GRADIENT-FREE DIFFUSE APPROXIMATIONS OF THE WILLMORE FUNCTIONAL AND WILLMORE FLOW NILS DABROCK SASCHA KNÜTTEL AND MATTHIAS RÖGER

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"GRADIENT-FREE" DIFFUSE APPROXIMATIONS OF THE

WILLMORE FUNCTIONAL AND WILLMORE FLOW

NILS DABROCK, SASCHA KNÜTTEL* AND MATTHIAS RÖGER

Abstract. We introduce new diﬀuse approximations of the Willmore functional and

the Willmore ﬂow. They are based on a corresponding approximation of the perimeter

that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We

identify the candidate for the Γ–convergence, prove the Γ–limsup statement and justify the

convergence to the Willmore ﬂow by an asymptotic expansion. Furthermore, we present

numerical simulations that are based on the new approximation.

1. Introduction

The Willmore functional

W(Γ) :=ZΓ

H2(y) dHn−1(y),

of a C2-regular hypersurface Γ⊆Rnwith mean curvature His one of the most prominent

examples of a curvature energy. Such energies have already been considered by Poisson [1]

and Germain [2] in the 19th century and appear in a variety of applications, for example

as a shape energy of bio membranes as proposed by Canham [3] and Helfrich [4]. The

Willmore functional in particular has been studied intensively in diﬀerential geometry and

geometric measure theory by Thomsen and Blaschke [5, 6] at the beginning of the last

century as well as by Willmore [7] and more recently by Simon [8], Kuwert and Schätzle

[9], Riviére [10]. The most spectacular recent contribution is the proof of the Willmore

conjecture on the minimal Willmore energy of immersed tori by Marques and Neves [11].

In the case of planar curves, the Willmore energy reduces to Euler’s Elastica energy for the

bending of a rod. This energy has an even longer history (see e.g. [12]), has been thoroughly

investigated in many contributions [13, 14] and still is an active ﬁeld of research.

Gradient ﬂows for curvature energies as steepest decent dynamics have also attracted a lot

of attention. The Willmore ﬂow, in particular, has been considered in many contributions

over the past decades, see for example Simonett [15] and Kuwert and Schätzle [16, 17, 18].

Motivated by phase separation problems and as a tool for numerical simulations, diﬀuse ap-

proximations of curvature energies and in particular the Willmore functional and Willmore

ﬂow are widely used. The most famous example is the phase ﬁeld approximation going

back to De Giorgi [19]. This approximation is based on the Van der Waals–Cahn-Hilliard

2010 Mathematics Subject Classiﬁcation. 35R35, 35K65, 65N30.

Key words and phrases. Willmore ﬂow, phase-ﬁeld model, diﬀuse interface, sharp interface limit.

All authors are aﬃliated with Technische Universität Dortmund, Fakultät für Mathematik,

Vogelpothsweg 87, D-44227 Dortmund, Germany

*Corresponding author. Tel.: +49 231 755 5163, Fax:+49 231 755 5942,

E-Mail: Sascha.Knuettel@math.tu-dortmund.de

arXiv:2210.06191v1 [math.AP] 12 Oct 2022

2 NILS DABROCK, SASCHA KNÜTTEL* AND MATTHIAS RÖGER

energy, given by

PCH

ε(u):=ZΩε

2|∇u|2+1

εW(u)dLn,(1)

where Wis a suitable double well potential and uis a smooth function on a domain Ω⊆Rn.

To achieve low energy values, the function uhas to be close to the wells of the potential

except for thin transition layers with thickness of order ε. The celebrated result by Modica

and Mortola [20, 21] states that this functionals Γ–converge in the sharp interface limit

ε→0to the perimeter functional P,

PCH

ε→cCH

0P, cCH

0=Z1

−1

√2WdL1.(2)

Since the mean curvature is the L2-gradient of the perimeter it is a natural approach to

take the L2-gradient of the diﬀuse perimeter (1) as a starting point for a diﬀuse Willmore

energy. This motivates a formal approximation of the Willmore functional, given by

WCH

ε(u):=ZΩ

2ε−ε∆u+1

εW0(u)2dLn,(3)

which is a modiﬁed version of De Giorgi’s proposal [19], introduced by Bellettini and Paolini

in [22]. The ε−1-factor compensates for the volume of the transition layer as we will see in

the calculations in chapter 3.

The Γ–lim sup property was proved in [22], the Γ–convergence was shown in dimensions

2 and 3 for smooth limit conﬁgurations in [23]. This gives a solid justiﬁcation for this

approximation, though many issues concerning the Γ–convergence are still to be resolved,

in particular concerning non-smooth limit conﬁgurations.

Corresponding diﬀuse approximations of the Willmore ﬂow have been introduced by [24]

and have been justiﬁed by formal asymptotic expansions in [25], see also [26]. In a recent

article [27] Fei and Liu prove the convergence of diﬀuse approximations to the Willmore

ﬂow for well-prepared initial data, as long as the smooth limit ﬂow exists.

Quite a number of numerical schemes for the simulation of the Willmore ﬂow have been

proposed. For the treatment in a sharp-interface approach, we refer to [28, 29, 30, 31, 32,

33, 34]. Level set techniques have been used in [35, 36]. Diﬀuse approximation have been

employed in a huge number of applications, see for example [37, 38, 39, 40, 41, 42, 43, 44,

45, 36].

A number of alternative diﬀuse approximations of the Willmore energy have been proposed,

in particular to enforce the Γ–convergence of approximations for non-smooth limit conﬁgu-

rations to the L1lower semi-continuous envelope of the Willmore functional [46, 47, 43, 48].

These approximations, however, often lack the simplicity of the standard approximation

and its direct relation to applications.

A class of nonlocal perimeter approximations can be derived from classical Ising-type mod-

els [49]. They involve a discrete gradient and a double integral

PAB

ε(u) = 1

εZΩ

W(u) dLn+ε

4ZΩZΩ

Jε(x−y)u(x)−u(y)

ε2dydx, (4)

where Jε=ε−nJ(·/ε)for a suitable kernel J. Alberti and Bellettini [49] proved the Γ–

convergence with ε→0to an (in general anisotropic) perimeter functional. At least on a

formal level, one might construct diﬀuse approximations of the Willmore energy starting

from the L2-gradient of PAB

ε. However, to the best of our knowledge this has not been

addressed yet and a rigorous justiﬁcation in a general framework seems to be diﬃcult. We

GRADIENT-FREE DIFFUSE APPROXIMATION OF THE WILLMORE FUNCTIONAL 3

refer to Braides [50] for some more details on the above mentioned models and an overview

over diﬀerent perimeter approximations.

In the present paper we consider an approximation that is in between the Cahn–Hilliard

model and the Ising-type models just described. It is motivated by an in general anisotropic

two-variable energy studied by Solci and Vitali in [51]. In the isotropic case the functional

is characterized as

Gε(u, v):=ZΩε

2|∇v|2+1

2ε(u−v)2+1

2εW(u)dLn.

Such an energy does also appear in a one-dimensional model for the longitudinal deforma-

tion of an elastic bar, proposed by Rogers and Truskinovsky [52], and can be connected to

certain two-variable models for phase separation processes, see the references in [53]. Solci

and Vitali proved that Gεdoes Γ–converge on L1(Ω)2towards a functional Gthat is only

ﬁnite on {(u, u)∈BV (Ω; {±1})2}with G(u, u) = c1P(u)for some c1>0.

We follow here the analysis of Amstutz and van Goethem [53] of a gradient-free approxi-

mation of the perimeter functional that is obtained by considering the marginal functional

of Gε, where for given uthe variable vis chosen as minimizer of Gε(u, ·). This leads to a

representation of the optimal vε=vε[u]as solution of

−ε2∆vε+vε=uin Ω,∇vε·νΩ= 0 on ∂Ω,(5)

and to the functional

PAG

ε(u):= inf

v∈H1(Ω) ZΩε

2|∇v|2+1

2ε(u−v)2+1

2εW(u)dLn

=ZΩε

2|∇vε|2+1

2ε(u−vε)2+1

2εW(u)dLn

=ZΩ

2εu(u−vε) + W(u)dLn.(6)

For the particular choice W(r)=1−r2in [−1,1] and locally constant linear growth

outside [−1,1] the Γ–convergence in L1(Ω) towards a multiple of the perimeter functional

was proved in [53]. This can be generalized to a bigger class of double-well potentials and in

particular to the potentials used below. For our analysis, however, it is important to have

some smoothness and quadratic behavior of Win the wells, see 1.2 below. This condition,

on the other hand, excludes the double well potential used in [53].

In [53] numerical simulations of some topology optimization problems were presented,

where the gradient-free structure of the functional (with respect to the variable u) proved

to be advantageous. We note that in the case Ω = Rnthe approximation PAG

εcorresponds

to PAB

εin (4), with a particular choice of J.

The solution operator induced by the PDE (5) is linear and self-adjoint. The L2-gradient

of PAG

εtherefore is given by

Hε:=∇L2PAG

ε(u) = 1

εu+1

2W0(u)−vε.(7)

In analogy to the sharp interface situation Hεcan be seen as a diﬀuse mean curvature.

This suggests the formal Willmore energy approximation

Wε(u):=WAG

ε(u):=ZΩ

ε3u+1

2W0(u)−vε2dLn,(8)

4 NILS DABROCK, SASCHA KNÜTTEL* AND MATTHIAS RÖGER

which is the main object of the current study. The additional factor ε−1in Wεagain

accounts for the small volume of the transition layer region. We remark that no gradients

appear (explicitly) in the functional and only the rather well-behaved solution operator

u7→ vεassociated to (5) enters the energy. This makes the above functional an interesting

candidate for numerical simulations.

Besides the theoretical interest in Willmore approximations and its use in numerical sim-

ulations, the analysis of the functional Wεis also important for the understanding of the

corresponding perimeter approximation (6), and in particular the associated L2-gradient

ﬂow. In fact, the Γ–convergence of Wεis one of the properties necessary to apply a general

result about convergence of gradient ﬂows proved by Sandier and Serfaty in [54].

The function vεthat appears in the functional PAG

εand that is characterized by (5) rep-

resents a particular regularization of u. Perimeter approximations based on other regu-

larizations are possible as well (also the functional PAB

εcan be represented this way) and

the results in [51] could be used to prove its Γ–convergence, which also was central for the

proof in [53]. This becomes less clear when dealing with approximations of the Willmore

energy. Here we exploit the very convenient PDE characterization of vε. We suspect that

the Γ–convergence towards the Willmore functional can be proved also for diﬀuse approx-

imations based on other regularizations of u. However, to the best of our knowledge no

such results are currently available. Developing a general theory therefore might be an

interesting ﬁeld for future research.

Besides the static functionals we are also interested in L2-type gradient ﬂows. In the

sharp interface setting we therefore consider evolutions of phases (E(t))t∈(0,T )and of the

associated boundaries Γ(t) = ∂E(t). In case of the perimeter functional the formal L2-

gradient ﬂow is given by the mean curvature ﬂow

V=H, (9)

where Hand Vdenote the scalar mean curvature and normal velocity of the evolution in

direction of the inner unit normal ﬁeld associated to E(t),t∈(0, T ). Mean curvature ﬂow

is one of the most prominent geometric ﬂows and has been studied extensively of the past

decades. We refer to [55] for a proper introduction to the subject.

The Willmore ﬂow is the formal L2-gradient ﬂow of Wand is given by

V=−∆ΓH+1

2H3−H|II|2,(10)

where II(·, t)denotes the second fundamental form of Γ(t)and ∆Γthe Laplace-Beltrami

operator on Γ(t). The Willmore ﬂow is a fourth order geometric evolution law, which

introduces quite some additional challenges for the analysis of the ﬂow. We refer to the

already mentioned fundamental contributions [15], [16, 17, 18]. For a derivation of the

formula for the L2-gradient of the Willmore functional see [56, sections 7.4 - 7.5].

In an analogue way we associate L2-gradient ﬂows to the diﬀuse perimeter PAG

εand Will-

more energy approximations Wεin (6), (8).

The L2-gradient of PAG

εhas already been characterized in (7) by Hε=1

ε−vε+u+1

2W0(u).

Taking the variational derivative of Wεwe ﬁnd

∇L2Wε(u) = 2

ε21 + 1

2W00(u)−(Id −ε2∆)−1Hε.(11)

GRADIENT-FREE DIFFUSE APPROXIMATION OF THE WILLMORE FUNCTIONAL 5

Appropriately rescaled this leads to the diﬀuse mean curvature ﬂow

ε∂tuε=Hε,(12)

and the diﬀuse Willmore ﬂow

ε∂tuε=−2

ε21 + 1

2W00(u)−(Id −ε2∆)−1Hε.(13)

We may expect that the diﬀuse ﬂows converge in the sharp interface limit ε→0to mean

curvature ﬂow and Willmore ﬂow, respectively.

The goal of this paper is to provide some justiﬁcation to the above mentioned formal ap-

proximation properties. In particular, we will identify the candidate for the Γ–convergence

of the functionals Wε, which in fact is proportional to the Willmore functional, with a

speciﬁc constant of proportionality that only depends on the choice of the double well

potential. For this candidate we prove the corresponding Γ–limsup construction. In ad-

dition, we give a rigorous lower bound in particular classes of phase ﬁeld approximations

that are described by suitable expansion properties. Moreover, we justify by a formal as-

ymptotic expansion the convergence of the ﬂow (13). For the proof we basically follow the

approach already used by Loreti and March [25] and Wang [26]. However, the operators

that deﬁne the gradient-free approximation are diﬀerent from the standard case and the

derivation of the convergence property is much more involved, in particular for the case of

the approximate Willmore ﬂow.

We ﬁnally use our approach for numerical simulations. We follow the implicit spectral

discretization scheme proposed by [36] for the standard diﬀuse approximation and compare

the new and the standard scheme.

Preliminaries 1.1.

We collect some basic notations, deﬁnitions and assumptions that we will use throughout

the paper.

Let n∈Nand Ω⊆Rnbe a bounded open domain with Lipschitz boundary. We denote by

Lnthe n-dimensional Lebesgue measure and by Hn−1the (n−1)-dimensional Hausdorﬀ

measure. The space BV (Ω; {±1})consists of all function of bounded variation with values

in {±1}almost everywhere. BV (Ω; {±1})can be identiﬁed with the sets of ﬁnite perimeter

in Ω, where we associate to a set Eof ﬁnite perimeter the rescaled characteristic function

u:= 2XE−1. The essential boundary of a set E⊆Ωwith ﬁnite perimeter in Ωis denoted

by ∂∗E. For u= 2XE−1we then have |∇u|= 2Hn−1Γ.

With these notations we deﬁne the perimeter functional PΩ:L1(Ω) −→ [0,∞]by

PΩ(u):=









2ZΩ|∇u|dLn,if u∈BV (Ω,{±1})

+∞,else.

(14)

Note PΩ(2XE−1) = Hn−1(∂∗E∩Ω) for E⊆Ωwith ﬁnite perimeter.

Furthermore we deﬁne the Willmore functional W:L1(Ω) −→ [0,∞],

WΩ(u):=







Z∂E∩Ω

H2dHn−1,if u= 2XE−1for E⊆Ωwith ∂E ∩Ω∈C2

+∞,else,

(15)

where the mean curvature His deﬁned as the sum of the principle curvatures of ∂E (taken

positive for convex E). We will drop the Ω-index for Pand Was Ωis ﬁxed. When we use

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"GRADIENT-FREE"DIFFUSEAPPROXIMATIONSOFTHEWILLMOREFUNCTIONALANDWILLMOREFLOWNILSDABROCK,SASCHAKNÜTTEL*ANDMATTHIASRÖGERAbstract.WeintroducenewdiuseapproximationsoftheWillmorefunctionalandtheWillmoreow.TheyarebasedonacorrespondingapproximationoftheperimeterthathasbeenstudiedbyAmstutz-vanGoethem[Interf...

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GRADIENT-FREE DIFFUSE APPROXIMATIONS OF THE WILLMORE FUNCTIONAL AND WILLMORE FLOW NILS DABROCK SASCHA KNÜTTEL AND MATTHIAS RÖGER

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