CLOSED CURVES OF PRESCRIBED CURVATURE AND A PINNING EFFECT

. We prove that for any H : R 2 → R which is Z 2 -periodic, there exists H ε , which is smooth, ε -close to H in L 1 , with L ∞ -norm controlled by the one of H , and with the same average of H , for which there exists a smooth closed curve γ ε whose curvature is H ε . A pinning phenomenon for curvature driven ﬂow with a periodic forcing term then follows. Namely, curves in ﬁne periodic media may be moved only by small amounts, of the order of the period.

1. Introduction. In this paper, curves in the plane with prescribed curvature are dealt with.
We show that, for a "generic" H, periodic, possibly with small L ∞ -size, and with prescribed (possibly zero) average, there exists a closed, convex curve whose curvature at any points agrees with H. The genericity is in the L 1 -sense.
We then apply this result to show a pinning phenomenon in an evolutionary problem driven by the curvature. More precisely, our result is the following: Theorem 1.1. For any H ∈ L ∞ (T 2 ), with H ≡ 0, and for any ε > 0 there exists

2) and
such that there exists a set E ε , with smooth compact boundary, whose curvature agrees with H ε at any point of ∂E ε . Moreover, we can choose E ε such that either E ε or R 2 \ E ε is a convex set (with the convention that the curvature of a convex set is positive).

MATTEO NOVAGA AND ENRICO VALDINOCI
We observe that Theorem 1.1 does not hold, in general, if we choose H ε := H, and H changes sign. However, it would be interesting to know: • whether a result analogous to Theorem 1.1 holds if we replace the L 1 norm in (1.2) with a stronger one (e.g., the L ∞ norm); • whether a result analogous to Theorem 1.1 holds in higher dimension; • under which conditions on H we can choose H ε := H in Theorem 1.1 (for instance, H strictly positive or chessboard-like); • whether the random setting, instead of the periodic one, exhibits similar phenomena; • whether a PDE analogue holds (for instance, whether there exists a mesoscopic phase transition [9] in the plane whose interface is a closed curve).
As a consequence of Theorem 1.1, we have a pinning phenomenon for the curvature flow.
Namely, given δ > 0, for an open interval I ⊆ R and a function H : T 2 → R, we say that a family of closed, smoothly embedded curves for any x ∈ Γ t and any t ∈ I. Here v, κ and ν denote, respectively, the normal velocity, the curvature and the exterior unit normal of E t at x ∈ Γ t . Notice that when H = 0, (1.4) reduces to the usual curvature flow [7]. Equation (1.4) has been studied for instance in [6], where a general existence result has been established. We denote by d H (A, B) the Hausdorff distance between two sets A, B ⊆ R 2 . With this notation, we have that solutions of (1.4) are, for a "typical" H, confined in a δ-neighborhood of their initial data, according to the following result: Theorem 1.2. Let H ∈ L ∞ (T 2 ) be such that both H + ≡ 0 and H − ≡ 0, where H ± denote respectively the positive and the negative part of H. Then, for any ε > 0 there exist H ε ∈ C ∞ (T 2 ), satisfying (1.1), (1.2) and (1.3), and C ε > 0 such that any {Γ t } t∈I , Γ t = ∂E t , which moves by δ-periodic H ε -curvature satisfies (1.5) Related pinning effects in the graph case have been studied in [4]. Theorem 1.2 should be compared with the results in [3,2], where the limit of the functionals is carefully investigated (as usual, in (1.6), we denoted by Per the perimeter of a Caccioppoli set), and it is shown that the functionals in (1.6) converge, in the sense of Γ-convergence, to an anisotropic perimeter, with anisotropy depending on H.
Since equation (1.4) corresponds to the gradient flow of (1.6), one may expect that the solutions of (1.4) converge, as δ → 0, to a solution of the gradient flow of the limit functional, that is to an anisotropic curvature flow. We refer to [10] for a presentation of a general framework of convergence of gradient flows, under suitable conditions on the energy. However, the result in Theorem 1.2 indicates that this is not always the case, as the solutions of (1.4) do not move in the limit due to the effect of the strong forcing term.

CLOSED CURVES OF PRESCRIBED CURVATURE AND A PINNING EFFECT 79
The rest of the paper is organized as follows: Section 2 contains the proof of Theorem 1.1, by making use of an auxiliary result, namely Proposition 1, which is proved in Section 3. The proof of Theorem 1.2 is given in Section 4.

2.
Proof of Theorem 1.1. The main step towards the proof of Theorem 1.1 consists in the following Suppose that there exist r and r > 0 in such a way that r ∈ [r, 1/4] and c > 0 for which and K(x) = 0 for any x outside Then, there exists a C ∞ closed, convex curve γ whose curvature at any points is equal to K.
We postpone the proof of Proposition 1 to Section 3 and we show now that Proposition 1 implies Theorem 1.1.
For this, we fix a small ε > 0 and we take H as in the statement of Theorem 1.1.
We consider a standard mollifier ρ ε and we define the mollification of H as where ρ ε is chosen in such a way that , since the other case can be treated analogously. Up to change of coordinates, we may suppose x o = 0. Then, by periodicity, We take a cut-off function and τ ε (x) = 0 for any x outside We set Then, by (2.2), and K ε ≥ 0 on R 2 . Thus, in both the cases considered above, we have found and for suitably small c ε , r ε > 0. We can thus apply Proposition 1 and obtain a C ∞ curve γ ε = ∂E ε , with E ε compact convex set, such that the curvature of γ ε is equal to K ε at any point. (2.5) We denote by Notice that π(γ ε ) is a closed set of zero Lebesgue measure in T 2 and so we can find a ball β ε , with Lebesgue measure b ε ∈ (0, 1), and open sets U We consider a cut-off function ψ ε ∈ C ∞ (T 2 , [0, 1]) such that ψ ε (x) = 1 for any x ∈ U (1) ε and ψ ε (x) = 0 for any x outside U (2) ε . Hence, we take α ε ∈ C ∞ (T 2 , [0, +∞)) to be a cut-off function such that α ε (x) = 0 for any x outside β ε and βε α ε (x) dx = 1.
By definition of b ε , we can also suppose that Let also Note that the curvature of γ ε agrees with H ε , due to (2.5), since the support of π(γ ε ) lies in U ε . Therefore, γ ε satisfies the claim of Theorem 1.1. We sow that H ε also satisfies the claims of Theorem 1.1. For this, we use (2.6) and (2.8) to get which proves (1.2). Finally, (2.8) gives (1.3) and H ε is C ∞ (T 2 ) by construction.
Notice that, if we have instead K ε ≤ −c ε on B 3rε (x o ), we can reason as above replacing the function H with −H. The only difference is that in this case we obtain a curve γ ε = ∂E ε , still satisfying (2.5), where E ε is unbounded and R 2 \ E ε is a compact convex set.
This completes the proof of Theorem 1.1 when Proposition 1 is in force.
3. Proof of Proposition 1. First of all, we fix α > 0, to be taken conveniently small in what follows, and we construct a closed convex polygon P α whose vertex are in Z 2 and such that the angles between its edges are in [π − α, π).
For this scope, we fix a small a > 0 and a point P 1 ∈ Z 2 . We take a half-line λ 1 with rational slope through P 1 whose angle with respect to the horizontal axis is in [a, 2a]. Say, for definiteness, that the angles we consider are taken to be oriented anticlockwisely. Due to the rationality of the slope of λ 1 , there exists P 2 ∈ Z 2 ∩ λ 1 . We then take a half-line λ 2 with rational slope through P 2 whose angle with respect to λ 1 is in [a, 2a].
We then iterate this procedure (see Figure 1) and we find a half-line λ n with rational slope through P n whose angle with respect to λ n−1 is in [a, 2a].
We denote by β n the angle between λ n and the horizontal axis. By construction, β n ∈ [β n−1 + a, β n−1 + 2a] (3.1) and therefore we can take m to be the first angle for which β m ≥ (π/2) − 3a.
In particular, the angle between λ m and the vertical axis is in (0, 3a]. The polygon P α is then obtained by the segments P 1 P 2 . . . P m+1 by even reflections along the horizontal and vertical axes. The reflections make P α closed. Since P n ∈ Z 2 for any n, the vertices of P α are in Z 2 . Also, if a is chosen suitably small, the angles of P α are close to π but less then π (thus, in particular, P α is convex).
We now take c and r > 0 as in the statement of Proposition 1 and we construct a closed C 1,1 curve Γ which consists in: • pieces of segments outside • arcs of circumferences with curvature less then c/2 in B r .
The curve Γ is constructed by modifying P α . Indeed, we take Γ to agree with P α outside B r .
Then, if P is a vertex of P α , we call Q and R to be the two points in ∂B r/2 (P )∩P α and we take Γ in B r/2 (P ) to be the arc of circumference passing through Q and R and tangent to P α from inside (see Figure 3). If we call 2θ the angle of P α in P , the radius ρ of such circumference satisfies ρ = r 2 tan θ, due to standard trigonometry (see Figure 4). Accordingly, the curvature of Γ inside B r/2 (P ) is of the order of 1/(r tan θ). Since we know that θ ∈ (π − α)/2, π/2 , such curvature is smaller than c/2, provided that α is small enough (possibly in dependence of r and c).
This ends the construction of the curve Γ satisfying the desired properties.
We define E to be the bounded set for which ∂E = Γ. Let also R ⊇ E to be a square, with horizontal/vertical edges, such that By (3.3) and our hypotheses on K, we have that We look at the following functional. Given any bounded Caccioppoli set F ⊂ R 2 (see [8] for the definition and the basic properties of such an F ), we define By standard compactness arguments (see, for instance, [8] or page 1425 in [1]), the functional I attains its minimum under the constraint that Let F be one of such minima. We have that the curvature of γ := ∂F is equal to K at any point in which γ does not touch ∂E ∪ ∂R (see, for instance, Section 11.1 in [1]). Then, the proof of Proposition 1 will be finished once we show that To prove (3.5), we first observe that the curvature of γ is bigger or equal to K in a neighborhood of ∂E .
Indeed, if we take a small perturbation F of F , supported in the neighborhood of ∂E , for which F ⊆ F , we know that We take ν to be the external normal of F and we write F as a normal deformation (see [8]), that is for some smooth compactly supported function ζ and > 0. Then, if π ∂F is the natural projection onto ∂F , we have where H n−1 is the (n − 1)-dimensional Hausdorff measure. Also (see formula (10.12 hence C ≥ K on ∂F , which proves (3.6). We now make an elementary observation of strong comparison principle type. Namely, for δ > 0, if u ∈ C 2 ((0, δ)) ∩ C 1 ([0, δ)) with u(t) ≥ 0 for any t ∈ [0, δ), u (0) = u(0) = 0 and div u (t) 1 + (u (t)) 2 ≤ 0 for any t ∈ (0, δ), then u(t) = 0 for any t ∈ [0, δ). Indeed, thanks to (3.6), the osculating circle of γ has curvature bigger than, or equal to, c in B r (j). Since the curvature of the osculating circle of ∂E in the interior of B r/2 (j) is at most c/2, we see that (3.11) holds true. Moreover, Indeed, if such a touching point P existed, since ∂E contains a segment passing through P , we would obtain from (3.10) that γ and ∂E agree as long as ∂E is flat, that is up to ∂B r/2 (j ), for some j ∈ Z 2 . But this would be in contradiction with (3.11) and it thus proves (3.12). Therefore, from (3.11) and (3.12), we have that Furthermore, γ cannot touch ∂R at its corner, since cutting the corner would decrease the perimeter and leave unchanged the term F K(x) dx, thanks to (3.4), thus decreasing I. Also, γ cannot touch ∂R at the other points as well, since otherwise it should be a straight line in a neighborhood of ∂R , due to (3.4).
These observations together with (3.13) imply (3.5) and so complete the proof of Proposition 1.

4.
Proof of Theorem 1.2. For all ε > 0, we let γ ± ε = ∂E ± ε be the smooth curves given by Theorem 1.1, which correspond to the forcing term ±H respectively. 1 Thanks to our assumptions on the function H, we may assume that the sets E ± ε are both compact and convex. Therefore, we can find a square with integer vertices containing γ ± ε , and we denote by C ε the sidelength of such square. Thus, we consider a tiling of R 2 made by squares of sides C ε each containing an integer translation of E ± ε (see Figure 5). 1 We remark that the construction of γ − ε may be performed consistently with the one of γ + ε , up to changing Hε in a small set. Indeed, if any of the straight segments of γ − ε enter a ball B r/2 (j) of (3.2), used in the construction of γ + ε , one takes a small neighborhood of such segment in B r/2 (j), resets Hε to be zero and γ + ε to be a segment there, with a smooth interpolation. In dealing with the proof of Theorem 1.2, up to a dilation of factor 1/δ, we may and do assume that δ := 1 in (1.4). Thus, we take any {Γ t } t∈I , with Γ t = ∂E t , that moves by 1-periodic H ε -curvature and we show that sup s,t∈I d H (Γ s , Γ t ) ≤ const C ε . (4.1) Dilating back by a factor δ the estimate in (4.1), we then obtain (1.5).
To prove (4.1), we observe that all the integer translations of E + ε and of R 2 \ E − ε (which is an unbounded set) are stationary solutions of (1.4), with δ := 1. Consequently, by comparison principle (see, for instance, page 18 in [5]), Γ t cannot travel neither through the translations z + γ + ε such that (z + E + ε ) ⊂ E t , z ∈ Z 2 , nor through the translations z + γ − ε such that E t ⊂ (z + R 2 \ E − ε ). Such confinement proves (4.1) and thus completes the proof of Theorem 1.2.