]. Also some existence results can be more readily achieved in this class, see, e.g. [7,12] or [14] and solutions to (1.4) can be analysed in greater depths, e.g. the behaviour of singularities in [16].
Furthermore, the addition of the orientation complicates and changes the situation in the Helfrich setting. For example, the class of invariances is notably smaller (see, e.g. [8]) and lower semi-continuity is in general false under varifold convergence; see [18, p. 550, Remark (ii)] for a counterexample. Nevertheless, some progress has been made: Existence of Helfrich surfaces with prescribed surface area near the sphere has been achieved in [24] by examining the corresponding L 2 -flow. This result has been extendend to a general existence result for spherical Helfrich immersions with prescribed area and enclosed volume in [29] by variational parametric methods. The axisymmetric case has been handled independently in [5] and [4]. A more general approach was used in [10] by working with .
Gauss-graphs. There lower semicontinuity was shown, but the limit has to be in C 2 , which is a priori not clear. In general, this is a hard problem, since the Helfrich energy for oriented varifolds lacks a variational characterisation (cf. (2.3) and cf. [13, p. 3]).
In this paper, we will show a lower-semicontinuity estimate for minimising sequences with an arbitrary but fixed topology by an ambient approach, i.e. geometric measure theory. In the case of spherical topology this has been achieved with a parametric approach in [29,Thm. 3.3]. The case of embeddings still remains open, since we do not have a Li-Yau type inequality. This prevents our arguments to be adaptable to this situation as explained in Remark 5.2.
For our minimising procedure, we introduce the following set for a given two-dimensional smooth connected manifold Σ without boundary, which fixes the topology.
for a positive parameter Area 0 > 0 and a nontrivial real parameter Vol 0 ∈ ℝ ⧵ {0} . Furthermore let us call Now we can state our main result in case M Area 0 ,Vol 0 ≠ ∅ (For unknown terminology please consult Sect. 2): k be the sequence of oriented integral 2 varifolds corresponding to f k by (2.4). Then there exists an oriented integral 2 varifold V 0 on ℝ 3 and a subsequence V 0 k j , such that More importantly, this subsequence enjoys a lower-semicontinuity estimate: Furthermore, there exist at most finitely many bad points, such that V 0 is locally a union of C 1, ∩ W 2,2 graphs outside these bad points. For a precise statement of the graphical decomposition please refer to Lemma 5.1.
The proof works in two major steps. The first is to show some preliminary regularity of V 0 (see Lemma 5.1). After this the lower-semicontinuity estimate follows precisely as in [13,Lemma 4.1], hence we will not include this step here.
The initial regularity is shown by comparing the Helfrich energy of the minimising sequence to a biharmonic replacement. This idea was first used by Simon in [39]. Since the biharmonic replacement does not have the same area and/or enclosed volume, we will correct these parameters by an idea of Schygulla [36]. We generalise Schygulla's techniques to the case of immersions and to two prescribed quantities. This idea essentially is adjusting these quantities of the biharmonic replacement outside of the replacement region by a two parameter diffeomorphism.
The paper is build up as follows: Sect. 2 is concerned with compactness in the class of oriented varifolds. Here the Helfrich energy, the area and the enclosed volume are formulated for these varifolds. Next in Sect. 3 we examine our definition of enclosed volume for oriented varifolds in greater detail and for example calculate an Euler-Lagrange equation. Afterwards we construct the aforementioned two parameter diffeomorphism and analyse these in 4. In Sect. 5 we finally show the initial C 1, -regularity, from which the lower-semicontinuity estimate follows. In Appendix A, we collect some usefull results for our reasoning.

Compactness
In this chapter, we recall the necessary results and objects to obtain compactness for a minimising sequence in measure theoretic terms: Since the Helfrich energy depends on the orientation, we will work with oriented varifolds in our variational framework. Oriented varifolds were introduced by Hutchinson in [22,Sect. 3]. We recall the necessary definitions here (see also [13,Appendix B]). First is called the oriented Grassmannian manifold of two-dimensional oriented linear subspaces in ℝ 3 . Since we need to connect orientations with normals, we also need the Hodge star operator * ∶ G 0 (2, 3) → B 1 (0) . In our setting, * becomes just the cross product, i.e. * ( 1 ∧ 2 ) = 1 × 2 . An oriented integral 2-varifold on an open set Ω ⊂ ℝ 3 is given by a countable 2-rectifiable set M ⊂ O , H 2 -measurable densities + , − ∶ M → ℕ 0 and an orientation ∶ M → G 0 (2, 3) , such that * ( (x)) ⟂ T x M for H 2 a.e. x ∈ M . Then the corresponding oriented varifold is a Radon measure on Ω × G 0 (2, 3) given for Φ ∈ C 0 0 (Ω × G 0 (2, 3)) by Furthermore, we need to define the Helfrich energy for such an oriented integral varifold. Hence we need to make sense of a mean curvature. For this, let 0 ∶ ℝ 3 × G 0 (2, 3) be given by 0 (x, ) = x . Then the mass of V 0 is defined as which is also an integral varifold in the sense of [38,Sect. 15]. The first variation of V 0 is defined as the first variation of V 0 , i.e. V 0 ∶= V 0 Thus we define the mean curvature vector of V 0 to be the mean curvature vector of V 0 (cf. [38,Sect. 16]). We say V 0 has a mean curvature vector This means that V 0 does not have a generalized boundary in the sense of [38,Sect. 39]. The integral 2 current associated with V 0 is given by We choose the same notations for currents as in [38,Chapter 6].
For an integral oriented varifolds V 0 = V 0 [M, + , − , ] the current satisfies and is therefore integral as well. We will call [|V 0 |] the boundary in the sense of currents of V 0 . Here is the boundary operator for currents (see, e.g. [38,Eq. (26.3)]). Convergence of oriented varifolds is defined as weak convergence of the corresponding Radon measures. That is, we say a sequence of oriented integral varifolds V 0 k on Ω ⊂ ℝ 3 converges weakly to an oriented integral varifold Then [22,Theorem 3.1] gives us, that the following set is sequentially compact with respect to oriented varifold convergence: Here, M Ω � (⋅) denotes the mass in the sense of currents. The Helfrich energy of V 0 is (see also [13, Eq. (2.1)]) Furthermore, we need to define the enclosed volume and the area of such oriented varifolds: Now let Σ be a smooth oriented two-dimensional manifold and f ∶ Σ → ℝ 3 a smooth immersion. To employ the compactness criterion (2.2) we need to define a corresponding oriented integral varifold (see [13,Sect. 2]): be an H 2 measurable orientation. Then the corresponding oriented integral varifold is To define the densities, let us denote the chosen continuous orientation of T x Σ by (x) . Here (2.5) sign − (df ( (x))⟋( f (y)).
Analogously, sign − (df ( (x))⟋( f (y)) = 1 , if f (y) is the opposite orientation of df ( (x)) . Please note that these densities are only well defined H 2 ⌊f (Σ) almost everywhere, which is enough to obtain a well defined oriented varifold. Let f k ∈ M Area 0 ,Vol 0 be a minimising sequence as in Theorem 1.1 with orientation Furthermore let A k be the second fundamental form of f k . By (2.6) and since the topology of f k is fixed, we get for some constant C > 0 independent of k (see also [35,Eq. (1.1)]). Hence, by [22,Thm. 5.3.2] ∶= V 0 has a weak second fundamental form A ∈ L 2 ( ) . By possibly extracting another subsequence, we obtain Without loss of generality we also have M = spt( ).
Before we can proceed, we need to ensure that the limit V 0 satisfies Area(V 0 ) = A 0 and Vol(V 0 ) = Vol 0 . For the first one, we use the following lemma Now let x ∈ spt( ) . For an arbitrary > 0 we obtain by, e.g. [27,Prop. 4.26] and the defintion of the support of a Radon measure Hence spt( k ) ∩ B (x) ≠ � for k big enough. Therefore, we can find x k ∈ spt( k ) such that x k → x . By (2.11), we finally obtain and the lemma is proven. ◻ Furthermore, (2.11) and (2.9) yield a constant N > 0 , such that for all k we have Now choose a smooth cut-off function ∈ C ∞ 0 (B 2N (0)) , such that = 1 on B N (0) . The varifold convergence of k now yields Also (x, ) ↦ (x)⟨x, * ( )⟩ defines a continuous function with compact support on ℝ 3 × G 0 (2, 3) . Hence, the oriented varifold convergence yields The enclosed volume will need more attention, since we have to calculate a first variation. We will do this in Sect. 3

First Variation of enclosed volume
In this section, we derive a suitable formula for the first variation of the enclosed volume with respect to a smooth vectorfield.
Now we claim the following equation, which we will prove afterwards: As a short remark: If V 0 would be given by an embedded closed surface, R would represent the open and bounded set with boundary V 0 . Since R is of bounded variation, we find a Borel measure |▿ R | and a Borel measurable We now claim The proof is as follows: We define ∶= −g 1 dx 2 ∧ dx 3 + g 2 dx 1 ∧ dx 3 − g 3 dx 1 ∧ dx 2 and then [38,Remark 26.28] yields: Furthermore, we have Hence, which yields (3.4).
Since R has compact support and by multiplying with a smooth cutoff function, (3.4) is valid for every smooth vectorfield. Hence, we can apply (3.4) to g(x) = 1 3 x and obtain which is (3.2). Under our assumptions, i.e. finite mass and compact support, the current R is unique. Let us prove this claim: Assume we have R 1 , R 2 ∈ D 3 (ℝ 3 ) with compact support and finite mass satisfying Then and Now the constancy theorem for currents (see, e.g. [38,Thm. 26.27]) yields a c ∈ ℝ , such that If we want to calculate the first derivative of the enclosed volume, we need to make sense of mapping an oriented integral varifold by a diffeomorphism. So let g ∶ ℝ 3 → ℝ 3 be a diffeomorphism. Then we define (3.6) R 1 = R 2 .

3
Then we have for every ∈ D 2 (ℝ 3 ) By [38,Remark 27.2] we also have and hence Now we are ready to calculate the first variation of the enclosed volume. Let The second equality follows from [38,Remark 27.2] and the fact, that x ↦ Φ(t, x) is orientation preserving for every t ∈ ℝ . If we now decompose R into positiv and negativ parts, we can employ the calculation for the first variation of varifolds (see, e.g. [38,Sect. 16]) and we finally obtain

Area and volume correction
In this chapter, we introduce a two parameter diffeomorphism of ℝ 3 to adjust an error appearing in the graphical decomposition method by Simon, see [39,Sect. 3] for the prescribed area and enclosed volume. This diffeomorphism will be constructed by two vector fields and their corresponding flows (cf. Fig. 1). This idea was introduced by Schygulla in [36] for a one parameter diffeomorphism and prescribed isoperimetric ratio. We will expand this method by using a version of the inverse function theorem (see Theorem A.2) with some explicit bounds on the size of the set of invertibility.
Before we can start working on our diffeomorphism, we have to define currents for the minimising sequence f k , with which we will be able to calculate the enclosed volume as in Sect. 3. So let (R k ) k∈ℕ ⊂ D 3 (ℝ 3 ) be a sequence of integral currents satisfying (see also (3.1)) Here, C > 0 is given by the isoperimetric inequality [   Since Φ does not depend on k, we obtain F 1 k → F 1 pointwise as k → ∞ . Furthermore, R k converges in the BV-sense and we therefore have L 1 -convergence R k → R . Hence, for all s, t ∈ ℝ which yields the pointwise convergence F k → F . Here −1 and D refer to the x-variable of Φ.
Since we want to locally invert F and F k , we have to derive F and find X, Y such that DF(0, 0) is invertible. Equation (3.11) and the usual calculation of the first variation of a varifold (see, e.g. [38,Sect. 16]) yields To find the necessary vectorfields, we work similarly to [36,Lemma 4] and prove the following two lemmas: Proof We proceed by contradiction and assume the statement is false. Then there exist sequences r k → 0 and x k ∈ spt( ) , such that for all ∈ spt( ) ⧵ B r k (x k ) there is an > 0 such that for all Y ∈ C ∞ 0 (B ( ), ℝ 3 ) . By Lemma 2.1 spt( ) is compact and thus we can extract a subsequence and relabel such that x k → x ∈ spt( ) . Hence we obtain Thus by (3.5) which is a contradiction to Vol(V 0 ) = Vol 0 ≠ 0 . ◻ Lemma 4.2 (cf. Lemma 4 in [36]) Either H ∈ L ∞ ( ) or there exists an r > 0 , such that for every z ∈ spt( ) there exist two points Y , X ∈ spt( ) ⧵ B r (z) , such that for every Proof Let us assume that we do not find an r > 0 as requested. Then we find sequences ) . Since spt( ) is compact, we find a convergent subsequence of z k and after relabeling we have z k → z ∈ spt( ) . Hence for ( X ), ℝ 3 ) satisfies (4.6). According to Lemma 4.1, we find an ∈ spt( ) ⧵ {z} and a Y ∈ C ∞ 0 (B Y, ( )) with From now on this Y is fixated. All in all we find for every X ∈ spt( ) ⧵ {z} a radius X such that for every X ∈ C ∞ 0 (B X, X ) we have Since X is arbitrary and ({z}) = 0 , we have Furthermore + + − ≥ 1 -a.e. and ± ∈ ℕ 0 finally yield H ∈ L ∞ ( ) . ◻ Since we like to apply Theorem A.2 to F k , we need to be able to estimate the difference of two values of the first derivative independently of k. To do this, we employ the mean value theorem and hence need to estimate the second derivative: Proof We start by estimating F 2 : Since (3.10) only needs the corresponding map to be a diffeomorphism, we obtain the same result for x ↦ Φ(s, t, x) . Furthermore, the usual substitution formula yields: Now there exists a constant C > 0 only dependend on T > 0 , X, Y and their respective derivatives, such that for every s, t ∈ (−T, T) we have Hence, for these s, t the isoperimetric inequality [ -a.e. . (4.9) Furthermore, in the Jacobian J k Φ(s, t, x) the measure k is independent of s and t. Also the derivatives d k Φ(s, t, x) appearing in the Jacobian, can be estimated by the full derivative of x ↦ Φ(s, t, x) . Therefore there is a constant C > 0 only dependend on T > 0 , X, Y and their respective derivatives, such that for every s, t ∈ (−T, T) we have This yields which is the desired conclusion. ◻ If we change the minimising sequence by Φ , we also have to be sure, that the Helfrich energy and the second fundamental form is controlled as well: Proof By a partition of unity on Σ and a rigid motion we can assume, that we can write x ↦ Φ(s, t, f k (x)) locally as a smooth graph with small Lipschitz norm. Hence, we have u k ∶ ℝ 2 ⊃ B r (0) → ℝ smooth with |∇u k | ≤ 1 and u k (0) = 0 , which satisfies for a small open set U ⊂ Σ: Now we can calculate the second fundamental form of Φ(s, t, f k (⋅)) by using u k and chain rule: Then the second derivatives are as follows: The unit normal can be expressed by for all (s, t) ∈ B T (0) and C is independent of k. Here f s,t k is the immersion Σ ∋ x ↦ Φ(s, t, f k (x)).

Partial regularity and lower semicontinuity
In this section, we adapt the partial regularity method introduced by Simon (see [39,Section 3]). This method is based on replacing parts of the minimising sequence with biharmonic graphs and compare the resulting energies. Here we use the idea of Schygulla [36,Lemma 5] to correct the area and enclosed volume of the modified sequence, so that they become competitors for the minimum of the Helfrich energy again. Let 0 > 0 be fixated. In dependence of this 0 we say x 0 ∈ spt( ) is a good point, iff (see (2.10)) In neighbourhoods of these good points we will show C 1, regularity and a graphical decomposition of . Since we work with immersions with possible self intersections, we also use the ideas of Schätzle (see [35,Prop. 2.2]) to implement Simon's regularity method (cf. [39,Section 3]). The lemma is now as follows (cf. also [13, Lemma 3.1]): Here, L i ⊂ ℝ 3 are two-dimensional affine spaces and i = 1, … , I x 0 ≤ C(Area 0 , Vol 0 , H 0 ) . Furthermore, the u i satisfy the following estimate Moreover, we have a power-decay for the second fundamental form, i.e. ∀x ∈ B x 0 4 (x 0 ) , Proof We start by applying the graphical decomposition lemma of Simon (see [39,Lemma 2.1]) to the minimising sequence f k ∶ Σ → ℝ 3 . This lemma is also applicable for immersions by an argument of Schätzle (see [35, p. 280] and cf. [13,Lemma A.6], in the beginning we work as in these papers). We repeat some steps of [35, (2.11)-(2.16)], which we will need later (see also [13, (3.5
Furthermore, for every i = 1, … , I k we have affine two-dimensional planes After taking a subsequence depending on x 0 , x 0 and relabeling we can assume I k = I and � . (5.10) (5.11) k,i → i weakly as varifolds in B x 0 (x 0 ).

3
As shown in [13,Eq. (3.13)] (see also [35, p. 281]) we get As in [35,Eq. (2.16)] we can pass to the limit in (5.8) and obtain We will apply Allard's regularity theorem A.1 to the i . Inequality (5.13) already takes care of the needed density estimate (A.3). The remainder of the proof will be about showing (A.2) or a similar L p -bound for the mean curvature. Here we need to make a distinction of cases given by Lemma 4.2.
First we assume H ∈ L ∞ : In this case, we will show that H i ∈ L p , p > 2 arbitrary. By the usual regularity Theorem of Allard (see, e.g. [38,Thm. 24.2]), the i will be C 1, graphs.
We need the definition of the tilt and height excess (see, e.g. [

4.3] is applicable and yields
Since H ∈ L ∞ ( ) and i ≤ we finally obtain which yields by the usual Allard regularity theorem (see, e.g. [38,Thm. 24.2]), that every i is a C 1, -graph with ∈ (0, 1) arbitrary. This case is therefore done.

3
Now we modify Schygulla's argument [36,Lemma 5] to our situation: The beginning of the argument is as in [39,Lemma 3.1]. For the readers convenience and because we need the notation, we repeat these steps here (see also [13, pp. 9-10]): Let us choose 0 < < x 0 fixed but arbitrary. We need to apply the graphical decomposition Lemma [39,Lemma 2.1] Since D k,i is topologically a disc, Σ and Σ k, are topologically equivalent. f graph k, does not have the same area or enclosed volume as f k , which we like to correct now. Therefore we need some estimates on these properties. We start with the area. Here we use (5.8) and Lemma A.3 to obtain Let us proceed with the enclosed volume. Since f k (Σ) ⊂ B N (0) , for N > 0 big enough and independent of k, we also get f graph k, . Hence the definition of the enclosed volume and the Cauchy-Schwartz inequality yield Now we apply Lemma 4.2: Hence, we find an r > 0 and points X , Y ∈ spt( ) ⧵ B r (x 0 ) (without loss of generality we assume 0 , x 0 ≤ r 2 ), which satisfy that for every X , Y > 0 , we find vectorfields X ∈ C ∞ 0 (B X ( X )) and Y ∈ C ∞ 0 (B Y ( Y )) , such that Here R denotes the 3-current defined in the beginning of Sect. 4 and R the corresponding BV-function. We also fix X = Y = . Furthermore, let Φ be defined as in (4.2) and F and F k be as in (4.3) but with respect to f graph k, instead. The results of Sect. 4 are still valid, since the diffeomorphism Φ does not influence the graphical comparison function (cf. Fig. 1). As in the beginning of Sect. 4 we obtain The oriented varifold convergence and the L 1 convergence of R k yield for a fixed constant c 0 > 0 . Hence, for k big enough, we get Hence, DF k (0, 0) is invertible. Furthermore, the mean value theorem and Lemma 4.3 yield for every T 0 > 0 a C = C(T 0 ) > 0 , such that for every (s, t), (s � , t � ) ∈ B T (0) we have f graph k, t).
if we choose 0 < T < T 0 arbitrary. By the formula for the inverse matrix via the adjunct matrix we obtain for k big enough The constant C is independent of k. Next we will apply Lemma A.2. Hence, we need to define a functions F and F k satisfying the assumptions of that Lemma: Here −1 is meant as the matrix inverse. Hence Let furthermore (s, t), (s � , t � ) ∈ B T (0) for 0 < T < T 0 . Then by (5.30) and (5.31) we have if we choose T 0 small enough. So Lemma A.2 is applicable to F k and therefore we find for every (ỹ,z) ∈ B (1− 0 )T (0) parameters (s k , t k ) with By (5.32) we obtain Since DF k (0, 0) is invertible, we obtain a > 0 (by (5.29) only dependend on T 0 ), such that for every (y, z) ∈ B (F k (0, 0)) we find (s k , t k ) ∈ B (1− 0 )T (0) satisfying Furthermore we may choose > 0 to be maximal, i.e. satisfying the following property: There is a (y 0 , z 0 ) ∈ B (0) and a (ỹ 0 ,z 0 ) ∈ B (1− 0 )T (0) with This yields Hence Now we choose T 0 ∶= 0 , T ∶= (by choosing 0 small enough our results are still true). For later purposes we also state that the inverse inequality of (5.34) is true as well, only with a bigger constant of course, i.e. we have with C 1 < C 2 independent of k   DF k (0, 0)(ỹ 0 ,z 0 ) = (y 0 , z 0 ).
Here k → 0 for k → ∞ . The estimates connecting D 2 w k,i, with the corresponding curvatures can be seen by, e.g. Hole filling yields a 0 < Θ < 1 independent of k, satisfying Next we formulate the lower-semicontinuity property of the minimising sequence:

Lemma 5.3
The minimising sequence V 0 k for the Helfrich problem (1.6) satisfies Proof Since is locally a graph of C 1, ∩ W 2,2 graphs outside of finitely many points (see Lemma 5.1), and these graphs are approximated by k,i , see (5.11), the proof of the lowersemicontinuity estimate is the same as in [13,Lemma 4.1]. ◻ Acknowledgements Open Access funding provided by Projekt DEAL.
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Auxilliary results
For the readers convenience we collect a few needed results: The following is a variant of Allard's regularity Theorem. A proof of this statement can be found in [39,Section 3] or [37,Korollar 20.3] (see also [35,Theorem B.1]).