The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues, the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity we build on the theory of Goldman, Chambolle and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the $L^2$ gradient flow is trivial, i.e. there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed by Novaga and Valdinoci for forced mean curvature flow.

1. Introduction 1.1. Sharp Interface Models. The primary focus of the paper is on the analysis of sharp and diffuse heterogeneous surface energy functionals. We start the exposition introducing the sharp interface energy on subsets S ⊂ R d where U ⊂ R d is an open bounded domain and a is Z d periodic and 1 ≤ a(x) ≤ Λ. The energy is well defined on locally finite perimeter subsets of R d , and can also be made sense of on a closed set with finite perimeter. The quantity of interest in this case is the effective surface tension which can be defined for each normal n bȳ σ(n, a) = lim In this note we show that the presence of gradient discontinuities at all rational directions appears not only in specially constructed examples, but is a generic feature in the topological sense. Theorem 1.1. For each n ∈ S d−1 , the sets of coefficients A n ⊂ C ∞ (T d ; [1, Λ]) for which the associated surface tensionσ has a gradient discontinuity at n is open and dense in the topologies induced by C(T d ; [1, Λ]) and W 1,p (T d ; [1, Λ]) for d < p < +∞. Furthermore, ∩ n∈S d−1 A n is also dense in those topologies.
This result will be proved below in Section 5. Note that the coefficients a in the theorem are necessarily not laminar. Indeed, [14] shows that the surface tension is necessarily C 1 in some non-empty open set of S d−1 when the underlying medium is laminar. From that point of view, the theorem shows that the general non-laminar case can be much less regular than the laminar setting.
Associated with this energy is the heterogeneous curvature flow of an evolving set S t where n is the outward normal to ∂S t , V n is the outward normal velocity of ∂S t , and κ is the mean curvature oriented so that convex S has κ ≥ 0.
In this case we are interested in the limiting behavior of the rescaled curvature flow of S ε t starting from some compact initial data S 0 , By analogy with what has been proved in related models (cf. [9], [10], [36]), one would expect that the limiting equation would be V n = −µ(n) div ∂S (σ(n)) where the additional anisotropic term µ : S d−1 → [0, ∞) is the mobility, the infinitesimal velocity of the system induced by additive forcing (see below for more details). At the very best, based on the examples ofσ, we are looking at something like a crystalline curvature flow. However the situation is more delicate than even this.
To start with, the construction of Theorem 1.1 also implies a certain kind of pathology at the level of the gradient flow, which we state next using the language of level set PDE for convenience: There is a family of coefficients F ⊂ C ∞ (T d ; [1, Λ]), which is dense in C(T d ; [1, Λ]), such that if a ∈ F, then the following statement holds: for each u 0 ∈ U C(R d ), if (u ε ) ε>0 are the solutions of the level set PDE (4) See Section 5 for the proof. The point is that wherever the front recedes some areas of the positive phase are left behind, and wherever the front advances some areas of the negative phase remain. We refer to this phenomenon as bubbling. Bubbling is well known to occur in these kinds of interface motions, see Cardaliaguet, Lions, and Souganidis [13], the only part which is possibly new about Corollary 1.2 is that the set of coefficients for which it holds is dense. Despite this somewhat pathological behavior, it is still conceivable that the "bulk" of the front moves by a limiting curvature flow. Simply instead of being a transition between the +1 and −1 phases it is actually a transition between some more complicated + and − phases that include the bubbles left behind by the bulk of the moving front. Remark 1.3. Note that we do not prove topological genericity in Corollary 1.2, only density. The set of coefficients F which we construct is dense but not open in C(T d ; [1, Λ]) and open but not dense in C 1 (T d ; [1, Λ]). It would be interesting to know whether the occurrence of gaps in the lamination by strong Birkhoff plane-like solutions at every lattice direction, which occurs generically in the uniform topology by Theorem 1.1, directly implies bubbling as in Corollary 1.2.
1.2. Interface pinning. In fact, it can happen that the effective dynamics are in a sense "worse" than this, the entire front can be pinned not only some compact bubbles. Through the construction of a specific class of examples in dimension d = 2, we show that it is possible that the mobility µ(n) ≡ 0, meaning the scaling limit is trivial. This is exactly the phenomenon known as pinning, which occurs ubiquitously in problems involving interface motion in heterogeneous media.
Discussing formally we explain the so-called Einstein relation [24,32,44], which identifies the friction term in the effective diffusivity as the mobility, the infinitesimal response of the system to an external volume forcing. Consider the solution S t (n) of the forced mean curvature flow for a constant large scale forcing F ∈ R with planar initial data (5) V n = −a(x)κ − Da(x) · n + F with S 0 = {x · n ≤ 0}.
Then associated with each propagation direction n there are minimal and maximal asymptotic speeds c * (n, F ) = lim x · n and c * (n, F ) = lim x · n which may not be the same. It is not difficult to check that both are monotone nondecreasing in F and c * (n, 0) = c * (n, 0) = 0. Ignoring, for now, the possibility that the two asymptotic propagation speeds do not agree in some cases we define c(n, F ) = c * (n, F ) = c * (n, F ) and then define the mobility µ(n) by Although both quantities are strictly monotone in the set where they are non-zero, there can occur a nontrivial pinning interval [−F * (n), F * (n)] 0 so that c * (n, F ) = 0 for F ≤ F * (n) and c * (n, F ) = 0 for F ≥ F * (n).
In this case µ(n) = 0, possibly at every direction n ∈ S d−1 . This is a wellknown phenomenon which occurs in many related models of interface propagation in heterogeneous media. See for example [20,21,25,26,39]. Physical intuition suggests that the existence of a nontrivial pinning interval at every direction is, in some sense, a generic feature. Again, special assumptions (e.g. laminarity) on the medium may produce positive mobility at some directions. Here we prove that there exists a medium in dimension d = 2 with nontrivial pinning interval at every direction, actually we make an even stronger statement. Theorem 1.4. There is a medium a ∈ C ∞ (T 2 ; [1, Λ]) and an F a > 0 such that, for each u 0 ∈ U C(R 2 ), if (u ε ) ε>0 are the solutions of the level set PDE associated with (3) and the initial datum u 0 and forcing F ∈ (−F a , F a ), that is, if they are the viscosity solutions of the equations Furthermore, given any ζ > 0, we can choose a so that a − 1 L ∞ (T 2 ) ≤ ζ.
Taking u 0 to be a linear function in the previous theorem, we see that a has a non-trivial pinning interval: Corollary 1.5. If a and F a are as in Theorem 1.4, then c * (n, F ) and c * (n, F ) as defined above are well-defined for each F ∈ (−F a , F a ) and n ∈ S 1 , and c * (n, F ) = c * (n, F ) = 0.
The proof of Theorem 1.4 is based on the construction of a medium a such that stationary strict supersolutions are plentiful. The following lemma is the main technical result used in its proof: Lemma 1.6. There is a medium a ∈ C ∞ (T 2 ; [1, Λ]), and a (numerical) constant C > 0 such that if K ⊂ R 2 satisfies an interior and exterior ball condition with large enough radius, then there is a strict stationary supersolution S of (2) such that {x ∈ K | dist(x, ∂K) ≥ C} ⊂ S, S ⊂ K + B C (0), d H (∂K, ∂S) ≤ C.
In terms of the evolution equation (3) on regions of R 2 , the lemma immediately implies a quantitative pinning result for C 2 compact sets in the homogenization limit: Corollary 1.7. If a is the medium of Lemma 1.6, K is any compact subset of R 2 with C 2 boundary, and S ε t is any solution of (8) V n = −a( x ε )κ − 1 ε Da( x ε ) · n + 1 ε F. with initial data K and forcing F ∈ (−F a , F a ), then there is an ε 0 (K) > 0 such that, for each ε ∈ (0, ε 0 (K)), All of the above sequence of results will be proved in Section 3.
Remark 1.8. The construction in Lemma 1.6 is stable with respect to uniform norm perturbations of a and Da so we can also conclude that µ(·) ≡ 0 on an open subset of a ∈ C 1 (T 2 ; [1, Λ]). It is possible that there is also an open subset of C 1 (T 2 ; [1, Λ]) on which there is no pinning at any direction, although we have no evidence to suggest such a set of data exists. The only explicitly understood case is that of laminar media of the form a(x) =ã(x · n) for some n ∈ S 1 , where the mobility is always zero at the laminar direction unless the medium is homogeneous, that property is again stable with respect to small perturbations in the C 1 (T 2 ; [1, Λ]) norm.
1.3. Diffuse Interface Models. In the diffuse interface setting, we obtain results analogous to those in the sharp interface model by exploiting the close connection between the two models as the diffuse interface width vanishes. To be more precise, we consider diffuse interface functionals AC δ θ defined on configurations u : R d → R of the form (9) AC δ θ (u; U ) =ˆU where θ is a Z d -periodic function satisfying 1 ≤ θ ≤ Λ 2 , W : R → [0, ∞) is a doublewell potential with {W = 0} = {−1, 1} satisfying standard assumptions (see (27), (28), (29) below), and δ > 0 is a parameter that, roughly speaking, encodes the typical width of a (minimizing) diffuse interface. In summary, the next results show that when δ is small enough, relative to the C 1 (T d ) norm of θ, F δ θ and its L 2 gradient flow exhibit the same large scale behavior as E √ θ and its flow. At equilibrium, the large scale behavior of F δ θ is described by a homogenized surface tension, as in the sharp interface case. The following formula for the effective surface tension can be derived Note that the boundary data is just the plane separation data χ {n·x≤0} − χ {n·x>0} but smoothed out at the unit scale to avoid technical difficulties associated with the discontinuous boundary condition. We expect the sub-differential of the effective surface tension to be characterized by the geometry of the plane-like minimizers of AC δ θ , just as in the sharp interface case. However, this has not yet been proved. Toward that end, we expect the next result will be of interest.
such that if θ ∈ G, then there is an open set I(θ) ⊂ (0, 1) with 0 ∈ I(θ) such that if δ ∈ I(θ), then the following statements hold: (i) For each n ∈ S d−1 , the family M δ θ (n) of strongly Birkhoff plane-like minimizers of AC δ θ in the n direction has gaps.
The statements above remain true if C(T d ; [1, Λ]) is replaced by W 1,p (T d ; [1, Λ]) for p ∈ (d, ∞). See Section 6 for the proof. Although the parameter δ is useful for the statements of our theorems, it is cumbersome for the following informal discussion, so we set δ = 1 for the next paragraph. The L 2 gradient flow of the Allen-Cahn energy functional is well known to be (10) u Considering the long time behavior of (10) in the parabolic scaling leads to the equation By analogy with what has been proved in related models (cf. [9], [10], [36]), one would expect that, in the limit ε → 0, the interface between the positive and negative phases evolves by a curvature flow with the mobility µ AC : S d−1 → [0, ∞), as before, being the infinitesimal response to additive forcing. We will show (see Theorem 1.10 below) that, on an open set of coefficients in C(T 2 ), this homogenization limit holds but results in a trivial flow µ AC (n) ≡ 0.
In fact, as in the sharp interface case, we give examples of coefficients θ for which the pinning interval associated to the gradient flow dynamics is uniformly bounded below with respect to the direction, i.e. the mobility is zero at every direction, and the homogenization scaling limit (11) results in a trivial flow even when an external force is added.
Remark 1.11. It is not hard to check that this result is stable with respect to small perturbations of the coefficients in the uniform norm. Thus we can say that this pinning phenomenon is not non-generic in the coefficient space C(T 2 ; [1, Λ 2 ]).
Remark 1.12. In the theorem, the reaction term θ(x)(W (u) − F ) appearing in (12) satisfies Thus, the result is a manifestation of the front-blocking phenomenon in the study of bistable reaction diffusion equations (cf. Lewis and Keener [33]). In fact, we show below that, for θ ∈ O, (12) has stationary, plane-like solutions for all forcing values F in this interval (see Remark 4.2).
Remark 1.13. Our arguments also apply to diffuse interface energies where the heterogeneity appears on the gradient rather than the potential term, as well as those where it multiplies both. That is, the same results apply to energies of the formˆU For more details, see Remarks 4.5 and 6.5 below.
1.4. Literature. The study of variational models in periodic media falls under the broad umbrella of Aubry-Mather theory, named after the fundamental contributions of Aubry and LeDaeron [2] and Mather [35], who investigated the (discrete) Frenkel-Kontorova model and twist maps. In Aubry-Mather theory, one of the main questions is the existence and structure of "plane-like" minimizers and its relation to the large-scale (or homogenized) behavior of the energy itself. The investigation of continuum models via PDE methods was initiated by Moser [38] with the fundamental structural theorems contributed by Bangert [3,4].
The results of Moser and Bangert concern graphical energies modeled on the Dirichlet energy. In more recent years, variational problems with more of a geometrical flavor have been shown to possess the same basic structure. Caffarelli and de la Llave [12] extended the basic existence results of Aubry-Mather theory to surface energies like those considered here. There has also been considerable interest in diffuse interface energies, including contributions by many authors. For references and connections to the work of Moser and Bangert, see the book of Rabinowitz and Stredulinsky [40] and the expository paper by Junginger-Gestrich and Valdinoci [30].
Chambolle, Goldman, Novaga [14] studied the effective energy for the sharp interface model, giving a precise characterization of the differentiability properties in terms of the existence (or not) of gaps in the corresponding laminations by plane-like solutions. They also gave specific examples where the effective surface tension has gradient discontinuities at every direction satisfying a rational relation. Ruggiero [42] and Pacheco and Ruggiero [41] showed that media with gaps in the lamination at every direction are residual (i.e. they form a dense G δ set) in two dimensions in the C 1 and C 1,β norms, respectively. Our result Theorem 1.1 shows that the gap phenomenon is residual in higher dimensions as well, at least for the rational directions, but only in the uniform norm. It would be interesting to obtain a similar result for all directions satisfying a rational relation or, even better, all directions, and in stronger topologies.
An analogous connection between gaps in the laminations by plane-like solutions and surface tension regularity has not yet been established in the case of diffuse interfaces, see [15,36] for partial results in this direction.
The front bubbling phenomenon, as discussed in Corollary 1.2, has also been known for some time, examples were constructed for forced mean curvature flow by Dirr,Karali and Yip [20] and by Cardaliaguet, Lions and Souganidis [13]. Novaga and Valdinoci [39], in the setting of the forced mean curvature flow with homogeneous perimeter, have shown that bubbling as in Corollary 1.2 occurs generically with respect to the L 1 distance on the coefficient field in dimension 2. Note that this type of genericity is quite similar to our result because their forcing term corresponds to Da in our setting.
Front pinning is another well known and fundamental feature of interface propagation in heterogeneous media, and has been studied for many related models in both periodic and random media [8, 16, 19, 21-23, 25, 31]. In the reaction diffusion literature this is referred to as wave blocking [28,33]. Examples of front pinning have been constructed for various models in both periodic and random media: for the forced quenched Edwards-Wilkinson equation, Dirr and Yip [21] have shown that pinning is generic and they have also constructed pinning examples for forced Allen-Cahn (homogeneous energy, but heterogeneous volume forcing) in one dimension. The first author [25] gave examples of front pinning at every direction for the Bernoulli free boundary problem in heterogeneous media. Novaga and Valdinoci [39], which showed a similar result to our Corollary 1.2 in the context of forced mean curvature flow (homogeneous surface energy, but heterogeneous volume forcing), does not explicitly give an example of pinning of the entire interface (as in our Corollary 1.7), we believe that a small modification of their ideas would also yield an example of pinning in 2-d. We also were recently made aware of a paper by Courte, Dondl and Ortiz [16] which considers a curvature driven motion with dry friction in random media with sparse obstacles. They show the occurrence of additional pinning by the Poissonian obstacles and establish the precise scaling exponent of the additional pinning in the sparse obstacle limit. Their fundamental barrier construction bears significant similarity to ours, patching together barrier pieces near concentrated obstacles (the example of [25,Section 5.2] is also similar, but patching barriers is easier due to the particular nature of that problem).
In the context of bistable reaction diffusion equations in one-dimensional periodic media, Xin [45] and Ding, Hamel, and Zhao [18] have constructed unbalanced reaction terms for which one nonetheless finds plane-like stationary solutions, a phenomenon associated with pinning. Our results provide (non-laminar) examples of this in two dimensions.
1.5. Organization of the paper. We begin in Section 2 with some background on viscosity solution theory of interface motions and related concepts. Next, in Section 3, we construct a special class of "highly pinning" media and use the construction to prove Theorem 1.4, Corollary 1.5, Lemma 1.6, and Corollary 1.7. In Section 4, we prove Theorem 1.10, obtaining analogous interface pinning results for diffuse interface models. In Section 5, we prove Theorem 1.1 and Corollary 1.2 on the genericity of surface tension gradient discontinuities for sharp interface models. In Section 6, we prove Theorem 1.9, the analogous result on genericity of surface tension singularities for diffuse interface models.

Preliminaries
2.1. Viscosity Solutions and Level-Set Formulation. Given a positive periodic function a ∈ C 1 (T d ) and a force F ∈ R, we are interested in sets moving with normal velocity given by (15) V n = −a(x)κ − Da(x) · n + F.
We will use the level-set formulation, which amounts to studying the following nonlinear PDE: Above Du is a shorthand for the normal vector Du = Du −1 Du. The level-set formulation allows one to define (weak) solutions of (15) using viscosity solutions of (16). Very roughly speaking, we say that a family of sets (S t ) t≥0 is a viscosity solution of (15) if the characteristic function u(x, t) = χ St (x) 1 determines a viscosity solution of (16). We will not make this completely precise here since the main technical constructions of the paper only require time-stationary solutions. For a full account of the level-set theory, see the original survey articles [9] and [6] and the textbook [27].

2.2.
Stationary Viscosity Solutions. The basic plan of attack of the article is the construction of (time-stationary) sets that act as barriers of the evolution, that is, we are interested in sets solving the equation (17) − a(x)κ − Da(x) · n + F = 0 or, equivalently, functions u solving the PDE This is made precise next. To simplify the notation here and in what follows, we will define the mean-curvature operators MC * , MC * : We begin by defining precisely what it means for a function to be a time-independent viscosity super-or subsolution of (16).
Given an open set U ⊂ R d , we say that a lower (resp. upper) semi-continuous function u : U → R is a viscosity supersolution (resp. subsolution) of (18) if the following property holds: given any point x 0 ∈ U and any smooth function ψ defined in a neighborhood of x 0 , if u − ψ has a local minimum (resp. maximum) at x 0 , then As an abbreviation, we write that −a(x)MC * (Du, D 2 u)−Da(x)·Du−F Du ≥ 0 holds in U if u is a viscosity supersolution of (18) in U . A similar convention will be used for subsolutions.
Roughly speaking, a set S ⊂ R d is now a subsolution or supersolution of (15) precisely when its characteristic function χ S is a viscosity subsolution or supersolution as above. The precise definition is given next.
Given an open set U ⊂ R d and a closed set S ⊂ U , we say that S is a supersolution of (17) in U if the characteristic function u = χ Int(S) is a viscosity super-solution of (18) Similarly, given an open set U ⊂ R d and an open set V ⊂ U , V is called a subsolution of (17) if the function v = χ V is a viscosity subsolution of (18) in U .
Finally, if S ⊂ R d is such that the closure S is a supersolution of (17) and the interior Int(S) is a subsolution of (17), then we say that S is a solution of (17).
It is useful to bear in mind that, in the case of sets, the differential inequalities involved in the definition of viscosity sub-and supersolution only need to be checked on the boundary.
Proof. The "only if" is immediate. To see that the "if" direction holds, recall that S = ∂S ∪ Int(S). Further, if x 0 ∈ Int(S), then χ Int(S) ≡ 1 in a neighborhood of x 0 and then the desired differential inequality follows directly from the second derivative test.
Finally, we make frequent use of the fact that the minimum of two supersolutions is once again a supersolution. The next result states this in the context of sets.

Proposition 2.4.
Given an open set U ⊂ R d , if the closed sets S 1 , S 2 ⊂ R d are supersolutions of (17) in U , then S 1 ∩ S 2 is also a supersolution in U .

Diffuse Interfaces.
In this section, we briefly provide definitions of solutions of our diffuse interface evolution equations. Recall that the equation of interest, after scaling out the and δ variables and restriction to time-stationary solutions, is As in the last section, we will only define time-stationary sub-and supersolutions since those are all that is needed for our main technical results. (For relevant definitions for the time-dependent equation, see, e.g., [17].) Given an open set U ⊂ R d , we say that a lower (resp. upper) semi-continuous function u : U → R is a viscosity supersolution (resp. subsolution) of (19) in U if, for any x 0 ∈ U and any smooth function ψ defined in a neighborhood of x 0 , if u − ψ has a local minimum (resp. maximum) at x 0 , then −∆ψ(x 0 ) + θ(x 0 ) (W (u(x 0 )) − F ) ≥ 0 (resp. ≤ 0).

2.4.
Half-relaxed Limits. We next recall the definition of half-relaxed limits, a notion that has been used extensively in the theory of viscosity solutions since it was introduced by Barles and Perthame [5].
Definition 2.6. Given a family of functions (u ) >0 in R d , the upper and lower half-relaxed limits, lim sup * u and lim inf * u , are the functions in R d defined by See, for instance, the notes by Barles [7] for properties of half-relaxed limits and some of their applications.

Sharp interfaces: medium with a non-trivial pinning interval at every direction
We will construct a medium a(x) ∈ C ∞ (T 2 ; [1, Λ]) which has a plane-like stationary solution S e of However, since we do not have any general theorem establishing the relationship between this mobility and the homogenization of (3), we prove a slightly more general result, stated above in Lemma 1.6 and Corollary 1.7: within a unit distance of any sufficiently regular set K ⊂ R 2 , there is a stationary solution of (20). The bulk of the work consists in proving the existence of certain sub-and supersolutions. Toward that end, here is a statement of the main result of this section: There is an R > 0 (a numerical constant) and an a ∈ C ∞ (T 2 ; [1, Λ]) for which there is an F a > 0 and a C > 0 such that the following holds: if K ⊂ R 2 satisfies an exterior and interior ball condition of radius R, then there is a supersolution S * (K) of (20) with F = F a and a subsolution S * (K) of (20) with F = −F a such that Furthermore, we can assume that S * (K) and S * (K) have piecewise smooth boundaries.
Once Lemma 3.1 is proved, we invoke a version of Perron's Method to establish the existence of solutions (Lemma 1.6). One could also construct a solution near K by constrained energy minimization and, as a result, find a pinned "local energy minimizer." While that is not the approach taken here, it is worth pointing out for the purpose of exposition that these pinned solutions are not just unstable energy critical points. Notice that the boundary of the supersolution consists of translated copies of certain basic curves, some of which are circular arcs connecting lattice points (called "edges" in what follows) and others that are simple closed curves surrounding one (called "nodes").
As the reader familiar with homogenization will likely realize, Lemma 3.1 implies homogenization of the gradient flow (Theorem 1.4). For completeness, the details are provided at the end of the section.
The construction of stationary solutions relies on an explicit construction of super and subsolutions -found below in Section 3.7. These two cases are symmetric so we will only need to handle constructing supersolutions. At a technical level this involves repeated patching together of supersolution pieces, in the d = 2 case we consider these are just concatenations of curves. However in service of a possible future generalization to higher dimensions, and to separate the arguments involving patching from the actual explicit construction, we will start by setting up a framework which handles the topological issues of the patching procedure. Additionally, since this patching will proceed using lattice cubes, we will use certain facts about sets consisting of unions of such cubes.
The idea of the construction is we show that it is possible to find a coefficient a together with certain curves that are stationary supersolutions of (2) on their boundaries. These curves are chosen so that they can be combined to bound any union of lattice cubes -see Figure 1 for the basic picture to keep in mind. The arguments showing that such an a can be found are in Section 3.7 -the reader may wish to start there -while the remainder of the section formalizes the construction.
3.1. Regular Z 2 * -measurable sets. The construction of sub-and supersolutions uses the fact that smooth subsets of R 2 admit nice discrete approximations. This leads us to define regular Z 2 * -measurable sets.
In what follows, Z 2 * is the dual lattice of Z 2 , that is, Z 2 * = Z 2 + (1/2, 1/2). This is a convenient way of indexing the lattice cubes {z + [−1/2, 1/2] 2 } z∈Z 2 * that will be used in our approximations of smooth sets. These approximations will consist of unions of such cubes, as in the next definition: The boundary of any Z 2 * -measurable set is a union of paths in a certain graph. This will be convenient in the formalism that follows. By the graph (Z 2 , E 2 ), we mean the set Z 2 ⊂ R 2 together with directed edges [x, y] consisting of the line segment connecting two points x, y ∈ Z 2 with x − y = 1. We identify [x, y] with the oriented line segment, oriented so that its tangent vector is parallel to y − x. The normal vector n [x,y] to this line segment is defined by rotating the tangent vector counter-clockwise (hence, e.g., n [(0,0),(1,0)] = (0, 1)).
In the discussion that follows, it will be useful to say that z, Given a Z 2 * -measurable set A corresponding to the points Z A ⊂ Z 2 * , we define the associated boundary cubes A b and interior cubes A int by We will restrict our attention to a particularly nice class of Z 2 * -measurable sets for which the boundary ∂A equals the image of simple paths in (Z 2 , E 2 ). Toward that end, the following definition will be convenient: and z is a wired neighbor of z, then there is a z ∈ Z 2 * such that z + [−1/2, 1/2] 2 ⊂ A and z is a neighbor of both z and z .
A topological argument proves that regular Z 2 * -measurable sets are determined by simple paths. Precisely, given an interval E ⊂ Z, we say that γ : we say that γ is closed. A path that is both simple and closed is called a simple closed path.
For convenience, we denote by {γ} ⊂ R 2 the path traced out by γ, that is, The next result shows that the boundary of a regular Z 2 * -measurable sets is the image of a disjoint union of simple paths in the graph (Z 2 , E 2 ). Later, the orientation of the paths will be important. Hence before stating the result, we define what it means for a path to traverse the boundary of a regular Z 2 * -measurable set in a clockwise fashion.
Given a Z 2 * -measurable set A and an edge [x, y] ∈ E 2 such that [x, y] ⊂ ∂A, we say that [x, y] traverses ∂A clockwise if n [x,y] is parallel to the outward normal vector to ∂A, or, more precisely, Otherwise Given a Z 2 * -measurable set A and an interval E ⊂ Z, we say that a path γ : and, for each j ∈ P , the path γ (j) traverses ∂A clockwise. Furthermore, the finite length paths in (γ (j) ) j∈P are all simple closed paths.
A sketch of the proof is given next; the details can be found in Appendix C.
Sketch of proof. Start at a boundary vertex x 0 ∈ ∂A ∩ Z 2 . By (ii) in the definition of regularity, there is a unique neighbor x 1 ∈ ∂A ∩ Z 2 of x 0 such that [x 0 , x 1 ] ⊂ ∂A and the (oriented) edge [x 0 , x 1 ] traverses ∂A clockwise. Apply this procedure again with x 1 replacing x 0 , furnishing a neighbor x 2 of x 1 so that [x 1 , x 2 ] traverses ∂A clockwise. Iterating this results in a path γ : N ∪ {0} → Z 2 given by γ(j) = x j . If the image {γ} is finite, then it is necessarily a simple closed path; otherwise, in the infinite case, it is a simple path.
If ∂A \ {γ} is nonempty, repeat the construction at a different boundary vertex. Since Z 2 is countable, this process eventually decomposes ∂A as a countable union of pairwise disjoint simple paths in Z 2 , as claimed.

3.2.
Abstract framework for patching super/subsolutions. We set up an abstract framework which is useful to compartmentalize arguments relating to patching together local smooth super/subsolutions to form a global super/subsolution. We consider only the two dimensional case, and we do not at all consider a fully general notion of admissible patching. We expect that, with significantly more topological work, these ideas could be generalized and would be useful for constructing examples in higher dimensions.
Since we will be combining sets that only satisfy the supersolution property locally, it will be convenient to track the domain together with the set. That is the purpose of the next definition. It will be convenient for us to impose some (but not too much) boundary regularity on the local supersolutions we work with. This is the purpose of the next definition of piecewise smooth local supersolution.  Given a local supersolution we call n S to be the outward normal to ∂S and τ S to be the corresponding tangent vector which is the outward normal rotated by 90 degrees clockwise. We say that a pair of local supersolutions (S 1 , If U is doubly connected (i.e., U is connected and R 2 \ U consists of exactly two connected components), we call ∂ out U to be the boundary of the unbounded component of the complement, ∂ in U to be the boundary of the bounded component of the complement, and we call fill(U ) to be the union of U with the bounded component of the complement. A piecewise smooth local supersolution (S, U ) is called a smooth patch if ∂S ∩ U equals the image of a single smooth curve.
As a basic building block, we will use so-called supersolution edges and supersolution nodes, defined next.  See Figure 2 for a graphic representation of a node and edge local supersolution. Now our goal is to define an appropriate notion of combining supersolutions. We begin with a patching operation that amounts to taking local intersections. We use the following notation, α∈I x∈Kα to avoid writing the more (respectively) unintuitive and lengthy formulae on the right.
Then patch((S e , U e ) e∈I ) := ( is a local supersolution in ∪ e∈I U e .
See Figure 3.2 for a visualization of the patch operation.
The idea of the condition in Lemma 3.8 is simply that a collection of supersolutions on several overlapping domains can be patched together by locally taking the minimum as long as each supersolution is not the minimal supersolution at any point of the boundary of its own domain (which is in the closure of one of the other domains).
Proof. Call (S * , U * ) = patch((S e , U e ) e∈I ). We just need to check, for any interior In that case, −χ S * is a minimum of supersolutions in N (x 0 ) and so it is a supersolution in N (x 0 ). More precisely, for a sufficiently small neighborhood N (x 0 ) of x 0 , we claim This is immediate unless x 0 ∈ ∂U e for some e.
In that case, call J = {e ∈ I : x 0 ∈ ∂U e }. For each e ∈ J, let V e be the relatively open set in U e such that x 0 ∈ V e and (21) holds. Note that we can fix an open ball N (x 0 ) containing x 0 such that the following hold: A direct argument involving (21) shows that Of course this patching procedure, does not require the supersolutions involved to be edges, however the hypothesis will typically not hold when one of the supersolutions involved is a node, see Figure 4. It is a bit more topologically delicate to explain how to join a pair of edges to a node. Toward that end, we begin by defining a criterion for the node and two edges to be admissible. Definition 3.9. Given a node (S, U ) and an edge (S , U ), we say that (S , U ) is consists of exactly two points, and the corresponding arc of ∂ out U separates U into two connected components, (ii) (∂S ∩ U ) ∩ ∂U consists of exactly two points, one in ∂ out U and the other, in ∂ in U . (In particular, ∂S contains a piecewise smooth path γ : If the path γ in (ii) is such that the velocityγ is parallel to the tangent vector τ S to ∂S , then we say that (S , U ) is incoming at (S, U ). Otherwise, ifγ is anti-parallel to τ S , we say that (S , U ) is outcoming at (S, U ). Figure 4. Two edges intersect a node, how to patch to get a supersolution. The darker shaded region is S join .
Given a node (S v , U v ) and a pair of disjoint incident edges (S e1 , U e1 ) and (S e2 , U e2 ), respectively incoming and outgoing, we now explain how to patch and define an edge supersolution (S join , U join ), where U join is defined by See Figure 4 for a graphical representation of the patching procedure; the figure will be used to describe S join . Let γ be the curve obtained by starting at point (I); traversing ∂S e1 ∩ U e1 until point (II); then traversing ∂S v counterclockwise until point (III); traversing ∂S e2 ∩ U e2 until point (IV); then following ∂U e2 clockwise to (V); following ∂U v to (VI); and, finally, proceeding to (I) along ∂U e1 clockwise. In other words, γ is the curve that bounds the dark gray shaded region in the figure. We let S join be the closure of the domain bounded by γ, i.e. the shaded region, which is well-defined by the Jordan Curve Theorem.
Note that although Figure 4 depicts a particular configuration of nodes and edges, the construction above makes sense whenever (S e1 , U e1 ) is incoming at The reason is the points (I), (II), (III), (IV), (V), and (VI) and the paths between them are well-defined due to the definitions. For instance, (II) is the unique point in (∂S e1 ∩U e1 )∩(∂S v ∩U v ), whose existence and uniqueness is guaranteed by Definition 3.9.
is a node and (S e1 , U e1 ), and (S e2 , U e2 ) are, respectively, incoming and outgoing edges admissibly incident on (S v , U v ), then the pair (S join , U join ) is a local supersolution edge.
Proof. First, to see that U join is simply connected, observe that it equals the bounded component of the simple closed curve obtained by starting at point (I) in Figure 4; proceeding clockwise around ∂U e1 until it first intersects ∂U v ; continuing around ∂U v until it first intersects ∂U e2 ; proceeding to (IV); then continuing on to (V), (VI), and back to (I). This follows from the definition of U join , Definition 3.9, and the Jordan Curve Theorem. Further, by construction, ∂S join ∩ U join equals the part of ∂S join that starts at point (I), then proceeds to (II) and (III) before ending at (IV). Parametrizing this path as γ : Finally, by Propositions 2.3 and 2.4, to prove that S join is a supersolution in U join , it suffices to verify that, for any We now show how to use the supersolution patching procedure to produce supersolutions that approximate regular cube sets. To abstract away some of the details, we start by defining a type of network that will allow us to associate a local supersolution to each edge and vertex of the graph (Z 2 , E 2 ). Definition 3.11. We say that a family of pairs (S e , U e ) e∈E 2 and (S v , U v ) v∈Z 2 forms a (Z 2 , E 2 )-compatible local supersolution network if: (i) There is an F ∈ R such that, for any x ∈ Z 2 and any e ∈ E 2 , the pairs (S x , U x ) and (S e , U e ) are piecewise smooth local supersolutions of (20). (ii) For any x ∈ Z d and e ∈ E 2 , (S x , U x ) and (S e , U e ) are smooth patches.
If e 1 , e 2 ∈ E 2 and the line segments e 1 and e 2 are disjoint, then (S e1 , U e1 ) and and (S y , U y ) are admissibly incident, the former outgoing at x and the latter, incoming at y.
Remark 3.12. The translation invariance assumption, that is, condition (viii) in the above definition, is useful for two reasons. First, it automatically implies that there is a constant C > 0 such that diam(U ν ) ≤ C for each ν ∈ Z 2 ∪E 2 . Further, the functions parametrizing the curves {∂S ν ∩U ν } ν∈Z 2 ∪E 2 satisfy uniform C 2 estimates.
Combining our abstract supersolution patching procedure with the notion of a local supersolution network, we now describe how to approximate an arbitrary regular Z 2 * -measurable set by a supersolution. We assume in the discussion that follows that we have fixed a (Z 2 , E 2 )-compatible local supersolution network consisting of edges (S e , U e ) e∈Z 2 and nodes (S v , U v ) v∈Z 2 .
By Theorem 3.4 the boundary of every regular Z 2 * -measurable set is a disjoint union of simple paths. Thus to create a supersolution approximating such a boundary, we start by describing the method to approximate a single simple path in (Z 2 , E 2 ). To begin with, given m, n ∈ Z, suppose that γ : {m, . . . , n} → Z 2 is a finite simple path in E 2 , simple paths with infinite length will be considered later. We can create an edge supersolution along γ by the following procedure: call ) and then, inductively, define which is incident on γ(m) and γ(n), respectively outgoing and incoming.
When γ is a simple closed path (S γ , U γ ), as defined above, joins all the nodes/edges along γ except misses the node at γ (m). Of course we can simply patch this node in using basically the same procedure as before, although it is slightly awkward to phrase in our terminology. Simply take (S * , U * ) to be the node.join of the ordered triple and then this can be joined with (S γ , U γ ) by the patch operation (Sγ, Uγ) = patch((S γ , U γ ), (S * , U * )).
In the future we will simply omit the bars and write (S γ , U γ ) abusing notation in the case when γ is a simple closed path. [m,n] denotes the restriction of γ to {m, . . . , n}, then the pair (S γ , U γ ) defined by also defines a local supersolution.
Proof. The first statement is a direct consequence of Lemmas 3.8 and 3.10. For the second statement, first, observe that γ is locally finite: that is, for each compact set K ⊂ R 2 , #{i ∈ Z | γ(i) ∈ K} < ∞. Combining this with the diameter bound in Remark 3.12, we find that ( Thus, as a local uniform limit of supersolutions, S γ is a supersolution in U γ . Now we describe how to approximate regular Z 2 * -measurable sets by supersolutions. If A is a regular Z 2 * -measurable set, it can be written as the sum of countably many connected components A = ∪ n∈J A n for some J ⊂ N, where (A n ) n∈N are regular Z 2 * -measurable sets. Let us thus start in the case that A is simply connected.
First, let γ : E → Z 2 be a simple path such that ∂A = {γ} and γ traverses ∂A clockwise; such a path exists by Theorem 3.4 and simple connectedness. Let (S γ , U γ ) be the local supersolution constructed as in Lemma 3.13.
It will be useful to know, and follows essentially from condition (vi) in Definition Lemma 3.14. If A is a simply connected, regular Z 2 -measurable set and the curve γ and local supersolution (S γ , U γ ) are as constructed above, then The proof is deferred to Appendix D. Next, for each cube As in the proof of Lemma 3.13, this defines a local supersolution. It only remains to "fill in" the rest of A. Let {q n } n∈N ⊂ Z 2 * be an enumeration of the cubes q + [−1/2, 1/2] 2 contained in A int . Define (S A , U A ) by Note that, with this definition, A int ⊂ Int(S A ). Since each cube q n + [−1/2, 1/2] 2 is surrounded by cubes in A, one readily checks that (S A , U A ) is a local supersolution. We claim that, in fact, S A is a supersolution in R 2 .
Lemma 3.15. S A is a supersolution in R 2 with piecewise smooth boundary.
Proof. By Proposition 2.3, it suffices to check that, for every We showed how to construct a supersolution S A in case A is a simply connected, regular Z 2 * -measurable set. If, on the other hand, A is only connected and not simply connected, we proceed by letting S = S Aj , where A j is chosen so that R 2 \ A j is the jth connected component of R 2 \ A. Since any compact set in R 2 sees at most finitely many boundary paths of A, ∩ j S Aj ∩ B(0, R) equals a finite intersection of supersolutions for any R > 0. Hence S A = ∩ j S Aj is a supersolution itself.
When A is not even connected, we let {S An } be the supersolutions associated to its connected components. By Definition 3.11 and the regularity of A, these supersolutions are pairwise disjoint. Hence the union S A := ∪ n S An is also a supersolution.
Summing up, we have Lemma 3.16. If (S e , U e ) e∈E 2 and (S v , U v ) v∈Z 2 form a (Z 2 , E 2 )-compatible local supersolution network, then there is a constant C > 0 such that, for each regular Z 2 * -measurable set A, there is a closed set S A , which is a supersolution in R 2 , such that A int ⊂ Int(S A ) and Proof. We only need to verify the distance bounds. Since any point in S A \ A is contained in the set S γ as defined above, the diameter bound in Remark 3.12 implies that S A ⊂ A + B C . At the same time, A int ⊂ Int(S A ) so, by condition (i) in the definition of regularity, we have A ⊂ S A + B √ 2 . This gives d H (S A , A) ≤ C. Since the union of the images of the paths γ defined above is precisely ∂A, the same reasoning shows d H (∂S A , ∂A) ≤ C.

3.4.
Approximating smooth sets. We still need to show that we can approximate sufficiently smooth sets by regular Z 2 * -measurable sets. Toward that end, the main technical result we need follows: Lemma 3.17. There is an R > 0 such that if K ⊂ R 2 satisfies an interior and exterior ball condition of radius R, then the Z 2 * -measurable approximation A K of K defined by Proof. To show that (i) and (ii) in Definition 3.3 hold for R > 0 large enough, we argue by contradiction. Where (ii) is concerned, if the lemma is not true, then, after translating and rotating, we can find sets (K n ) n∈N such that, for each n ∈ N, K n satisfies an interior and exterior ball condition of radius n and By compactness of [−1/2, 1/2] 2 , we conclude that there is a half-space K ∞ ⊂ R 2 still satisfying (22). This is readily shown to be impossible. A similar approach establishes (i). The remaining claims follow directly from the definitions.
3.5. Proofs of the Main Lemmas. All that remains to prove Lemmas 1.6 and 3.1 is to show that the preceding discussion is not vacuous. In other words, we need to show there is a medium a ∈ C ∞ (T 2 ; [1, Λ]) for which a local supersolution network can be constructed. This is true, and it will be proved in Section 3.7. Let us state it as its own proposition for now: There is an a ∈ C ∞ (T 2 ; [1, Λ]) for which a (Z 2 , E 2 )-compatible local supersolution network (Definition 3.11) exists for some force F > 0. In fact, given ζ > 0, this can be done so that a − 1 L ∞ (T d ) ≤ ζ.
Combining this with Lemmas 3.16 and 3.17, we obtain Lemmas 1.6 and 3.1: Proof of Lemma 3.1. Let R be the constant of Lemma 3.17 and a be a medium as in Proposition 3.18. Let F a = F > 0 be the associated force. Given a set K satisfying an interior and exterior ball condition of radius R, let A K be the approximation of Lemma 3.17. By applying Lemma 3.16 to a and F a , we obtain a closed set S K = S A K which is a supersolution of (20) with F = F a . We note that S * (K) = S K has all the desired properties by concatenating the bounds and inclusions of the two lemmas.
To obtain a subsolution with the desired properties, we repeat the previous procedure with K replaced by R 2 \ K, which still satisfies interior and exterior ball conditions of radius R since K does. This leads to a closed set S R 2 \K containing R 2 \ K which is a supersolution of (20) with F = F a . We conclude by defining the open set S * (K) = R 2 \ S R 2 \K and noting that S * (K) is a subsolution of (20) with F = −F a < 0.
Proof of Lemma 1.6. Let a be a medium as in Lemma 3.1 and let R and F a be the constants of that same lemma. Given a set K ⊂ R 2 satisfying exterior and interior ball conditions of radius R, and given any F ∈ (−F a , F a ), let S * (K) and S * (K) be the respective super-and subsolution guaranteed by Lemma 3.1. The conclusions of Lemma 3.1 readily imply that the hypotheses of Proposition A.1, which is Perron's Method in this context, apply in this situation. Thus, we obtain a solution S ⊂ R 2 of (20) such that S * (K) ⊂ S ⊂ S * (K). Further, a quick computation shows that the claimed inclusions and distance bound also hold.
Finally, Corollary 1.7 follows by scaling: Proof of Corollary 1.7. If K is compact with C 2 boundary, then there is a δ > 0 such that K satisfies exterior and interior ball conditions of radius δ. Hence there is an ε 0 (K) > 0 such that ε −1 K satisfies exterior and interior ball conditions of radius R for any ε ∈ (0, ε 0 (K)). Applying Lemma 3.1 and blowing down space by a factor ε, we obtain, for each ε > 0, a stationary subsolution S ε * and a stationary supersolution S * ,ε of (20) such that is the solution flow of (20) with initial datum K, then, for each t > 0, the comparison principle implies S ε * (K) ⊂ S ε t ⊂ S * ,ε . We then use the distance bounds on S ε * and S * ,ε to deduce those for S ε t and K.
3.6. Homogenization of the Level Set PDE. In view of Corollary 1.7, it is straightforward to conclude that solutions of the level set PDE are also pinned.
Step 1: Edge supersolutions. We begin with the edge supersolutions, corresponding to each directed edge of Z 2 (near which the node supersolutions will be constructed) via certain (oriented) circular arcs. The circular arcs will have small positive curvature which will create the positivity needed for the supersolution property. We just need to ensure that the interiors of any two distinct edges are disjoint.
By translation invariance, we can fix our attention on (0, 0) and the 8 directed edges incident on (0, 0) connecting to its Z 2 neighbors (±1, 0) and (0, ±1). It is convenient to start by defining the arcs connecting (0, 0) to (1, 0) and (1, 0) to (0, 0) since the other arcs are constructed the same way. We will construct two arcs γ ± , γ + for the directed edge Notice that the point (1/2, −t) is equidistant to (0, 0) and (1, 0), no matter the choice of t, the distance being R(t) = t 2 + 1/4. For t > 0 sufficiently large (to be fixed later), let γ + be the circular arc connecting (0, 0) to (1, 0) with radius of curvature R(t) and center (1/2, −t) and γ − be the "reflected" arc obtained by doing the same construction, but with center (1/2, t). Each arc γ ± is oriented by the outward normal vector to the corresponding ball. In particular each arc has small positive mean curvature .
Choosing R large enough we can guarantee that the circular arcs are close to the line segment [(0, 0), (1, 0)] The interiors of these arcs are clearly disjoint, being separated by the chord connecting (0, 0) and (1, 0). Notice, further, that the angle formed between the  Repeat the same construction between (0, 0) and each of its other Z 2 nearest neighbors. This results in 8 distinct arcs. By the previous observation on tangent lines at (0, 0) all 8 arcs are disjoint, except for possible intersections at their endpoints, provided the radius of curvature R is chosen large enough. This condition along with (23) fixes our choice of R.
Analogously, for any directed lattice edge e ∈ E 2 , we define an associated arc γ e by translating the corresponding arc incident at the origin. (Note that this hides the ± notation we have used here in Step 1 inside of the directed edge e.) Step 2: Node shape. We add nodes to our network at each Z 2 vertex. The same construction will be repeated at each one so we restrict attention to (0, 0). The key point is to create large radial gradients to allow for a node supersolution but also to enforce that the incoming/outgoing edge supersolutions are tangential to Da so that the large gradients will not destroy their supersolution property.
To start with, let B r denote the disk centered at (0, 0) with radius r 1. Each of the eight arcs γ incident on (0, 0) passes through ∂B r at some point. As just discussed, the injectivity of the map sending arcs to tangent vectors shows that each arc is associated to a unique intersection point on ∂B r provided r is small enough. By making a small perturbation of R 2 \ B r , we can construct a smooth region O with the property that each arc emanating from 0 intersects ∂O at a unique intersection point, and the normal vector of ∂O is parallel to the tangent line of the arc at the intersection point; see Figure 7. We can also choose O to be symmetric with respect to π/2 rotations and R 2 \ O ⊂ B 1/4 (0), so that ∂O only Define the parameter which we will need to choose large below using our freedom to choose η. Let n ∂O be the outward pointing normal to O and κ ∂O , the mean curvature (following the sign convention κ ∂O = −tr(D 2 d O )). Modify η if necessary so that A satisfies (24) A > 2 (κ ∂O ) − L ∞ (∂O) .
We then find, for each x ∈ ∂O, In other words O is a supersolution of (20) for some F 1 > 0 or, in level set form, u = χ O is a supersolution of the equation Step 4: Construction of ϕ, part 2. We proceed to ensure that the edges of the network satisfy the necessary differential inequalities outside of k∈Z d (k + R 2 \ O) (actually outside of a neighborhood of the closure). Given an edge γ in the network, orient it so that its normal vector points away from the center of the corresponding circle. If O 1 and O 2 are the two regions intersecting γ at either end, first, assume that x ∈ γ ∩ {d O1 ≥ ν} ∩ {d O2 ≥ ν}. It follows that ϕ vanishes in a neighborhood of x and, thus, where R is the radius of curvature fixed earlier.
It remains to check the requisite inequalities near a vertex, which we can take to be (0, 0) by symmetry. Assume that x ∈ γ ∩ {|d O | ≤ ν}. We are only interested in the part of γ in a small neighborhood of O, so as long as we can prove the requisite supersolution property for d O (x) in a neighborhood of [0, ν] we will be done (in particular for small negative values of d O since values ≥ ν have already been handled).
We start at the intersection pointx ∈ γ ∩ ∂O and work outwards. By construction, n γ (x) · n O (x) = 0. Thus, by continuity, there is a ν ∈ (0, ν/2) such that Hence, for such x, we find Next, we consider the case when x ∈ γ ∩ {ν ≥ d O1 ≥ −ν }. Recall that in the construction of ϕ through η, so far we have only needed to know that η (0) = A = η L ∞ ([−ν,ν]) with A a fixed constant satisfying (24). Hence, we are still free at this stage to require the following condition on η: With this condition in hand, we find Also notice that the restrictions on η are loose enough that we can still require η L ∞ ([−ν,ν]) ≤ ζ, where ζ was the small parameter fixed at the start of the proof.
To summarize, in this step of the proof, we have shown that there is a constant F 2 > 0 such that, for any x ∈ γ satisfying d k+O (x) > −ν for all k ∈ Z 2 , we have Defining F , (S e , U e ) e∈E 2 , and (S v , U v ) v∈Z 2 . Finally, we define the local supersolutions that comprise our network. To begin with, let F = min{F 1 , F 2 }.
Recall from condition (viii) in Definition 3.11 that we require translation invariance, i.e., (S ν+x , U ν+x ) = x + (S ν , U ν ) for all x ∈ Z 2 and ν ∈ Z 2 ∪ E 2 , hence we only need to construct the supersolution node (S 0 , U 0 ) associated with the origin and the supersolution edges (S e , U e ) for edges e containing the origin.
Let us begin with the node (S 0 , U 0 ). Recall the smooth open set O defined above.
Note that ∂S 0 ∩ U 0 = ∂O so (S 0 , U 0 ) is a local supersolution of (20) by Step 3 of the proof and Proposition 2.3.
Next, we construct the edge supersolutions (S e , U e ) for edges e containing the origin. We begin with e = [(0, 0), (1, 0)]. Let γ + be the circular arc constructed above connecting (0, 0) to (1, 0). Let δ 1 and define (S e , U e ) as follows: Notice that ∂S e ∩ U e = γ ∩ U e by construction, and hence (S e , U e ) is a local supersolution of (20) with F = F a . Furthermore, U e \ S e is disjoint from the cube It is not hard to verify that if δ and c are small enough, then the constructed network satisfies the conditions in Definition 3.11. The details are left to the reader.

Pinning in a Diffuse Interface Model
In this section, we treat the diffuse interface setting, completing a construction analogous to that in Section 3. The basic idea is straightforward, we have proven the existence of pinned super/subsolutions for a non-trivial interval of forcing parameters in the sharp interface model, this gives us the room to approximate these solutions by a diffuse interface in the natural way and maintain the strict sub/supersolution property.
At a technical level there are two main issues that we need to address. First, the super/subsolutions we constructed in the previous section are not smooth, they have corner-type gradient discontinuities at a discrete set of points.
Further, as we will see, (2) differs from (12) by a square root. Hence, in what follows, we let a be as in Section 3 and define θ ∈ C ∞ (T 2 ) by (25) θ(x) = a(x) 2 .
Before proceeding further, notice that the first equations in (12) are related through the scaling (x, t) → (ε −1 x, ε −2 t). Accordingly, in what follows, we will be frequently interested in the unscaled equation: Lastly, we need to make explicit our assumptions on W : Here is the main technical result of this section, which will be the key component of the proof of Theorem 1.10: If a is as in Lemma 1.6, then there are constants δ 0 , β,F , C > 0 such that, for each K ⊂ R 2 satisfying an interior and exterior ball condition with large enough radius and each δ ∈ (0, δ 0 ), there is a continuous, stationary supersolution u + of (26) with F =F and a continuous, stationary subsolution u − of (26) with In particular, for each F ∈ [−F ,F ], there is a stationary solution u of (26) taking values in [−(1 + βδ), 1 + βδ] such that The construction of the supersolution u + proceeds in three steps. First, for a sharp interface supersolution E as in Section 3, we construct a "level set function" d with the property that the interfaces {d = s} are sharp interface supersolutions close to ∂E for all s close enough to zero. The second and third steps follow [9]. In the second step, we use d and the standing wave solution of the homogeneous Allen-Cahn equation to build a diffuse interface supersolution in the domain {|d| < γ} for a suitable γ > 0. Lastly, we extend this diffuse interface supersolution to the entire space.
The subsolution u − is built analogously. These sub-and supersolutions will be used to prove that the scaled problem (12) is pinned (Theorem 1.10).
By taking K to be a half space, we establish the existence of plane-like stationary solutions: Remark 4.2. Given e ∈ S 1 , let K = {x ∈ R 2 | x · e ≤ 0} in Lemma 4.1 and let u be the associated stationary solution. If δ is small enough, then it is not hard to show that u satisfies lim s→±∞ sup |u(x) − u ± (αδ)| | ±(x · e) ≥ s = 0, where u − (αδ) < u + (αδ) are the unique stable critical points of W (u) + αδu. Hence u is a plane-like solution heteroclinic to the two spatially homogeneous stationary solutions.

Preliminaries.
In what follows, we let D a : R 2 × R 2 → [0, ∞) be the metric induced by a. Specifically, this is the function defined by Recall that D a is a metric on R 2 equivalent to the Euclidean metric. Furthermore, D a is invariant under integer translations in the following sense: Given a (non-empty) set A ⊂ R 2 , define the a-distance dist a (·, A) : We also introduce the Allen-Cahn one-dimensional transition front associated with the homogeneous energy function with θ ≡ 1. We call q : R → R to be the solution of the second order ODE and from this first order ODE plus the previous boundary conditions at ±∞ it is easy to see q ∈ (−1, 1) andq > 0.

4.2.
Modifying the Interfaces. Given a set K satisfying exterior and interior ball conditions of radius R, let E = E(K) be the supersolution of (20) constructed by the algorithm of Section 3. Let d E : R 2 → R be the signed distance to E, that is, the function given by If E were smooth and compact, then it would be easy to see that, at least close to E, d E is a supersolution of a stationary level set PDE. Our setting complicates things slightly, but not irredeemably.
The following property about E is sufficient for our immediate purposes: There is a collection of local supersolutions (S i , U i )| i∈I of (20) with some positive forcing F = F 0 > 0 such that the sets ∂S i ∩ U i are smooth uniformly in i and there is an r > 0 so that, for all x 0 ∈ ∂E, there is a finite sub-collection In words, the supersolutions constructed in Section 3 are, locally, an intersection of a finite number of local smooth supersolutions. See the proofs of Lemma 3.8 and Lemma 3.10, which show that the patch and node.join operations create a supersolution which is, locally, an intersection of the input supersolutions. (Recall from Definition 3.11 and Remark 3.12 after that the "basic building block" supersolution edges and nodes in the network are uniformly smooth.) Proposition 4.4. There is an r > 0 depending on the network constructed in Section 3, but not the particular choice of E, such that d E satisfies the following viscosity inequalities: Proof. Let (S i , U i ) be the collection of supersolutions from Property 4.3. The S i have smooth boundary in U i uniformly in i so there is r 1 > 0 such that the signed a-distance functions d Si in the tubular neighborhoods {|d Si | < r 1 } ∩ U i are smooth and satisfy the following differential inequalities in the classical sense: Let x 0 ∈ ∂E. By Property 4.3 there is r 2 > 0 (independent of x 0 and, without loss, smaller than r1 2Λ ) so that for some subcollection I ⊂ I. We can also add the following requirement without loss: ∂S i ∩ B r2 (x 0 ) = ∅ for all I ∈ I . With this additional property, and since and so d Si are supersolutions of (32) in B r2 (x 0 ) for all i ∈ I . Further note that for x ∈ B αr (x 0 ) the unsigned distance satisfies Let us call r 3 = r2 1+Λ . By the formula (33), and since the minimum of supersolutions is a supersolution, we find that d E is a supersolution of (32) in the region Since x 0 ∈ ∂E was arbitrary and the radius r 3 did not depend on the particular x 0 we have that d E is a supersolution of (32) in the region

4.3.
Diffuse interface near ∂E. It will be convenient in what follows to recenter around d E = r/2 by Hence {−r/2 < d < r/2} = {0 < d E < r}. Note this changes none of the viscosity inequalities we have proven above since they are all invariant under addition of constants.
For the moment, let δ, β > 0 be free variables. Define v δ : where q(s) is the solution of (31). We claim that if δ and β are chosen appropriately, then v δ is a strict supersolution of (26) in {−r/2 < d < r/2} for sufficiently small α > 0 . Note that q is increasing and smooth so if a smooth test function ϕ touches v δ from below at some point x 0 , then δq −1 (ϕ − 2βδ) touches d from below at x 0 . We will compute as if d is smooth, but technically one does the computations on a smooth touching test function as is standard in viscosity solution theory (also the specific d under consideration is smooth at any point where it can be touched from below by a smooth test function in the neighborhood considered). We compute where the O(β 2 δ) error term can be bounded, more precisely, by |W |.
Since Dd 2 = a 2 in a neighborhood of x, it follows that Recalling that W is bounded from below away from 0 in a neighborhood N of {−1, 1}, we can choose β > 0 small so that Then, as in [9,Lemma 4.3], we deduce that there is F 1 (F 0 , W ) > 0 such that, for any δ < 1, Remark 4.5. This section is the only part of the argument where we use the specific form of (9). If instead we wanted to build sub-or supersolutions for the Euler-Lagrange equation associated with the energy model (13), the L 2 gradient flow is Hence when we invoke an ansatz of the form u δ ( . . Notice that, in this case, the highest order term suggests the identity a(x) 2 Dd(x) 2 = θ(x) Dd(x) 2 = 1. Thus, the only change necessary is to replace the Riemannian metric D a above by D a −1 (i.e. interchange a with a −1 ).
Where (14) is concerned, since a = √ θ, the gradient flow is Employing the ansatz u δ (x, t) = q(δ −1 d(x)) + . . . , we obtain . . Accordingly, in this case, the Euclidean distance should replace D a in the definition of d.

4.4.
Diffuse interface outside of {−r/2 < d < r/2}. We proceed to extend v δ to the whole space. On the one hand, when d ≥ r/2, the function v δ as defined above is almost 1 so we can simply take the minimum. When d ≤ −r/2, we interpolate between v δ and −1 using a partition of unity.
Most of the work is in the interpolation. Let λ : R → [0, 1] be a smooth, increasing function such that λ(u) = 0 if u ≤ − 3r 8 and λ(u) = 1 if u ≥ − r 8 . We wish to define u δ : for some suitable smoothed function d approximating d.
Lemma 4.6. There is a smooth function d : {−r/2 < d < r/2} → R with bounded first and second derivatives such that the following inclusions hold: Proof. Given a mollifying family (ρ ζ ) ζ>0 , define d by for small constants c, ζ > 0. The boundedness of Dd E implies d has bounded first and second derivatives with bounds depending only on ζ. Further, for the same reason, ζ can be chosen independently of E (or K).
It is now a more-or-less straightforward adaptation of [9] to show that u δ is a supersolution in {−r/2 < d(x) < 0}.
Lemma 4.7. There is a δ 1 > 0 andF > 0 depending only on θ, W , and the choice of the network in Section 3 such that if δ ∈ (0, δ 1 ) is sufficiently small, then u δ is a supersolution of which is also supersolution of (36) for small enough δ since W (−1) > 0.
This leaves to check the region {−7r/16 ≤ d(x) ≤ −r/8}. Note that the supersolution property (34) for v δ does hold in this region.
As is standard in viscosity solution theory we can carry over the computations which rely on differentiability to a touching test function. The only issue is that x → λ(d(x)) is not strictly positive so some care is required at points this function vanishes.
To address this, for n ∈ N, define u Now if ϕ is a smooth test function touching u (n) δ from below at some point theñ will touch v δ from below at the same point. Arguing as in [37,Lemma 6], we see that there is a δ 1 > 0 such that if δ ∈ (0, δ 1 ), then u (n) δ is a supersolution of (35) as soon as n is large enough. Sending n → ∞, we deduce that u δ is a supersolution by stability.
As for the dependence of δ 1 , we only need to be able to eliminate the error terms in the construction above; and these depend only on Λ, W , and the size of the derivatives of d, which are determined by ζ.
Finally, we extend u δ to a supersolutionū δ as follows: Proposition 4.8. There are constantsδ > 0 andF > 0 depending on the network constructed in Section 3 and on W , but not on E, such that if δ ∈ (0,δ) is sufficiently small, thenū δ is continuous in R 2 and satisfies the following differential inequality in the viscosity sense: Proof. Given Lemma 4.7 we just need to check that the constant function 1 + βδ is a supersolution of (36) and thatū δ is identically equal to 1 + βδ in a neighborhood of {d(x) = r/2}.
If x ∈ {d(x) > r 8 } then, from the exponential convergence of q, for δ > 0 sufficiently small. Therefore, The next remark puts the computations above into some context. Remark 4.9. By reprising the arguments just presented, one can show that, as δ → 0 + , solutions of the Cauchy problem concentrate along interfaces whose motion is governed by the level set PDE This can be seen using the ansatz u δ (x, t) = q(δ −1 d(x, t)), where d is the signed distance to {u * (·, t) = 0} with respect to the Riemannian metric D a above. Similarly, arguing as in [1], one can show that the energy (9) Γ-converges to (1) as δ → 0 + . Proof of Lemma 4.1. Given K, Lemma 1.6 furnishes a stationary supersolution E of (20) with positive forcing F 0 > 0 such that K ⊂ E and d H (K, E)+d H (∂K, ∂E) ≤ C. The arguments of the previous subsection show that, for δ > 0 small enough depending only on the coefficient a, there is a stationary supersolution u + of (26) with α =F > 0 such that We are taking r < 1 so we have As in the sharp interface setting, the existence of supersolutions implies that of subsolutions. To see this, notice that u is a subsolution of (26) if and only if −u is a supersolution of (26) with W replaced by the function u → −W (−u) and α replaced by −α. This does not change a (and the equation for E is correspondingly changed) so the construction goes through.
Finally, given α ∈ [−F ,F ], we construct a stationary solution u with the desired properties employing Perron's Method with u + and u − serving as barriers. 4.6. Sharp Interface Limit. Using the stationary solutions furnished by Lemma 4.1, we now prove that, for any sufficiently small external force α ∈ [−F ,F ], the macroscopic interfaces associated with (26) are pinned, i.e. we prove Theorem 1. 10.
Proof of Theorem 1.10. Let a be the coefficient field constructed in Section 3 and θ(x) = a(x) 2 . Also recallδ,F > 0 from Proposition 4.8 and F 1 from the discussion preceding Proposition 4.10. If necessary we can makeF andδ smaller so that F < F 1 andδ < 1.
Finally, here is a sketch of the proof of Proposition 4.10: Sketch of Proof of Proposition 4.10. We argue as in [9, Proposition 4.1] constructing the desired subsolution as in [37, Appendix A]. In the notation of the latter reference, in the present setting, f is defined by

Surface tension with gradient discontinuities at all directions satisfying a rational relation
In this section, we prove Theorem 1.1 concerning generic discontinuities of Dσ and Theorem 1.4, which proves that "bubbling" is a generic feature of the gradient flow. The basic strategy involves building compact subsolution barriers and the results apply in all dimensions d ≥ 2.
On the one hand, where the behavior of Dσ is concerned, we avail ourselves of the work of Chambolle, Goldman, and Novaga [14]. They prove that the behavior of the subdifferential of the surface tension σ closely mirrors the structure of the plane-like minimizers of the energy. In particular, the key question is whether or not the planelike minimizers in a given direction foliate space or not. At least philosophically, it is clear that sliding arguments can be used to show that the existence of barriers is an obstruction to the formation of foliations. This is precisely the strategy taken in what follows.
At the level of the gradient flow, on the other hand, the maximum principle implies that if a smooth open subset is a strict subsolution of the flow, then any set that initially contains the subsolution continues to do so at later times. Accordingly, such subsolutions are also relevant for the dynamics. 5.1. Plane-like minimizers. We need to consider plane-like minimizers which only have locally finite perimeter, it is natural to consider the class of sets which minimize E a under compact perturbations. In the literature these are referred to as Class A minimizers and we repeat the definition here for clarity: Definition 5.1. A set S of locally finite perimeter is called a Class A minimizer of E a if S minimizes E a (· ; B R ) with respect to compact perturbations in B R for all R > 1. More precisely, given any R > 0, if S ⊂ R d is a set of locally finite perimeter and the symmetric difference S ∆S satisfies S ∆S ⊂⊂ B R , then Following [14], given n ∈ S d−1 , we say that an open set of locally finite perimeter S ⊂ R d is a strongly Birkhoff plane-like minimizer in the n direction if (i) S is a Class A minimizer of (1), (ii) S equals its set of Lebesgue density one points, (iii) there is a c ∈ R and an M > 0 such that and (iv) S has the strong Birkhoff property, that is, We denote the family of all such sets by M(n).
We will need the following properties of M(n): Concerning (iii), Caffarelli and Cordoba [11] show the viscosity solution property just for the perimeter functional, but small modifications of their arguments work for our heterogeneous energy (1) as well.
In [14], the authors observe that, for energies of the form (1) (isotropic), a result of Simon [43] implies that the interfaces {∂S | S ∈ M(n)} are disjoint so that M(n) is a lamination. For more general types of surface energy (anisotropic) it is only known that no intersections can occur at regular points [14,Proposition 3.4]. Although it is convenient for sliding-type arguments, we will avoid using this fact below so that our arguments apply to other forms of energy as well (cf. Remark 5.10 below).

5.2.
Gaps. Before proceeding to the proof of Theorem 1.1, we define the notion of a gap and recall the main result of [14]. Definition 5.3. We say that a compact set K ⊂ R d with non-empty interior is a gap at direction n for the medium a if ∂S ∩ K = ∅ for every S ∈ M(n).
In the next result, we show that the property of having a gap at a direction n ∈ S d−1 is an open condition with respect to uniform norm perturbations of the medium.
Lemma 5.4. If a compact set K ⊂ R d with non-empty interior is a gap for the medium a at direction n then there exists δ > 0 so that if b ∈ C(T d ; [1, Λ] In the proof of the lemma, we will need to know that Class A minimizers are well-behaved under perturbation of the coefficient a. More precisely, we will use the following fact, which is proved in Appendix B. Lemma 5.5. Suppose that (a k ) k∈N ⊂ C(T d ) converges uniformly to some positive function a ∈ C(T d ) and S and (S k ) k∈N are sets of locally finite perimeter such that Proof of Lemma 5.4. We will prove the contrapositive, that is, that the set of coefficients for which K is not a gap is closed. Suppose that there is a sequence a k → a uniformly on T d and S k ∈ M(n, a k ) with ∂S k ∩ K = ∅. We claim that there is an S ∈ M(n, a) such that S ∩ K = ∅.
S can be obtained as a subsequential limit of (S k ) k∈N . To see this, recall that standard local perimeter bounds give that the S k have uniformly bounded perimeter on any compact region. Thus, by standard BV compactness results, we can choose a subsequence, not relabeled, so that S k → L 1 loc S and, for any R > 0, By Lemma 5.5, S is a Class A minimizer of E a . As for the strong Birkhoff property, for each k ∈ N, S k + y ⊃ S k if y ∈ Z d and y · n ≥ 0 and S k + y ⊂ S k if y ∈ Z d and y · n ≤ 0 so the same holds for the limit S. It remains to prove that ∂S ∩ K = ∅. By density estimates (e.g. [14, Proposition 3.1]), for any x ∈ ∂S k and any r > 0, for a positive constant c depending only on d, Λ. By assumption, we can fix x k ∈ ∂S k ∩ K for all k ∈ N. Let x * ∈ K be any limit point of the sequence x k . By the L 1 loc convergence, for any r > 0, Since r > 0 was arbitrary, we conclude that x * ∈ ∂S.
Next, we recall the main result of [14], which gives a direct relationship between regularity of the effective surface tension and the existence of gaps in M(n).
Theorem 5.6 (Chambolle, Goldman and Novaga). Let n ∈ S d−1 and let V (n) be the subspace of R d spanned by the rational relations satisfied by n. If dim(V (n)) = 0, we say n is totally irrational.
• If n is totally irrational, then Dσ(n) exists.
• The same holds if M(n) has no gaps.
• If n is not totally irrational and M(n) has a gap, then ∂σ(n) is a convex subset of V (n) of full dimension.
In [14,Section 6], the authors give some examples of media where σ is not differentiable at any direction satisfying a rational relation. We will show that this phenomenon is generic in the topological sense.
The strategy of proof uses the Euler-Lagrange equation.
The key observation is that if the equation associated to a admits a smooth, bounded open set as a strict subsolution, or the complement of a smooth, bounded open set as a strict subsolution, then these will act as a barrier to foliations.
Lemma 5.7. Given a medium a ∈ C(T d ; [1, Λ]), if there is a nonempty C 2 bounded open set Ω ⊂ R d such that the indicator function χ Ω is a strict subsolution of (38), then, for each n ∈ S d−1 , the family of strongly Birkhoff plane-like minimizers of (1) has a gap.
The main result of this section is compact barriers exist generically: Lemma 5.8. For any medium a ∈ C(T d ; [1, Λ]) and any δ > 0, there exists a medium a δ ∈ C 1 (T d ; [1, Λ]) with a − a δ C(T d ) ≤ δ and a C 2 open set Ω, which is bounded and nonempty, such that χ Ω is a strict subsolution of (38). If, in addition, a ∈ C 2 (T d ; [1, Λ]) and p ∈ [1, ∞), then this estimate can be improved to a − a δ W 1,p (T d ) ≤ δ.
Once Lemmas 5.7 and 5.8 are proved, Theorem 1.1 follows easily, as we now show.
Proof of Theorem 1.1. Given n ∈ S d−1 , let A n be the family of coefficients a given by A n = {a ∈ C ∞ (T d ; [1, Λ]) : there is a gap at direction n for a} By Lemma 5.4, A n is open in C ∞ (T d ; [1, Λ]) with the C(T d ) norm topology. Since the inclusion W 1,p (T d ) → C(T d ) is continuous for p ∈ (d, ∞), A n is also open in the W 1,p (T d ) norm topology. Combining Lemma 5.7, Lemma 5.8, and Theorem 5.6, we see that ∩ n∈S d−1 A n is dense in either topology.
The remainder of this section is devoted to the proofs of Lemmas 5.7 and 5.8 and Theorem 1.4.

Gap barriers.
We now show that compact subsolution barriers occur generically. The proof proceeds by exploiting the structure of the level sets of a generic medium. We start with a few preliminary reductions.
First of all, we make some room by observing that any function in C(T d ; [1, Λ]) can be approximated by functions (a n ) n∈N in C ∞ (T d , [1, Λ)) satisfying (39) max T d a n < Λ for each n ∈ N.
Therefore, in what follows, we always assume (39) holds. The next lemma shows we can also assume that a attains its maximum at unique, non-degenerate critical points: Lemma 5.9. If a ∈ C 2 (T d ) satisfies (39) and δ > 0, then there is an a δ ∈ C 2 (T d ) satisfying (39) such that the following holds: Proof. Let x 0 ∈ T d be a point where a achieves its maximum. Let f ∈ C ∞ c (R d ) be a radially decreasing bump function satisfying It is easy to see that a δ = a + δf has the desired properties provided δ is small enough.
With these preliminaries out of the way, we are prepared for the proof. The strategy is as follows: replacing a by a δ if necessary, we assume that a attains its maximum at a unique, non-degenerate critical point. This implies that there is c > 0 close to max a such that {a = c} is a topologically trivial hypersurface in T d .
Using a tubular neighborhood of ∂Ω = ∂{a > c}, we define a function ϕ such that It follows that the set Ω = {a > c} is a strict subsolution associated to the coefficient a ϕ = (1 + ϕ) · a. The complement of Ω will, correspondingly, be a strict supersolution.
Proof of Lemma 5.8. By the previous considerations, we can assume that a ∈ C 2 (T d ) satisfies (39) and attains its maximum at a unique, non-degenerate critical point x 0 . Fix ε > 0 and p ∈ (d, ∞). We will find a function a ε ∈ C 2 (T d ) satisfying (39) such that a ε satisfies the conclusions of the theorem and a ε − a W 1,p (T d ) < (2 + Da L ∞ (T d ) )ε. Notice that this is enough to obtain an estimate in C(T d ) by Morrey's inequality.
To start with, notice that if c is close enough to a(x 0 ), then {a > c} is an open, simply connected subset of T d with C 2 boundary. Let Ω = {a > c}.
Fix r > 0 such that the signed distance d to ∂Ω = {a = c}, positive in Ω and negative outside, is smooth in an r-neighborhood of the surface. Letting ν ∈ (0, r) be a small constant to be determined, choose a smooth function η : This is a C 2 function by the choice of η. Let a ϕ = (1 + ϕ) · a. Notice that a ϕ − a L ∞ (T d ) ≤ ε and, by the coarea formula, Thus, if ν is sufficiently small, we obtain Finally, we claim that Ω has the desired properties for the medium a ϕ . To see this, start by noting that Da and Dϕ are aligned with the outward normals to Ω along ∂Ω, i.e., for x ∈ ∂Ω, Thus, the indicator function χ Ω is a strict subsolution of (38). Remark 5.10. The approach above provides a general strategy for showing that the plane-like minimizers of a given surface energy has gaps, even when the energy does not have the form (1). For example, given a ψ ∈ C ∞ (T d ; R d ) such that ψ L ∞ (T d ) < 1, consider the energy given by By the divergence theorem, the Euler-Lagrange equation associated with this energy is κ + div ψ = 0 Up to making a small perturbation, we can assume that div ψ ≡ 0. Hence there is a ball B such that div ψ < 0 in B, and then we can find a smooth perturbationψ, which is arbitrarily close to ψ in C(T d ), such that Thus, B is a smooth compact subsolution and we deduce that a small perturbation of (40) has gaps in every direction. In particular, by [14], typically, the associated surface tension is non-differentiable at every lattice direction.

Existence of Gaps.
Once a smooth compact barrier is known to exist, no plane-like minimizer can touch it if the subsolution property is strict. By the monotonicity of the family of strongly Birkhoff plane-like minimizers, this means the barrier has to be contained in a gap. As we will see below, proving this is somewhat technical compared to the diffuse interface case -the basic issue being that, for the sake of generality, we will not use the fact that {∂E | E ∈ M(n)} is pairwise disjoint.
We will need the following lemma: Suppose that x 0 ∈ S * \ S. Applying density estimates again, we can find a ball B ⊂ S * \ S close to x 0 . IfS ∈ M(n) and ∂S ∩ B = ∅, thenS must be a strict subset of S * and a strict superset of S. In particular, S S S * ⊂ S * * , in violation of the definition of S * * . Hence B is a gap according to Definition 5.3, contradicting the hypothesis. Now we show how to use the lemma in a sliding argument.
Proof of Lemma 5.7. We argue by contradiction. Fix n ∈ S d−1 and let M(n) denote the family of strongly Birkhoff plane-like minimizers in the n direction. Assume M(n) has no gaps.
Let Ω be the bounded strict subsolution which was assumed to exist in the statement. Define If Ω \ S = ∅ for all S S * then, by compactness (cf. [14,Proposition 3.2]), there is an x ∈ Ω ∩ ∂S * = ∂S * ∩ ∂Ω. Otherwise Ω ⊂ S for some S S * , which contradicts the definition of S * . Thus, henceforth we can fix x 0 ∈ ∂S * ∩ ∂Ω.
Let d Ω be the signed distance function to Ω with {d Ω > 0} = Ω. Since χ Ω is a strict subsolution of (38), it follows that there is an r > 0 such that d Ω is smooth in {|d Ω | < r } and It is straightforward to check that there is an r > 0 such that χ S * − d Ω achieves its minimum in B r (x 0 ) at x 0 . Since S * is a plane-like minimizer, χ S * is a viscosity supersolution of (38) by Proposition 5.2. Thus, the following inequality holds: However, this directly contradicts (41).
We conclude that M(n) has gaps, no matter the choice of n ∈ S d−1 .

5.5.
Proof of Corollary 1.2. The previous arguments show that the existence of a smooth, compact strict subsolution of (20) forces the surface tensionσ to have corners. It also has consequences for the gradient flow, as we now show. While we do not know if it implies pinning in the strongest sense (i.e. pinning of the entire interface as considered in the example of Section 3) it does seem to rule out the possibility of homogenization in the usual way by pinning some compact connected components of the negative-phase. This phenomenon has been observed many times in the study of interface homogenization, see, for example, Cardaliaguet, Lions and Souganidis [13].
The statement forū * follows the same way using the complement compact supersolution R d \ Ω. 6. Gaps in the plane-like minimizer lamination are generic: diffuse interface case In this section, we prove results on the existence of gaps and weak pinning analogous to those of the previous one. Once again, we proceed by perturbing around the sharp interface δ = 0 setting. The existence of a compact strict subsolution for the sharp interface model will imply the same for the diffuse interface model when δ > 0 is small. By a sliding argument the existence of such barriers causes a gap in the family of strong Birkhoff plane-like minimizers just as in the sharp interface case.
We stop short of proving any results concerning the gradient of the diffuse surface tensionσ AC . The reason is there is currently no proven analogue of the result of [14] in the diffuse interface case. We believe that such an analogue does hold and leave its proof to future work.
As in the previous section, the results presented here apply in all dimensions d ≥ 2.
6.1. Plane-like minimizers and gaps. Let us introduce some notation and terminology to be used in what follows. To begin with, as in the sharp interface case, we recall the definition of a Class A minimizer of the diffuse interface energy AC δ θ (see (9)). Definition 6.1. A function u : R d → [−1, 1] is said to be a Class A minimizer of the energy functional AC δ θ if, for any R > 0 and any v ∈ u + H 1 0 (B R ), Given θ ∈ C(T d ; [1, Λ 2 ]), δ > 0, and n ∈ S d−1 , we say that a Class A minimizer u : R d → (−1, 1) of AC δ θ is a strongly Birkhoff plane-like minimizer in the direction n if, for each k ∈ Z d , Notice that lim x·n→±∞ u(x) = ∓1 automatically holds since the only periodic Class A minimizers are the constants 1 and −1.
We let M δ θ (n) denote the family of all strongly Birkhoff plane-like minimizers.
Arguing as in [4] (cf. [29]), one can prove that M δ θ (n) forms a lamination. That is, for each u 1 , u 2 ∈ M δ θ (n), either (For the connection between Moser-Bangert theory and (9), which allows us to invoke results from [4], see [30] and the introduction of [40]. ) We will say that M δ θ (n) has a gap if the graphs of its elements fail to foliate 6.2. Parametrizations of M δ θ (n). As in the sharp interface case, it will be convenient to know that M δ θ (n) has no gaps if and only if it admits a continuous parametrization.
In fact, M δ θ (n) has no gaps if and only if, in the terminology of [36], there is a pulsating standing wave U n ∈ U C(R × T d ) in the direction n (cf. Remark 6.6 below). To keep things short, we will not prove this stronger statement here.
The existence of such an element follows from the assumption that M δ θ (n) has no gaps; uniqueness follows from the fact that it forms a lamination.
The monotonicity of γ → u(· ; γ) also follows from the lamination property. It remains to check the continuity.
6.3. Obstruction. We saw above that if there are no gaps, we can continuously parametrize M δ θ (n). Hence, in that case, classical sliding techniques can be used to rule out the existence of certain (sub-or super-) solutions of (10). In particular, bump (strict) subsolutions cannot occur: and an F > 0 such that is non-empty, and u δ is smooth in a neighborhood of {u δ ≥ −1}, then, for each n ∈ S d−1 , M δ θ (n) has gaps.
Proof. To start with, observe that there is a constant c ∈ (0, 1) such that u δ ≤ 1 − c in R d . Indeed, were this not the case, then, by the compactness of {u δ ≥ −1}, we could find an x 0 ∈ R d such that u δ (x 0 ) = 1 = max R d u δ , but this would contradict the strict subsolution property. We argue by contradiction. If M δ θ (n) has no gaps, then Proposition 6.2 implies that there is a continuous, increasing parametrization γ → u(· ; γ) of M δ θ (n). Define Since u δ ≤ 1 − c, {u δ ≥ −1} is compact, and {u δ > −1} is non-empty, it follows that T < ∞. We claim that u(· ; T ) touches u δ above at some pointx ∈ R d . Indeed, this follows from the fact that the parametrization is continuous and increasing. Note that u δ (x) = u(x; T ) > −1. Since u δ is smooth in a neighborhood of {u δ ≥ −1}, the viscosity solution property of u(·; T ) yields This contradicts the strict subsolution property of u δ . 6.4. Dynamics. As in the sharp interface case, the previous construction also has a dynamical interpretation. (Below we once again use the half-relaxed limit notation from Definition 2.6.) Similarly if there is a smooth v δ : R d → [−1, 2] and an F > 0 such that and {v δ ≤ 1} is compact and {v δ < 0} is non-empty, then there is a symmetrical conclusion for the ε scaled problem above: Proof. Note, as in the proof of Proposition 6.3, that max R d u δ ≤ 1 − c for some c ∈ (0, 1).
is compact, and {u δ > 0} is non-empty, we conclude the proof by combining ideas from the proofs of Corollary 1.2 and Theorem 1.10 (especially Proposition 4.10).
6.5. Proof of Theorem 1. 9. In what follows, we let A Λ denote the family of all a ∈ C 1 (T d ; [1, Λ]) such that there is a smooth, bounded open set Ω ⊂ R d such that χ Ω is a strict subsolution of (20). By Lemma 5.8, A Λ is a dense subset of C(T d ; [1, Λ]) and W 1,p (T d ; [1, Λ]) for each p ∈ (d, ∞).
If θ ∈ Θ Λ , then there is a smooth, bounded open set Ω such that χ Ω is a strict subsolution and χ R d \Ω is a strict supersolution of (2). Arguing exactly as in Section 4, this implies we can find an F θ > 0, a δ θ ∈ (0, 1), and continuous functions (u Observe that G n is dense in C(T d ; [1, Λ 2 ]) since Θ Λ is. Next, notice that ifθ ∈ G n for some n ∈ N, then there is a θ ∈ Θ Λ such that θ ∈ G n (θ). In particular, for each δ ∈ [ δ θ n , δ θ ), −δ∆u Accordingly, for such choices of δ, Proposition 6.3 and Proposition 6.4 (in both subsolution and supersolution form) apply toθ. Let G = ∩ n∈N G n . This is dense in C(T d ; [1, Λ 2 ]) since G ⊃ Θ Λ . If θ ∈ G, then there is a sequence (θ (n) ) n∈N such that θ ∈ G n (θ (n) ) for each n. Hence θ satisfies the conclusions of the theorem with I(θ) given by n , δ θ (n) .
Finally, we observe that the same considerations apply to W 1,p (T d ; [1, Λ]) since, for each n, the set G n (θ) ∩ W 1,p (T d ; [1, Λ]) is open and Θ Λ remains dense in this topology.
Remark 6.5. Theorem 1.9 remains true if (9) is replaced by the variants (13) or (14). As in Remark 4.5, smooth diffuse interface subsolutions can be constructed from the sharp interface subsolutions of Lemma 5.8. The only difference in the proof is that since θ appears multiplied by derivatives of u δ in the PDE, we need to change the definition of G n (θ) accordingly. This is not a problem since the construction of Section 4 implies that u δ has bounded second order derivatives in the set {u δ > −(1 + βδ)}. Remark 6.6. Theorem 1.9 provides examples of diffuse interface models in periodic media in which, in every direction n ∈ S d−1 , there is no continuous pulsating standing wave. See [36] for a discussion of the relevance of pulsating standing waves to the analysis of the energy (9) and the homogenization of its gradient flow.
Given an n ∈ S d−1 , a pulsating standing wave of (9) is a function U n ∈ L ∞ (R × T d ) that is a distributional solution of the PDE (n∂ s + D y ) * (n∂ s + D y )U n + a(x)W (U n ) = 0 in R × T d , lim s→±∞ U n (s, y) = ∓1, U n L ∞ (R×T d ) ≤ 1, ∂ s U n ≤ 0.
A pulsating standing wave can be interpreted as a generating function (or hull function) for the plane-like minimizers in the n direction (see [36,Section 6]). Such functions always exist, but they can be discontinuous. Indeed, if U n is a pulsating standing wave and it is a continuous function in R × T d , then the plane-like minimizers of (9) in the n direction form a foliation by [36,Proposition 1]. Therefore, Theorem 1.9 shows that it is possible that there are no continuous pulsating standing waves in any direction.

Appendix A. Perron's Method
In this appendix, for the sake of completeness, we prove a version of Perron's Method for sharp interfaces: It shows that provided there are sufficiently regular (but not necessarily smooth) sets E * ⊂ E * defining stationary sub-and supersolutions, it is possible to find a stationary solution E between them. If v is a supersolution of (42) and v, a subsolution, then there is an open set E ⊂ R d satisfying E * ⊂ E ⊂ E * such that χ E is a discontinuous viscosity solution of (42) (i.e., χ E is a viscosity supersolution and χ E is a viscosity subsolution).
In the proof, we will use semi-continuous envelopes. In particular, given a locally bounded function w : R d → R, we denote by w * , w * : R d → R the upper and lower semi-continuous envelopes defined by w * (x) = lim δ→0 + sup {w(y) | x − y < δ} , w * (x) = lim δ→0 + inf {w(y) | x − y < δ} . The proof of this level-set version of Perron's Method rests on the fact that if an open set defines a subsolution but fails to be a supersolution, then it is possible to find a larger subsolution containing it. More precisely, we have Proposition A.2. Suppose that w ∈ U SC(R d ; {0, 1}) is a subsolution of (42), x 0 ∈ R d , r > 0, and there is a smooth function ψ such that w * − ψ has a strict local maximum at x 0 in B r (x 0 ) and Dψ(x 0 ) > 0. If ψ satisfies the following differential inequality at x 0 −a(x 0 )tr Id − Dψ(x 0 ) ⊗ Dψ(x 0 ) D 2 ψ(x 0 ) − Da(x 0 ) · Dψ(x 0 ) < F Dψ(x 0 ) , then there is aw ∈ U SC(R d ; {0, 1}), which is a subsolution of (42), such that w ≥ w in R d ,w = w in R d \ B r (x 0 ), andw ≡ w.
Proof. The construction follows along the lines of the usual proof, see, e.g., [17,Section 4]. A little care is needed to ensure that the gradient of the smooth subsolution built in the argument never vanishes. At the end of the argument, we will have a subsolutionŵ taking values in R. The proof is completed upon definingw bỹ w(x) = 1, ifŵ(x) ≥ δ, 0, otherwise, for some suitable small δ > 0.
Proof of Proposition A.1. To start with, observe that the identity R d \ E * = R d \ E * implies that, at the level of semi-continuous envelopes, we have (v * ) * = v. This will be needed later in the argument. Let S denote the family of subsolutions w ∈ U SC(R d ; {0, 1}) of (42) satisfying v ≤ w ≤ v in R d . Note that S is nonempty precisely because E * ⊂ E * . Let v : R d → {0, 1} be the pointwise maximum of this family: v(x) = sup {w(x) | w ∈ S} .
As the supremum of a family of subsolutions, v * is also a subsolution.
We claim that v * is a supersolution. To see this, assume that x 0 ∈ R d and ψ is a smooth function such that v * − ψ has a strict local minimum at x 0 and Dψ(x 0 ) > 0. There are two cases to consider: (i) v * (x 0 ) = v(x 0 ) and (ii) v * (x 0 ) < v(x 0 ).
In case (ii), it necessarily follows that v(x 0 ) = 1. Since E * is open, there is an r > 0 such that {v = 1} = E * ⊃ B r (x 0 ). With this wiggle room, we can argue by employing a geometric version of the standard Perron argument: if ψ does not satisfy the desired differential inequality at x 0 , then Proposition A.2 implies that We claim that either {x 0 , x 1 , x 2 , . . . } is a simple closed curve (in case it is bounded) or else it is an infinite simple path for which x i → ∞ as i → ∞.
In particular, x j = x j+M for some M ∈ N. Observe that M ≥ 2 necessarily holds as x j = x j+1 by construction.
We claim that j = 0. To see this, we assume that j ≥ 1 and argue by contradiction. Toward that end, note that j + M − 1 ≥ 1 since M ≥ 2. Thus, x j+M −1 is well-defined.
The construction implies that the edge [x j+M −1 , x j ] = [x j+M −1 , x j+M ] is contained in ∂A and it traverses ∂A clockwise. In particular, [x j , x j+M −1 ] ⊂ ∂A and it traverses ∂A counterclockwise. Thus, by Lemma C.1 and the property (43) with i = j − 1, we must have x j−1 = x j+M −1 . Yet j − 1 < j so this contradicts the definition of j.
We conclude that j = 0 as claimed. From the identity x 0 = x M , we deduce by construction that x 1 = x M +1 , and then this implies that x k = x M +k for all k ∈ N by induction. In particular, {x 0 , x 1 , . . . , x M } = {x 0 , x 1 , . . . }. Furthermore, by taking M to be as small as possible, we find that the path γ (k+1) : {0, 1, . . . , M } → Z 2 given by γ (k+1) (i) = x i is a simple closed path in Z 2 , as claimed.
In this case, we set aside the infinite path {x 0 , x 1 , . . . } for the moment. Let us return to x 0 , but this time we proceed in reverse. By Lemma C.1, we can let x −1 ∈ ∂A ∩ Z 2 denote the unique neighbor of x 0 for which [x 0 , x −1 ] ⊂ ∂A and such that [x 0 , x −1 ] traverses ∂A counterclockwise. We then turn our attention to x −1 , letting x −2 be the unique element of ∂A ∩ Z 2 with [x −1 , x −2 ] ⊂ ∂A and such that [x −1 , x −2 ] traverses the boundary counterclockwise. Continuing in this way, we obtain a path {x 0 , x −1 , x −2 , . . . } such that, for any i ≥ 0, the following property holds: We claim that {x 0 , x −1 , x −2 , . . . } must also be unbounded. Indeed, were this not the case, then we could argue as above to deduce that {x 0 , x −1 , x −2 , . . . } is a simple closed curve {x 0 , x −1 , x −2 , . . . } = {x 0 , x −1 , . . . , x −L } for some L ∈ N, with x −(i+L) = x −i for every i ≥ 0. Yet, by (43) and (44) Therefore, by Lemma C.1, x 1 = x −(L−1) . Applying the lemma again with x 1 in place of x 0 , we see that x 2 = x −(L−2) . By recursion, we conclude that x k = x −(L−k) for every k ≤ L. This implies that x 0 = x L , hence x k = x L+k for every k ∈ N by construction, but then this would contradict the assumption that {x 0 , x 1 , . . . } is infinite. We conclude that {x 0 , x −1 , x −2 , . . . } is unbounded.
In sum, we showed that {x 0 , x 1 , . . . } and {x 0 , x −1 , . . . } define two infinite paths emanating from x 0 . By arguments similar to those in the previous paragraph, To begin the proof that η bottom (R) ⊂ A, we start by observing that either (a) η top (R) ⊂ R 2 \A and η bottom (R) ⊂ Int(A) or (b) η top (R) ⊂ Int(A) and η bottom (R) ⊂ R 2 \ A. To understand why, first, note that the line segment [γ(i), γ(i + 1)] ⊂ fill(U γ(i) ) ∪ U [γ(i),γ(i+1)] ∪ fill(U γ(i+1) ) ⊂ U γ for any i ∈ Z by condition (vi) in Definition 3.11. Thus, ∂A ⊂ U γ and hence the boundary curves of U γ do not intersect ∂A, that is, γ * (R) ∩ ∂A = ∅ for * ∈ {top, bottom}. From this and the connectedness of A, we conclude that one boundary curve must be in A and the other must be in R 2 \ A. To conclude the proof, we establish that (a) holds.