Phase field models for thin elastic structures with topological constraint
This article is concerned with the problem of minimising the Willmore energy in the class of connected surfaces with prescribed area which are confined to a small container. We propose a phase field approximation based on De Giorgi’s diffuse Willmore functional to this variational problem. Our main contribution is a penalisation term which ensures connectedness in the sharp interface limit. The penalisation of disconnectedness is based on a geodesic distance chosen to be small between two points that lie on the same connected component of the transition layer of the phase field.
We prove that in two dimensions, sequences of phase fields with uniformly bounded diffuse Willmore energy and diffuse area converge uniformly to the zeros of a double-well potential away from the support of a limiting measure. In three dimensions, we show that they converge -almost everywhere on curves. This enables us to show -convergence to a sharp interface problem that only allows for connected structures. The results also imply Hausdorff convergence of the level sets in two dimensions and a similar result in three dimensions.
We furthermore present numerical evidence of the effectiveness of our model. The implementation relies on a coupling of Dijkstra’s algorithm in order to compute the topological penalty to a finite element approach for the Willmore term.
Key words and phrases:Willmore energy, phase field approximation, topological constraint, Gamma-convergence
2010 Mathematics Subject Classification:49Q10; 74G65; 65M60
- 1 Introduction
- 2 Phase Fields
- 3 Proofs
- 4 Computer Implementation
- 5 Conclusions
In this article, we consider a diffuse interface approximation of a variational problem arising in the study of thin elastic structures. Our particular question is motivated by the problem of predicting the shape of certain biological objects, such as mitochondria, which consist of an elastic lipid bilayer and are confined by an additional outer mitochondrial membrane of significantly smaller surface area.
In order to describe the the locally optimal shape of thin elastic structures, we will rely on suitable bending energies. A well-known example of such a variational characterisation of biomembranes is given by the Helfrich functional [Hel73, Can70]
where denotes the two-dimensional membrane surface in and and denote its mean and Gaussian curvatures. The parameters , and are the bending moduli and the spontaneous curvature of the membrane and are, in the simplest case, constants. The integral is performed with respect to the two-dimensional Hausdorff measure . The special case when and is known as Willmore’s energy
While our results extend to certain Helfrich-type functionals of more general form (see section 5), the basic task we have set for ourselves is the following: Minimise Willmore’s energy in the class of connected -surfaces which are embedded into a domain such that the embedding has prescribed surface area . This problem does not necessarily admit a solution, but a minimiser exists in the varifold closure of these surfaces [DMR14].
In this version of the problem, the outer mitochondrial membrane is the rigid boundary , but the results also apply if an elastic container is used, see section 5.
We propose a phase field approach to this problem. For computations, this has some advantages to the naive approach of taking a gradient flow of Willmore’s energy (or an area preserving version of it).
First, Willmore flow is a fully non-linear degenerate parabolic PDE of fourth order on a moving surface, which makes both analytic and numerical treatment difficult. In contrast, our phase-field flow equations are fourth order quasi-linear parabolic with constant coefficients for the fourth derivatives. The price we have to pay for this relative luxury is, as usual, that we have to solve equations in one higher dimension.
Second, Willmore flow is a fourth order PDE and as such does not admit a maximum principle. In particular, a surface originally confined to even a convex container could easily leave it along the gradient flow, whereas for us the confinement to is simply given by the choice of boundary condition and the domain of the phase field.
The main challenge for phase fields is to control the topology of a limiting surface (in some sense) from quantities of diffuse interfaces. In this paper, we give a suitable concept of connectedness that admits a rigorous variational statement as and can be implemented efficiently in a simulation.
Heuristically, we introduce a quantitative notion of path-connectedness for the interfacial region of a phase field. The disconnectedness for two points in the interface layer is measured by a geodesic distance (the infimum length of connecting paths) where walking in the interfacial region does not add any length, but leaving the interface adds a positive amount. The total disconnectedness is then computed as a double integral over the distances of interface points.
This functional poses analytic challenges since it is not a priori clear how ’two-dimensional’ the interfacial layer of a phase field is. Since the distance function uses the length of curves, away from a limiting surface we need the phase-fields to converge almost everywhere on curves, i.e. on objects of codimension . Also, the structure of the interface has to be understood in a precise way in order to estimate the aforementioned double integral. More rigorous statements are given below.
Our analysis builds on ideas in [RS06]. Along the proof of our Theorem, we obtain a number of useful technical results about the convergence of phase field approximations to the limit problem which appear to be new in this context. Namely, for sequences along of uniformly bounded diffuse energy, in two dimensions we prove convergence of the transition layers to the support of a limiting measure in the Hausdorff distance and uniform convergence of the phase field to away from the limit curve. This is reminiscent of a result for minimisers of the Modica-Mortola functional among functions with prescribed integral [CC95] and more generally for stationary states [HT00].
In three dimensions, we show that neither result is true, but prove -convergence on curves and that is contained in the Hausdorff limit of interfacial regions. This enables us to precisely control our topological energy term. We also show that phase fields are uniformly -bounded in terms of their Willmore energy, also in three dimensions.
The numerical implementation of our functional will be discussed further in a forth-coming article [DW16]. We use a variant of Dijkstra’s algorithm similar to the one of [BCPS10] to compute the geodesic distance function used in the topological term of our energy functional. The weight in the geodesic distance is chosen to be exactly zero along connected components of the transition layer, which implies that it only needs to be computed once per detected connected component. This makes the functional efficient from an implementation point of view.
The article is organised as follows. In the remainder of this section we give some further context for our problem. In section 2 we give a brief introduction to phase fields for Willmore’s problem in general (sections 2.1 and 2.2) and to our approach to connectedness for the transition layers (section 2.3). Our main results are subsequently listed in section 2.4.
Section 3 is entirely dedicated to the proofs of our main results. We directly proceed to show -convergence of our functionals (section 3.1). In sections 3.2 and 3.3, we produce all the auxiliary estimates that will be needed in the later proofs. These are then used to show convergence of the phase fields, Hausdorff-convergence of the transition layers, and connectedness of the support of the limit measure (section 3.4).
In section 4 we show numerical evidence of the effectiveness and efficiency of our approach. To that end, we compare it to diffuse Willmore flow in two dimensions without a topological term and to the penalisation proposed in [DMR11]. We have been unable to implement the functional developed in [DMR14] in practice, so a comparison with that could not be drawn. In section 5 we discuss a few easy extensions of our main results.
1.1. Topology and Phase Fields
An often cited advantage of phase fields is that they are capable of changing their topology; in that sense our endeavour is non-standard. It should be noted that our phase fields may still change their topology (at least in three dimensions), only connectedness is enforced.
In [BLS15], a geodesic distance function has been used to minimise the length of a connected set containing a prescribed set of points in two dimensions (Steiner’s problem). Our setting is different in two ways: 1. Steiner’s problem has a finite number of a priori known points which need to be contained in while the transition layer of the phase field has no special points and 2. the phase field approximation of Steiner’s problem works in dimension , while we work in ambient space of dimension where the curves used in the definition of the distance function have codimension .
Previous work in [DMR11] provides a first attempt at an implementation of a topological constraint in a phase field model for elastic strings modelled by the one-dimensional version of the Willmore energy, Euler’s elastica. This technique prevents transitions in simulations for simple situations, but may fail in more complex cases, see section 4. Similar numerical approaches to controlling the topology using a diffuse Euler number are discussed in [DLW05, DLRW07].
This approach was complemented by a method put forth in [DMR14], which relies on a second phase field subject to an auxiliary minimisation problem used to identify connected components of the transition layer. While our functional can be seen as using a diffuse measure of path-connectedness, the functional in [DMR14] generalises more directly the notion of connectedness. For this model, a -convergence result was obtained, showing that limits of bounded-energy sequences must describe a connected structure. Unfortunately, the complicated nested minimisation problem makes it unsuitable for computation.
Approaches of regularising limit interfaces have been developed by Bellettini in [Bel97] and investigated analytically and numerically in [ERR14]. The approaches work by introducing non-linear terms of the phase field in order to control the Willmore energies of the level sets individually and exclude transversal crossings (which phase fields for De Giorgi’s functional can develop). These regularisations may prevent loss of connectedness along a gradient flow in practice, but do not lead to a variational statement via -convergence.
Furthermore, we would like to emphasise that we can easily describe a weakly* continuous evolution of varifolds along which connectedness is lost. Except at one singular time, the varifolds are embedded -manifolds and the evolution is -smooth. Details can be found in a forthcoming paper [Woj16].
It thus is not clear whether the approach of [Bel97] does prevent topological transitions, in particular, the loss of connectedness, in three ambient space dimensions. At least, it is more difficult to implement due to the highly non-linear term including the Willmore energies of level sets.
Also in [Woj16], we show that topological genus is not continuous under varifold convergence and that minimising sequences of a constrained minimisation problem with fixed genus may change topological type in the limit.
At this point, we thus know of no other model which can control the topology of phase field limits. Furthermore, our results are optimal since they allow us to control as much of the topology as can be controlled even for a sharp interface and they allow for efficient implementation.
1.2. Biological Membranes
Willmore’s and Helfrich’s energies are widely used in the modelling of thin elastic structures. The first mention in that context goes back to Sophie Germain [Ger21]. Later, their importance has been suggested heuristically [Hel73] based on the principle that when a membrane is written as a graph over its tangent space, only derivatives of at most second degree should occur in at most quadratic expressions at the base point. Another biological motivation in the context of red blood cells is given in [Can70].
A Helfrich type functional has also been obtained as a macroscopic limit of certain mesoscale models for lipid bilayers [PR09, LPR14]. Here bilayers are modelled using functions to express locations of hydrophilic heads and hydrophobic tails of lipid molecules. These functions are coupled through a Monge-Kantorovich distance to be close together, constrained to satisfy and the perimeter of is penalised. This energy prefers a bilayer structure and suitably rescaled versions -converge to a Helfrich functional with , and .
A heuristic way to motivate the occurrence of Willmore’s energy in this context comes from thin shell theory. In [FJM02b, FJM02a] Friesecke, James and Müller proved -convergence of non-linear three-dimensional elasticity to geometric bending energies including those of Willmore- or Helfrich-type in the vanishing thickness limit of thin plates. The admissible class here are isometric embeddings of domains in . The restriction to isometries stems from the fact that in-plane stretching energy scales with thickness of the plate and dominates out of-plane bending, which scales with the third power of the thickness parameter. Thus in the (second order) vanishing thickness limit, we are led to minimise Willmore’s energy in a class of isometric immersions. The case of shells (i.e. non-flat structures) has been treated in [FJMM03].
Lipid bilayers differ from shells in that they are liquid not solid and as such do not have a reference configuration. Assuming inextensibility, we must therefore also minimise over the space of Riemannian metrics on the bilayer with fixed area and the isometry constraint turns into the fixed area constraint.
1.3. Further Context
Without any claim of completeness, let us give a bit more context of our topic. Willmore’s energy is named for T.J. Willmore who studied it in a series of publications [Wil65, Wil71, Wil92, Wil00] and popularised it in his textbook [Wil93]. Independently of Willmore’s work, the energy had already been considered as a bending energy for thins plates in [Ger21] and as a conformal invariant of surfaces embedded in in [BT29, Tho23].
As mentioned above, Willmore’s energy is famously conformally invariant, in particular scale invariant. This shows that both the embedding constraint into and the area constraint are needed to give a non-trivial constraint.
Extrema (in particular, local minimisers) of the Willmore functional are of interest in models for biological membranes, but they also arise naturally in pure differential geometry as the stereographic projections of compact minimal surfaces in , see e.g. [PS87], also for examples.
Even for Willmore’s energy rigorous results are hard to obtain. From the point of view of the calculus of variations, a natural approach to energies as the ones above is via varifolds [All72], [Hut86], where existence of minimisers for certain curvature functionals is proved. The existence of smooth minimising Tori was proved by Simon [Sim93], and later generalised to surfaces of arbitrary genus in [BK03].
The long-standing Willmore conjecture that for all Tori embedded in was recently established in [MN14], and the large limit genus of the minimal Willmore energy for closed orientable surfaces in has been investigated in [KLS10]. The existence of smooth minimising surfaces under isoperimetric constraints has been established in [Sch12]. A good account of the Willmore functional in this context can be found in [KS12].
The case of surfaces constrained to the unit ball was studied in [MR14] and a scaling law for the Willmore energy was found in the regimes of surface area just exceeding and the large area limit. While the above papers adopt an external approach in the language of varifold geometry, a parametrised approach has been developed in [Riv14] and related papers. In [KMR14] this framework is used to solve the Willmore minimisation problem with prescribed genus and prescribed isoperimetric type. [DGR15] gives a study of the Willmore functional on -graphs and its -lower semi-continuous envelope.
In the class of closed surfaces, if is constant, the second term in the Helfrich functional is of topological nature due to the Gauss-Bonnet theorem. So, if the minimisation problem is considered only among surfaces of prescribed topological type it can be neglected. The spontaneous curvature is realistically expected to be non-zero and can have tremendous influence. It should be noted that the full Helfrich energy depends also on the orientation of a surfaces for and not only on its induced (unoriented) varifold. Große-Brauckmann [GB93] gives an example of (obviously non-compact) surfaces of constant mean curvature converging to a doubly covered plane. This demonstrates that, unlike the Willmore energy, the Helfrich energy need not be lower semi-continuous under varifold convergence for certain parameters.
Recently, existence of minimisers for certain Helfrich-type energies among axially symmetric surfaces under an isoperimetric constraint was proved by Choksi and Veneroni [CV13]. Lower semi-continuity for the Helfrich functional on -boundaries with respect to the -topology of the enclosed sets was established by means of Gauss graphs in [Del97].
Results for the gradient flow of the Willmore functional are still few. Short time existence for sufficiently smooth initial data has been shown in [Sim01, KS02] (see also [MS03]) and long time existence for small initial energy and convergence to a round sphere has been demonstrated in [KS01, KS12]. Kuwert and Schätzle’s lower bound on existence times in terms of initial curvature concentration in space has been generalised to Willmore flow in Riemannian manifolds of bounded geometry in [Lin13]. It has been shown that Willmore flow can drive smooth initial surfaces to self-intersections in finite time in [MS03]. This issue seems to be prevented by our connectedness functional on the phase field level, although we do not have a rigorous statement on this. Numerical simulations suggest that singularities can occur in Willmore flow in finite time [MS02]. [DW07, Figure 2] gives a numerical example of a disc pinching off to a torus. A level set approach to Willmore flow is discussed in [DR04].
Studies of numerical implementations of Willmore flow are, for example, due to Garcke, Dziuk, Elliott et al. in [BGN08, Dzi08, DE07]. Particularly interesting here is also an implementation of a two-step time-discretisation algorithm due to Rumpf and Balzani [BR12].
Phase field approximations of the functional in (1.1), on the other hand, often provide a more convenient approach to gradient flows or minimisation of the Willmore or Helfrich functionals. In particular if a coupling of the surface to a bulk term is desired, phase field models can provide an excellent alternative to a parametric discretisation [Du10].
The idea for applying a phase field approach to Willmore’s energy goes back to De Giorgi [DG91]. For a slight modification of De Giorgi’s functional, reading
-convergence to the sum of Willmore’s energy and the -fold perimeter functional () was finally proved by Röger and Schätzle [RS06] in dimensions, providing the lower energy bound in the varifold context, after Bellettini and Paolini had provided the recovery sequence over a decade earlier [BP93]. First analytic evidence had previously been presented in the form of asymptotic expansions in [DLRW05]. This and other phase field approximations of Willmore’s energy and their -gradient flows are reviewed in [BMO13]. A convergence result for a diffuse approximation of certain more general Helfrich-type functionals has also been derived by Belletini and Mugnai [BM10].
Regarding implementation of phase field models for Willmore’s and Helfrich’s energy, we refer to the work by Misbah et al. in [BKM05] and Du et al. [DLRW05, DLW05, DLW06, DW07, DLRW07, DLRW09, Du10, WD07]. In [FRW13], the two-step algorithm for surface evolution has been extended to phase fields. A numerical implementation of the Helfrich functional can be found, for example, in the work by Campelo and Hernandez-Machado [CHM06].
2. Phase Fields
2.1. General Background
A phase field approach is a method which lifts problems of -dimensional manifolds which are the boundaries of sets to problems of scalar fields on -dimensional space. Namely, instead of looking at , we study smooth approximations of . Such models are generally based on a competition between a multi-well functional penalising deviation of the phase field function from the minima (usually instead of the characteristic function values , ) and a gradient-penalising term preventing an overly sharp transition. Fix , open. Our model space is
If , this can be identified with , otherwise the first formulation is technically easier to work with. The boundary conditions are chosen to model sets which are totally contained in and whose boundaries may only touch tangentially. Then (as well as in more general settings) it is known [Mod87] that the Modica-Mortola or Cahn-Hilliard functionals
with and , -converge to
with respect to strong -convergence for functions (and else). The limit agrees with the perimeter functional . The double-well potential forces sequences of bounded energy to converge to a function of the above form as . By Young’s inequality we can control the -norms of through
So converges strongly in for all for some by the compactness theorem for -functions (up to a subsequence), so in particular pointwise almost everywhere (further subsequence). , so weakly in and pointwise a.e. which together implies that strongly in for all . The functional has originally been studied in the context of minimal surfaces, see e.g. [Mod87, CC95, CC06]. We will view them in a slightly different light. If is a finite energy sequence, we denote the Radon measure given by the approximations of the area functional as
Taking the first variation of this measure and integrating by parts, we get
is a diffuse analogue of the mean curvature as the variation of surface area. A modified version of a conjecture by De Giorgi then proposes that squaring , integrating over and dividing by should be a -convergent approximation of for -regular sets , again with respect to strong -convergence. Dividing by is needed to convert the volume integral into a diffuse surface integral over the transition layer with width of order . This is true at least in dimensions [RS06]. From now on, we will restrict ourselves to these dimensions. We thus introduce the diffuse Willmore functional
and the Radon measures associated with a sequence with bounded energy
Since the measures are uniformly bounded, we can invoke the compactness theorem for Radon measures to see that there are finite Radon measures with support in such that, up to a subsequence,
A finer comparison shows that . Clearly by construction for some , and from the above argument is not difficult to see that is a Caccioppoli set. In three dimensions, however, may well be empty. To see this, one can use the construction of [MR14], where two concentric spheres of very similar radius are connected by a modified catenoid. As the catenoid is a minimal surface, this keeps the Willmore energy bounded. Taking the radii to converge to the same quantity and approximating them with the recovery sequence of a phase field, we see that is the -limit of the enclosed set between the two original sphere, so empty.
In [RS06] it has been shown that is in fact the mass measure of an integral -varifold with and for the generalised mean curvature of . Unlike , can still behave wildly.
We aim to minimise the Willmore energy among connected surfaces with given area . To prescribe area , we include a penalisation term of
in the energy functional. So we have a good approximation of Willmore’s energy available, the area constraint can be approximated through an easy penalisation, and confinement to comes automatically from the choice of function space . The main challenge left is controlling the connectedness of which we interpret as a sharp interface membrane.
2.2. Equipartition of Energy
Intuitively, the Cahn-Hilliard energy prefers phase fields to be close to in phase and make transitions between the phases along layers of width proportional to . In particular, to attain the minimum, equality is needed in Young’s inequality on most of . So it makes sense that the terms and should make an equal contribution to the measure . The difference of their contributions is controlled by the discrepancy measures
their positive parts and their total variation measures . We will see that for a suitable recovery sequence, vanishes exponentially in ; more generally the discrepancy goes to zero in general sequences along which stays bounded as shown in [RS06, Propositions 4.4, 4.9]. The result has been improved along subsequences in [BM10, Theorem 4.6] to -convergence for and convergence of Radon measures for the gradients of the discrepancy densities for an approximation of Helfrich’s energy.
The control of these measures is extremely important since they appear in the derivative of the diffuse -densities with respect to the radius .
In order to ensure that the support of the limiting measure is connected we include an auxiliary term in the energy functional. The heuristic idea behind this is that if the support of the limiting measure is connected, then so should the set for . These level sets away from can be heuristically viewed as approximations of , and in other situations, they can be seen to Hausdorff-converge to it [CC95]. This is not quite true in our situation (in three dimensions, see Remark 2.8), but the intuition still holds.
Our concept is to introduce a quantitative notion of path-connectedness and penalise the measured disconnectedness. Take a weight function such that
and the associated geodesic distance
In particular, is not excluded. If is connected, we can connect any two points by a curve of length zero. If it is not, then gives a quantitative notion of how badly path-connectedness fails between these two points. To obtain a global notion, we take a second weight function resembling a bump, i.e.,
and take the double integral
So, if is connected, we can connect any two points such that with a curve of length zero, hence and both the integrand and the double integral vanish.
If on the other hand is disconnected, then we expect that should be able to discern different connected components such that . The core part of our proof is concerned with precisely that. We need to show that detects components of the interface and that distinguishes them. For the first result, we need to understand the structure of the interfaces converging to and make sure they cannot be so steep in that the double integral does not see them in the limit.
For the second part, the challenge is to understand how phase fields converge away from the interface. It is clear that they converge pointwise almost everywhere, but it is not a priori clear whether they can stay away from along a curve in three dimensions. Namely, if the support of a limiting measure of had two components , but the approximating sequence satisfied in a neighbourhood of a curve connecting and , then could not distinguish between the components.
In two ambient dimensions, this is excluded relatively easily. In three-dimensional space however, the curve would not be seen by the limiting measure , which is an integral -varifold. Thus we need to show that in analogy to the sharp interface, a long and narrow tunnel-like strucure around blows up in diffuse Willmore energy. We like to think of this as a convergence localised on curves, i.e. sets of co-dimension .
2.4. Main Results
We have the following results for phase fields with bounded Willmore energy and perimeter. In the following we assume that denotes the continuous representative.
Let and a sequence such that
Denote for a subsequence along which converge weakly as Radon measures to a limit and take any . Then uniformly on .
This result has already been proved by Nagase and Tonegawa in [NT07], where they also show a -inequality for in two space dimensions. The use of a monotonicity formula in their proof resembles ours with differences in the estimate on the discrepancy measures. We show our proof here to illustrate some ideas in the more involved argument for .
Let and . Then in the Hausdorff distance. If , up to a subsequence, Hausdorff-converges to a set which contains .
In three dimensions, is possible, see Remark 2.8. As an illustration, the following statement describes the convergence of phase fields on lower dimensional objects that we need. In the proofs a slightly different version of the theorem is used to account for changing curves as .
Let and a sequence such that
Denote for a subsequence along which converges weakly as Radon measures to a limit. Take any such that . Then converges -almost everywhere.
For our application to connectedness, we define the total energy of an -phase field as
If , existence of mininimisers for the functional is a simple exercise in the direct method of the calculus of variations, since uniform convergence of a minimising sequence for fixed guarantees convergence of the distance term.
Let be a sequence such that for some and , Radon measures such that , . If is disconnected, then
Using these results, we can show the following.
Let and a sequence such that . Then the diffuse mass measures converge weakly* to a measure with connected support and area .
Let , , and , with smooth boundary with area . Then
Here we have adopted the notation of [RS06] where -convergence is said to hold at a point if the - and -inequalities hold at that point. This distinction is necessary since it is not clear what -convergence properties hold at other points since does not converge to the -lower semi-continuous envelope of at non-embedded sets even in two dimensions.
This issue stems from the fact that the stationary Allen-Cahn equation admits global saddle-solutions which have vanishing sets asymptotic to a union of lines. The best known example vanishes along the coordinate axes and is positive in the first and third quadrants and negative in the second and fourth ones, see [dPKPW10]. These solutions can be used to approximate transversal crossings with zero Willmore energy, while the lower semi-continuous envelope of Willmore’s energy/Euler’s elastica energy becomes infinite at those points. Interestingly enough, the crossing has zero Willmore-energy as a varifold, so that the energy can still be justified, despite the fact that it does not give the sensible result in our situation.
While heuristic considerations suggest – and numerical simulations appear to confirm – that at least in dimensions and when , our additional term in the energy might prevent saddles, we do not investigate convergence at non-embedded points further.
Uniform convergence generally fails in three dimensions. This can be seen by taking a sequence as the optimal interface approximation which will be given in the proof of Corollary 2.7 and adding to it a perturbation for any . It can easily be verified that the energy will remain finite (at least if ) but clearly the functions do not converge uniformly.
Heuristic arguments show that for modifications , the coefficients should be optimal in three dimensions and , if , so we expect a convergence rate of in the two-dimensional case.
Since uniform convergence does not hold, it follows that the pre-images of certain level sets can have additional points in the Hausdorff limit.
Since is uniformly bounded, a standard argument for the Modica-Mortola model shows that in for . We are going to show in Lemma 3.1 that the sequence is uniformly bounded in when additionally , which implies -convergence for all .
The proofs are organised in the following way. First, anticipating the results of section 3.4, we show -convergence of to . We decided to move the proof to the beginning since it is virtually independent of all the other proofs in this article and can be isolated, but introduces the optimal interface transitions which will be needed later. After that, we proceed chronologically with technical Lemmata (sections 3.2 and 3.3) and the proof of Theorems 2.1, 2.3, 2.2, 2.5 and 2.6 in section 3.4.
3.1. Proof of -Convergence
We now proceed to prove Corollary 2.7.
Proof of the -inequality:.
Proof of the -inequality:.
We may restrict our analysis to the case of connected boundaries with area . In this proof, we will construct a sequence such that
The construction is standard and holds for arbitrary . We take the solution of the one-dimensional problem and apply it to the signed distance function of , thus recreating the shape of with an (approximate) optimal profile for the transition from to . For consider
Since , for all sufficiently small , and since is embedded, there is such that
is a diffeomorphism. We have the closest point projection and the signed distance function from which is positive inside and negative outside . Both are -smooth on and they satisfy and (see e.g. [GT01, Section 14.6])
where is a constant depending on the base projection point . Let us consider the optimal transition between and in one dimension. This optimal profile is a stationary point of , i.e. a solution of (and thus a zero energy point of ) with the side conditions that . Note that the optimal profile satisfies
so we find that . We look for transitions where both and are integrable over the whole real line, and since , we see that and . This gives us equipartition of energy already before integration. For simplicity, we focus on which has the optimal interface . Note that the functional rescales appropriately under dilations of the parameter space so that we have equipartition of energy before integration also in the -problem. The disadvantage of the hyperbolic tangent is that it makes the transition between the roots of only in infinite space, so we choose to work with approximations such that
for all .
Such a function is easily found as converges to exponentially fast. Then we set
A direct calculation establishes that with this choice of , we have for all , for all and . It remains to show that . We will show that even along this sequence. Since is connected and is a diffeomorphism, all the level sets
are connected manifolds for and small enough such that around . We know that
and pick any . Now let , . We can construct a curve from to by setting piecewise
and any curve connecting to in . This curve exists since connected manifolds are path-connected. The curve connects and and satisfies by construction , so . Therefore we deduce
if since the connecting curves have uniformly bounded length and . Thus in particular
3.2. Auxiliary Results I
A lot of our proofs will be inspired by [RS06] which again draws from [HT00]. We will generally cite [RS06] because it treats the more relevant case for our study. In this section, we will prove a number of auxiliary results which concern either general properties of phase fields or properties away from the support of the limiting measure which will enable us to investigate their convergence later. Results concerning phase interfaces are postponed until section 3.3.
We start with an optimal regularity lemma for phase fields on small balls. This is an improvement upon [RS06, Proposition 3.6] where in three dimensions, only boundedness in for finite and a slow growth in could be obtained. The new -bound also implies local Hölder continuity which is used extensively later.
Let , and such that . Then there exists such that for all
where is a constant depending only on and .
is -Hölder continuous on -balls, i.e.
Again, the constant depends only on and .
We will argue using Sobolev embeddings for blow ups of onto the natural length scale. In the first step, we show the set to be small. In the second step, we estimate the -norm of the blow ups, in the third we estimate the full -norm and use suitable embedding theorems to conclude the proof of regularity.
Step 1. First observe that
using that for . The boundary integral vanishes if is of finite perimeter since is continuous and . If this is not the case, take converging from above such that is of finite perimeter. This holds for almost all . The sign of the boundary integral can be determined since on the boundary of and for so that the same inequality can still be established. By symmetry, the same argument works for .
Step 2. Let be an arbitrary sequence and define the blow up sequence by
Then we observe that
As usual, denotes the volume of the -dimensional unit ball. In exactly the same way with a slightly simpler argument we obtain
Step 3. Now a direct calculation shows that