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1 Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig Γ-convergence of the Allen Cahn energy with an oscillating forcing term by Marcello Lucia, Nicolas Dirr, and Matteo Novaga Preprint no.:

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3 Γ-convergence of the Allen Cahn energy with an oscillating forcing term N. Dirr Max-Planck Institute for Mathematics in the Sciences Inselstr. 22, D Leipzig, Germany M. Lucia Dept. of Mathematics, Hill Center Rutgers University Piscataway, NJ 08854, USA M. Novaga Dipartimento di Matematica Università dipisa Via Buonarotti 2, Pisa, Italy Abstract We consider a standard functional in the mesoscopic theory of phase transitions, consisting of a gradient term with a double-well potential, and we add to it a bulk term modeling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness leading to a nontrivial interaction of forcing and concentration when 0. We show that the functionals Γ converge after additive renormalization to an anisotropic surface energy, if the period of the oscillation is larger than the interface thickness. Difficulties arise from the fact that the functionals have non constant absolute minimizers and are not uniformly bounded from below. 1 Introduction We briefly review the classical theory of phase transitions. Given Ω R N, let u :Ω R be an order parameter, i.e. a function which describes to what extend the physical system in a given point x Ωisinthe + or phase. Pure phases correspond to the two minimizers (for instance ±1 of a double-well potential W, which can be derived from atomistic considerations as mean-field free energy, and whose main property is to be convex in a neighborhood of ±1. The resulting free energy functional is characterized by a competition between a gradient term, modelling interaction energy, and the potential W. Such a functional is given by: { M (u := u 2 + W (u } dx, u H 1 (Ω, (1.1 Ω Supported by the German Academic Exchange Service (DAAD and the Department of Mathematics of the University of Texas at Austin 1

4 where >0 is a small parameter related to the interface thickness. If the system is prevented from staying close to +1 or to 1 everywhere(forexamplebyavolume constraint, then the transition layer (roughly speaking the set separating the positive and negative regions, will formally be of order. Moreover, sequences of finite energy for 0 should converge to ±1 almost everywhere. A suitable mathematical setup to make this rigorous is the notion of Γ-convergence. In [14] (see also [13] the authors characterize the Γ-convergence with respect to the L 1 (Ω topology of the family M, and they obtain the sharp interface limit, which is the area of the interface with surface tension c W (which is related to the double-well potential. More precisely, by setting c W := 1 1 W (tdt and B := {u BV (Ω : u(x { 1, 1} a.e. in Ω}, they prove that the Γ-limit of the functionals in (1.1, extended by + to all L 1 (Ω, is given by M 0 (u := { cw P (E,Ω if u = χ E B, + if u L 1 (Ω \B. (1.2 This convergence could be perturbed by rapidly oscillating spatial inhomogeneities modeling for example the interaction with a substrate. The result will depend on whether the scale on which the inhomogeneities oscillate is of order of the interface thickness, smaller or larger. One way to introduce spatial inhomogeneities is to consider an x-dependent gradient term, i.e. replace the term u 2 in (1.1 by A ( x u 2, where A(x is a positive definite symmetric matrix, periodically depending on x (a general version of this case is studied in [1]. In our paper the α energy in (1.1 is perturbed by a strong, rapidly oscillating field with zero average. More precisely, in our paper, we shall consider the functional { G (u := u(x 2 + W (u(x + 1 ( x } α g α u(x dx u H 1 (Ω, Ω where g L (R N is a periodic function with cell domain Q := ( 1/2, 1/2 N. This periodic term g has the effect of creating many local minima. Systems of this type are of relevance in material science, e.g the evolution of microstructures or the motion of magnetic walls. When α = 0, it follows from the results in [13, 14] (see also [7, Proposition 6.21] that the Γ-limit is the sum of the functional (1.2 and the volume term g(xu(x dx. When α>0, both amplitude and frequency of g become large as 0, hence the infimum of the functional over H 1 (Ω can be negative or even converge to as 0(forexamplewhenα>1/2, see Prop Therefore, to fit in the framework of Γ-convergence, we need to introduce an additive renormalization. However, in order to get a nontrivial Γ-limit, we need the renormalization to be of the same order of the perimeter and this can happen only if Q g dx = 0. We show for 0 <α<1 that the renormalized functionals Γ-converge to an anisotropic surface energy (see Theorem

5 There are similarities with the result in [1] but in many respects our setting requires new techniques. The main difficulties (beyond those encountered in [13, 14] and [1] arise from this renormalization and the (related facts that the functionals have non constant global minimizers whose energy is not uniformly bounded from below. To explain the main points, let us first note that the Euler-Lagrange equation is u W (u 2 = 1 2 α g ( x α on Ω, u =0 on Ω, (1.3 n i.e. the function g appears as a forcing term. There are two solutions of (1.3 u +,closeto+1,andu,closeto 1 (see Proposition 3.7 and Corollary 3.8 which are local minimizers of the energy and which are nonconstant if g 0, whereas in the unperturbed case or in [1] one gets u + 1, u 1. As their energy is strictly negative, and typically is of order Ω 1 2α, the aforementioned additive renormalization is necessary. The appearance of such a renormalization is in fact quite natural for phase transitions problems. The energy associated with an interface is the excess free energy due to the fact that more than one phase is present, so it is actually a difference of energies, determined only up to adding constants. If the pure phases, i.e. the global minimizers, are constants, then in order to ensure that the energy of the minimizers is zero, it is enough to choose min R W (u =0. In our case the minimizers are not constants, so we must compute their energy and show that it is proportional to the volume of the domain Ω (up to smaller order, as we want a local functional as Γ-limit. Moreover (again up to smaller order the energy of u + and u must be the same. Conditions on W and g will ensure both these properties. Now we consider the different scalings, i.e. the oscillation of g in relation to the interface thickness. In this paper we treat rigorously the case of slow oscillations, i.e. 0 <α<1, leaving the case α 1 to further investigation. Let γ : R [ 1, 1] be the unique increasing solution of 2γ = W (γ (1.4 which converges exponentially to ±1 at ±, andsuchthatγ(0 = 0. Performing the change of variables y = x α and letting ũ(y =u(x α, (1.3 becomes 1 α ũ W (ũ 2 1 α = 1 g (y. (1.5 2 Then a formal asymptotic expansion for solutions of (1.5 gives ( ( d(y ũ(y =γ 1 α + 1 α ũ 1 σ(y, d(y,y + o( 1 α, 1 α where d(x is the signed distance from the zero-level set of ũ (which we assume to be a smooth hypersurface and σ(y := y d(y d(y is the proection of y onto {ũ =0}. It follows c W d(x =g (xon{ũ =0}, which on the original scale becomes c W κ = 1 ( x α g α, (1.6 3

6 where κ is the mean curvature of the zero-level set of u. Hence, for α<1the problem is related to singular homogenization for the prescribed mean curvature equation. Indeed, in this case there is a splitting of the Γ-limit into a more standard limit, similar to [13, 14] with a g-term which does not depend on, and a prescribed mean curvature problem (see Theorems 2.1and 5.9. (1.6 shows that the chosen relation between amplitude and frequency of the forcing is interesting, since the interface will change its shape significantly within one unit cell. For a stronger amplitude we expect to see small bubbles everywhere, as the minimizers on a cell are no longer of constant sign, whereas for a weaker forcing the limit will be isotropic. Now we are able to summarize our results. Any sequence of bounded energy has a subsequence which converges in L 1 to a BV-function, which takes its values in { 1, 1}. The Γ-limit with respect to L 1 -convergence has the form ϕ(ν E dh N 1, (1.7 E Ω where E is a finite perimeter set on which lim u =1andν is the unit normal to E. Thanks to the aforementioned splitting, the anisotropy ϕ can be explicitly characterized (see Theorem 5.9, and it holds ϕ(ν =ϕ( ν andϕ c W for any forcing term g satisfying certain bounds and a symmetry condition, see Proposition We add a few remarks on the case of fast oscillations, i.e. α 1. For α =1there is no splitting of scales which makes this case more difficult, but under stronger conditions on g we still expect to obtain Γ-convergence towards anisotropic surface energy, even if without an explicit characterization of the energy as in (5.17. For α>1, we expect that the limit will be isotropic: When α 1, we can rewrite the functional G with the help of the periodic function Ψ(x whichsolves Ψ=g, and obtain that the problem is related to that of x-dependent potentials, ( W (v, x, =W v α 1 Ψ(x/ α A formal analysis of (1.3 shows that absolute minimizers will be of the form ±1 1 2 β Ψ( α x+o( β. Formally, the function γ( 1 x 1 2 β Ψ( α x connects them at a cost of c W, if the renormalization is taken into account, and it solves the Euler- Lagrange equation at the first order. The paper is organized as follows. In Section 2, we state the assumptions W and g must fulfill, we give a precise definition of the renormalized functional and we give a precise statement of our results. In Section 3 we show the existence of the minimizers u ± and estimate the cost of having a transition within a cube. In Section 4, we show that any sequence with bounded energy has a subsequence converging in L 1 (Ω to a BV-function taking values only in { 1, 1}. Using the estimates of Section 3, we derive the so-called fundamental estimate which is a localization property and show also that the limit energy of our functional is bounded from above and below by area functionals, if the functional is evaluated on sequences converging to characteristic functions of smooth sets. General principles allow to derive from these estimates a first Γ-limit theorem, which is valid up to a subsequence (see 4

7 Proposition In Section 5, we derive further properties of the limit functional and obtain, in particular, a representation formula (see Theorem 5.9, which implies that the Γ-limit is independent of the subsequence and of the scale parameter α. 2 Notations and main results Let N 2. We denote by A the class of all bounded open subsets of R N and by Q := ( 1/2, 1/2 N the open unit cube in R N centered at 0. For each E R N,the characteristic function χ E of E and the signed distance function d E to E are defined respectively by: { 1 if x E, χ E (x := 1 otherwise, { dist(x, R d E (x := N \ E if x E, dist(x, E otherwise. Moreover, if E Ω Awith χ E BV (Ω, the reduced boundary of E will be denoted by E (see [10]. Given Ω Aand >0, we consider the following functional { } Ω u 2 + W (u dx + 1 Ω g ( x G (u, Ω := α u dx, if u H 1 (Ω, α + otherwise. We require that g and W satisfy the following assumptions: (2.1 (H1 g L (R N is a periodic function with cell domain Q, satisfying Q g dx =0; (H2 W Lip loc (R, W 0, W (s =0iffs { 1, 1} and W (s =W ( s; (H3 There exist δ 0 (0, 1 and C 0 > 0 such that W is strictly convex on the interval (1 δ 0, + and C 1 0 (s 12 W (s C 0 (s 1 2, s (1 δ 0, + ; (H4 There exists ρ>0 such that W (1 + s W ( 1+s = 0 whenever s 1 <ρ; (H5 g(x 1,...,x i,...,x N =g(x 1,..., x i,...,x N for any i {1,...,N} (in this case we say that g is symmetric. A typical example of function satisfying (H2 and (H3 but not (H4 is given by the double-well potential defined by W (s =(1 s 2 2 /2. Assumption (H4 ensures that the two local minimizers around ±1, i.e. the pure phases, have exactly the same energy (hence they are both global minimizers of the energy. Without that condition, the Γ-limit could become trivial (equal to 0 or +. We observe that (H4 is not necessary in order to get the Γ-limit result when α<2/3 (see Remark 4.10, whereas it is necessary if α>2/3. 5

8 Notice also that assumption (H3 implies W (x C0 1 x 1 for x 1 δ 0, W (x C0 1 x +1 for x 1+δ ( We will see that in general lim 0 inf H 1 (Ω G (, Ω = for α>1/2, hence we shall introduce an additive renormalization for the functionals. Let R be the family of all set of the form R =int( z I α {Q + z}, where I is a finite subset of Z N. Given Ω Aand u L 1 (Ω, we define the renormalized functionals as { } sup G (u, R inf F (u, Ω := G (,R if {R R : R Ω}, R R,R Ω H 1 (R 0 otherwise. Note that inf L 1 (Ω F = 0 and since inf H 1 (R G (,R 0 (by comparison with constant functions, we also have F G. Our main result is the following: Theorem 2.1. Let 0 <α<1. Under the assumptions (H2, (H3 and (H4 whenever α 2/3, there exists a constant c 0 := c 0 (W such for any g satisfying (H1, (H5 and g L N c 0,theΓ-limit (with respect to the L 1 -topology of F (, Ω exists for each Ω Awith Lipschitz boundary. Furthermore, we have ϕ(ν E dh N 1 if u = χ E BV (Ω, Γ lim F (u, Ω = 0 E Ω (2.3 + otherwise, where ϕ : S N 1 (0,, independent of α, satisfies 0 <C ϕ(ν c W for all ν S N 1, (2.4 for some constant C>0, and its one-homogeneous extension ϕ : R N [0,, x { x ϕ(x/ x if x 0, 0 if x =0, (2.5 is convex. Remark 2.2. The function ϕ can be computed as a limit of the averaged minimum energy on large boxes of the functional Fg A (χ E:=c W P (E,A+ g(xχ E (xdx, (2.6 defined for each Borel set A Ωandeachχ E BV (Ω (see Theorem 5.9. Remark 2.3. We point out that the results of this section can be generalized to functionals with an x-dependence in the gradient term (see also [5], like for example { ( x Ĝ (u, Ω := A Ω β u 2 + W (u } 1 ( x dx + Ω α g α u dx, where α (0, 1, β 0andA(x is a positive definite symmetric matrix, periodically depending on x. A 6

9 3 Estimates for the minimizers In the following, unless otherwise stated, we shall always take α (0, 1. As we are interested in a local Γ-limit, we ultimately have to show that the renormalization is proportional to Ω. This will be done by comparing with minimizers on a cube. We need the following definitions. Definition 3.1. Let G (u, Ω := Ω ( u 2 + W (u Notice that, by the change of variables y = α x and setting dx + gu dx, u H 1 (Ω. (3.1 Ω v(y =u( α y, Ω := {y R N : α y Ω}, for Ω R we obtain the identity G (u, Ω = α(n 1 z Z N G 1 α ( v, (z + Q α Ω. (3.2 Thanks to condition (H5, in order to study the structure of minimizers of G on R, it is enough to analyze the minimizers on the cube with Neumann boundary conditions (which, again by condition (H5, are equivalent to periodic boundary conditions. Let us set c W := 1 1 and consider the functional { } W (t dt, B := u BV (Q :u(x { 1, 1}, a.e., { Fg Q (u := cw P (E,Q+ Q gχ E dx, if u = χ E B, + if u L 1 (Q \B. From the result of [13, 14] we have Γ lim G (,Q=F Q g. This fact gives some hint on the asymptotic behavior of the minimizers of the functionals G (,Q. To see this, let us recall the following isoperimetric inequalities [9, Section 5.6]. Proposition 3.2. Let Ω Awith Lipschitz boundary. Then, there exists a constant I(Ω > 0 such that 1. P (E,Ω I(Ω (min{ Ω E, Ω \ E } N 1 N for any E Ω; 2. Ω Du 2I(Ω u u N/(N 1 for any u BV (Ω, whereu := 1 Ω Based on this result, we can derive: Ω u. Proposition 3.3. Let Ω Awith Lipschitz boundary. then the minimizers of Fg Q are given by u ±1. If g L N (Q 2c W I(Q, 7

10 Proof. Since Fg Q (1 = Fg Q ( 1 = 0, it is enough to show Fg Q (u 0 for all u B. We have c W Du c W 2I(Q u u N/(N 1, Q gu = g(u u g N u u N/(N 1. Thus, Q Q Fg Q (u c W 2I(Q u u N/(N 1 g N u u N/(N 1, = u u N/(N 1 (c W 2I(Q g N, and the last term is nonnegative by assumption. Proposition 3.3 implies that if the minimizers of G (,Q exist and converge in L 1, they must converge to ±1. We need now to quantify this information, i.e. to obtain rates in. Proposition 3.4. Assume (H1 to (H3. Then, for any u H 1 (Ω we have G (t u ( t, Ω < G (u, Ω t >1+C 0 g. (3.3 Proof. By setting Ω t := { u >t}, from (H2 and (2.2, we get G (u, Ω G (t u ( t, Ω 1 W (u W (t dx + g(u sgn(ut dx, Ω t Ω t 1 (W (t g ( u t dx, Ω t 1 (C0 1 (t 1 g ( u t dx, Ω t and the last expression is positive whenever t>1+c 0 g. The following definition introduces a cutting and reflection procedure, which gives a function u t assuming values only in one of the convex regions of the potential W. Definition 3.5. Given u H 1 (Ω and t>0, we define { u t, if {u >0} 1 u t 2 := Ω ; ( u t, if {u >0} < 1 2 Ω. We are going to use this cutting to give an estimate of the energy required to have a sign change of the function u. 8

11 Proposition 3.6. Let Ω Awith Lipschitz boundary. Assume (H1 to (H3 and g < 1 2 C 1 0 δ 0. Then, there exist a constant t 0 with max{ 1 2, 1 δ 0} <t 0 < 1 and ω 0 > 0 (t 0, ω 0 depending only on W such that, G (u, Ω G (u t, Ω ( ω 0 8 g t/2 L N P ({u <s}, Ω ds. (3.4 t 0 I(Ω t/2 whenever u H 1 (Ω and t (t 0, 1 2C 0 g. Moreover, the inequality is strict if { u <t} > 0. Proof. Assume w.l.o.g. that {u >0} Ω /2 and, in the light of Proposition 3.4, that u 2 t. Recall that W (u =W ( u and compute G (u, Ω G (u t, Ω = + 2 { t<u<t} {u t} u 2 + W (u W (t gu dx = G 1 + G 2 + G 3, + g(u t dx where G 1 := G 2 := G 3 := { t u<t} { t<u<t} { t<u< t/2} ( u 2 + W (u W (t 2 W (u W (t dx, 2 dx + g(u t dx, g(u t dx +2 { t/2 u<t} {u t} gu dx. Let us first observe that (H2 and (H3 imply the existence of a value t 0 (depending only on W withmax{ 1 2, 1 δ 0} <t 0 < 1 such that, for all t (t 0, 1, we have W (s W (t+w (t(s t s > 1 2, (3.5 W (s W (t 0 s <t and inf s <1/2 {W (s W (t 0} > 0. (3.6 Let us also define ω 0 := inf 2{W (s W (t0 }. s <1/2 1. By using Schwarz inequality and Co-Area formula, we estimate G 1 as follows G 1 since { t u<t} t/2 2{W (u W (t} u dx ω0 P ({u <s}, Ωds, (3.7 inf 2{W (s W (t} s <t/2 t/2 inf 2{W (s W (t0 } = ω 0. s <1/2 9

12 2. We show that G 2 0. Using (3.5, we get for all t 0 <t<1 2C 0 g that W (u W (t G 2 + g(u t dx { t/2 u<t} 2 W (t 2g (t udx { t/2 u<t} 2 C0 1 (1 t 2 g (t u dx { t/2 u<t} 2 0 (3.8 and G 2 > 0if {u <t} > In order to estimate G 3,weuse u 2 t and Hölder to get G 3 2t g dx +2(2 t g dx { t<u< t/2} {u< t} 4 g dx {u< t/2} { 4 g L N u< t } N 1 N. (3.9 2 From the fact that {u < s} is a nondecreasing function of s, and using Proposition 3.2 together with the assumption {u >0} Ω /2, we get t 2 {u< t N 1 2 } N 0 Therefore, (3.9 gives t 2 G 3 8 t {u <s} N N ds P ({u <s}, Ωds. I(Ω t 2 g L N I(Ω 8 t 0 g L N I(Ω 0 t 2 0 t 2 P ({u <s}, Ωds, P ({u <s}, Ωds. ( Finally, from (3.7, (3.8 and (3.10 we obtain ( G 1 + G 2 + G 3 ω 0 8 t g L N 2 P ({u <s}, Ωds. t 0 I(Ω t 2 Moreover (3.8 implies that the inequality is strict if {u <t} > 0. In the following proposition, we show that the functional G admits global minimizers which are close to +1 or 1 ofanorder (see [11] for a similar result in case of minimizers of (1.1 with a volume constraint. 10

13 Proposition 3.7. Let Ω Awith Lipschitz boundary. Assume (H1 to (H3 and g < (1/2C 1 0 δ 0. Then the following holds. 1. The functional (3.1 admits a global minimizer u in H 1 (Ω. 2. Let H±(Ω 1 := {u H 1 (Ω : ±u 0a.e. in Ω}. Then, there exist positive constants c 0 (Ω,W, C 1 (Ω,W and 0 (Ω,W such that for g L N c 0 any global minimizer u must be contained in H+ 1 or H. 1 Moreover, any minimizer u ± H± 1 has the following property: u + 1 C 1, u +1 C 1 for < 0. Since the restriction of G (, Ω to B δ 0 (+1 (respectively to B δ 0 ( 1 is convex, Proposition 3.7 implies Corollary 3.8. Let Ω Awith Lipschitz boundary. Assume (H1 to (H3. and g L N c 0 (W, Ω. Then, for any such that g <C0 1 δ 0, the functional G (, Ω has exactly one absolute minimizer u + in H+(Ω 1 and one absolute minimizer u in H 1 (Ω. there exists a t 0 (1 δ 0, 1 such that for all u H 1 (Ω ( G (u, Ω min G (u +, Ω, G t0 (u /2, Ω C P ({u <s}, Ω ds. (3.11 t 0 /2 If W satisfies (H4, we also have u + =2+u and G (u +, Ω = G (u, Ω, and u ± are the only global minimizers in H 1 (Ω. Now we prove Proposition 3.7. Proof. The existence of a global minimizer follows from classical results (see for example Thm 2.6, [7]. From Proposition 3.4 we get immediately that the global minimizer u fulfills u 1+C or u 1 C for some C depending only on Ω and W. Assume now w.l.o.g. that {u > 0} Ω /2. Proposition 3.6 tells us that for a minimizer there exists a t with 1 δ 0 <t<1 such that the { t/2 <u <t} =0. Moreover it implies that P ({u <s}, Ω = 0 for some s ( t/2,t/2. Hence the isoperimetric inequality implies that also {u < t/2} = 0 is empty. Therefore u (x (1 δ 0, 1+δ 0 almostsurely. Proposition 3.9. Assume (H1 to (H3 with g 0. Then, 0 > min { G (,Q} 2C 0 g 2. (3.12 H 1 (Q Moreover, let Ω A. Then, for any (, α and any R R with R Ω, we have 0 > min {G (,R } 2 Ω C 0 g 2 1 2α. (3.13 H 1 (R In particular, as 0, we have { o(1 if α (0, 1/2, min {G (,R } = (3.14 H 1 (R O(1 if α =1/2. If α>1/2, R R with R Ω such that lim min {G 0 (,R } =. H 1 (R 11

14 Proof. Let v be a global minimizer of G on H 1 (Q. By Proposition 3.7, we assume w.l.o.g. that v 1 2C 0 g 2. This estimate together with the assumption that g is of average zero on Q yield G (v, Q gvdy g v + 1 2C 0 g 2. Q This proves (3.12. Now, note that the number of cubes of size α contained in R is equal to R. Hence, by using (3.2, we get for each u H 1 (R αn G (u, R R αn α(n 1 min G 1 α(,q= R H 1 (Q α min G 1 α(,q. (3.15 H 1 (Q Hence, from (3.15, (3.12 and the fact that R Ω, we derive (3.13. Consider now the case when α>1/2. Choose a function v Cc 1 (Q such that Q gv dx <0 (which is always possible if g 0 and extend it periodically on RN. Consider R R with R Ω /2. Then, using as before (3.2, we get G ( v( x α,r = R G α 1 α( v, Q Ω ( 2(1 α v 2 + C 0 v α gv dx 2 Q for 0. Above proposition shows that F and G have the same Γ-limit whenever α<1/2 and so the renormalization is not needed in such a case. We introduce the following definition in order to express the additive renormalization in a more convenient way. Definition Let u ± 1 α denote the minimizer of G 1 α on H 1 (Q {±u 0}. 2. Let c := α inf G v H 1 (Q 1 α(v, Q. Proposition Assume (H1 to (H3. If furthermore (H5 holds, i.e. if g is symmetric, then the functions which minimize min G H 1 (Q (,Q are periodic. Moreover, if (H4 holds then min G = R H 1 (R G α 1 α(u ±,Q= R c 1 α. Moreover, the functional F is additive on disoint sets contained in R. Proof. Let us denote by Hp 1(Q the class of periodic H1 - functions on the unit cube. Recall that the minimizers u + (resp. u are unique in the class of positive (resp. negative H 1 -functions. By symmetry of g, u + (x 1,..., x i,...x n is also a minimizer and thus equal to u +. The same holds for u. In particular the traces of u ± on opposite facets of the cube coincide, so u ± Hp 1(Q. 12

15 4 Γ convergence In this section, we establish the Γ-convergence of the functionals F for 0. In order to proceed, we need to distinguish between cubes in which a function u is mostly positive and those in which u is mostly negative. Definition 4.1. Given (R,u R H 1 (R, we define { Z + := z Z N : α (Q + z R, {u > 0} α (Q + z 1 } 2 α (Q + z, { Z := z Z N : α (Q + z R, {u > 0} α (Q + z } 1 < 2 α (Q + z, R ± := α (Q + z. z Z ± Using the notations introduced in the above definition, we show: Lemma 4.2. There exists C > 0 such that for any (R,u R H 1 (R,the following holds: {u 1/2} R + + {u 1/2} R C α F (u, R, (4.1 R { W (u + u ( x } α g α dx C { F (u, R + R 1 2α}, (4.2 W (u dx C{F (u, R + R 1 2α }. (4.3 R Proof. We first show (4.1. By setting v(x =u( α x, we have F (u, R + { (N 1α G 1 α(v, z + Q G 1 α(u +,z+ Q}. (4.4 z Z + Lemma 3.8 and the isoperimetric inequality applied to (4.4 yield F (u, R + C(N 1α {v 1/2} (z + Q N 1 N. (4.5 z Z + Using in the relation above the inequality m i=1 A i max i {1,...,m} { A i 1/N } m i=1 A i N 1 N (available for any m N and any A 1,..., A m R, we derive F (u, R + C(N 1α {v 1/2} (z + Q = C α {u 1/2} R +. z Z + Hence, arguing in the same way on R, we finally derive α F (u, R ± C {u 1/2} R ±. (4.6 13

16 Therefore, (4.6 together with F (u, R F (u, R + +F (u, R imply (4.1. To prove (4.2 and (4.3, we will show W (u + u ( x R 2 α g α dx C { F (u, R + R 1 2α}. (4.7 First let us introduce the notation B ± := {x R ± : ±u (x < 1/2}. (4.8 We note that by (H2 and (H3 we can find a constant c with 0 <c<c 1 0 such that W (u c(u 1 2 for u [ 1/2,. Moreover, there exist C, 0 > 0 such that W (u α (u 1g( x α > 0 for u >C,< 0. Hence { W (u + u ( x } { W (u R + 2 α g α dx = + u 1 ( x } R + 2 α g α dx { W (u = + u 1 ( x } { W (u R + \B + 2 α g α dx + + u 1 ( x } B + 2 α g α dx { c(u u 1 ( x } α g α dx C α g B + R + \B + 1 R+ (1 α g 2 C g α F (u, R α (by 4.1 4c C { F (u, R + + R + 1 2α}. (4.9 The corresponding estimate holds for R as well and so we get (4.7. From (4.7, we derive immediately (4.2. Furthermore, since the renormalization per unit volume c is negative and using (4.7, we can estimate { 1 W (u W (u dx F (u, R + u ( x } 2 R 2 α g α R C{F (u, R + R 1 2α }. As a first step we show that the Γ-limit (if it exists concentrates exactly on the class of characteristic functions of sets of finite perimeter. Proposition 4.3. Let Ω Aand u L 1 (Ω be such that lim sup F (u <. 0 Then the following holds: (a If u n u in L 1 (Ω for any subsequence n 0, then u =1a.e. in Ω; (b there exists a subsequence n 0, u BV (Ω with u =1a.e. in Ω such that u n u L 1 loc (Ω 0. Moreover, there exists C := C(W, g > 0 such that u C lim inf F n (u n, Ω. (4.10 n 0 Ω 14

17 Proof. Let R R be such that F (u, Ω = F (u,r. (a From Lusin s and Egoroff s Theorems (see [9], we deduce the existence of a compact set K Ω such that (up to a subsequence: K 0, u K continuous, u n u in L (K. Since u 1 we can further assume the existence of a constant η>0 such that un (x 1 η>0 x K, n N. Letting now c := min s 1 η W (s > 0, for n large enough we have W (u n F n (u n, Ω G n (u n + 1 ( x K n α g n Ω α n c K u +. n g α n (b By refering to the Definition 4.1, we set Ω u n { 1 if z Z + σ(u,z=, 1 if z Z, We shall show that [Hu ](x = { 1 if x R +, 1 if x R. (4.11 Let us set u Hu L 1 (R 0(as 0 and Hu BV (R C. (4.12 B,δ := {x R : u (x < 1 δ} (δ >0. Note that for 0 <δ 1 u Hu L 1 (R δ R +3( B + + B +2 B,δ +2 { u >1+δ} u dx By applying Lemma 4.2, we get B + + B C α and so B + + B 0. By (H2, (H3 and the bound on the energy, ( B,δ + u dx =0, lim 0 { u >1+δ} this shows the first statement in (4.12. To prove the second one, we note that, by construction, the total variation of Hu can be estimated by: R [Hu ] (N 1α 4 z i z =1 σ (z i σ (z 2. Now consider a pair of cubes Q i := α (z i +Q (i =1, 2 such that (z 1,z 2 Z + Z and z 1 z 2 = 1 (i.e the cubes are adacent. By setting C := int(q 1 Q 2, we claim that there exists C>0 such that F (u, C C (N 1α. (

18 Case 1: Q 1 {0 <u < 1/2} > Q 1 4 or Q 2 {0 < u < 1/2} > Q 2 4. In such a case, (H3 implies there exists a constant c such that these cubes contribute c Nα 1 c (N 1α to the energy. Case 2: C {u > 1/2}, C {u < 1/2} Q 1 Q 2 8. In this case, as in the proof of Lemma 4.2, by applying (3.11 (on two adacent cubes C and the isoperimetric inequality (prop. 3.2, we deduce the existence of a constant c>0 such that G(u, C inf G (, C ( N 1 1 N c H 1 (C 8 Nα. Hence each such C contributes at least c α(n 1 to the energy. Since each cube has 2N nearest neighbors, we get R [Hu ] CF (u,r. Therefore Hu is bounded in BV and so it has a subsequence converging strongly in L 1 to a function u BV. As a consequence of the lower semicontinuity of the BV -norm with respect to L 1 -convergence we obtain K [Hu ] CF (u, Ω for any compact set K Ω. (4.10 follows by letting K Ω. By (4.12, the corresponding subsequence of the original sequence u converges to u as well. The fact that the Γ-limit is a measure relies on the following Proposition, which gives the so-called fundamental estimate [7]. Notice that in our case the proof is quite different from the usual one, due to the fact that G is not positive. Proposition 4.4. Assume (H1-(H3 and (H5. For any U, U,V A, U U, and for any u, v L 1 (R N there exists a function ϕ such that F (ϕu +(1 ϕv, U V F (u, U +F (v, V +δ (u, v, U, U,V (4.14 where δ has the property lim 0 δ (u,v,u,u,v=0, whenever { u v L 1 (S 0, with S := (U \ U V, sup {F (u,u +F (v,v+ u + v } < +. (4.15 Remark 4.5. Assumption (4.15 is stronger than the one made in [7], since we also require u and v to be bounded in L (R N. However, from hypothesis (H3 it follows that we can assume that a Γ-realizing sequence is bounded in L, hence the Γ-limit does not change if we redefine F + outside a suitable ball of L (R N. Let us define a sequence of strips as follows. Set U 0 := U and define by recurrence for each i N: { } Z i := z Z N : α (Q + z U, dist( α (Q + z,u i α, 2 16

19 U i+1 = α (Q + z, S i := (U i+1 \ U i V. (4.16 z Z i The proof is splitted in three parts. We start with the following Lemma whose proof is more general than needed, so that it can easily be modified for the case α 1. Lemma 4.6. Let U, U,V, u and v be as in Proposition 4.4. Assume there exist some S i0 defined by (4.16, S S i0 (S i0, S andϕ C (R N, [0, 1] such that F (u,s i0 +F (v,s i0 0, (4.17 S i0 u v α dx + S i0 \ e S S i0 u v 2 dx 0, es u v W (u +W (v dx 0, (4.18 dx 0, (4.19 S i0 { u 2 + v 2} dx C, (4.20 supp { ϕ} S, ϕ =1on U i0, ϕ =0on R N \ U i0 +1, ϕ C 1, (4.21 where C is independent of. Then,lim 0 F (ϕu +(1 ϕv,s i0 =0. Proof. In order to simplify notation, we shall write u, v instead of u,v and set z := ϕu +(1 ϕv. Wehave F (z, S i0 = F (u, S i0 +{G (z,s i0 G (u, S i0 } { = F (u, S i0 + ( z 2 u 2 W (z W (u + + g S i0 = F (u, S i0 +I 1 + I 2 + I 3. S i0 ( x } z u α α dx By (4.17 F (u, S i0 0 while (4.18 implies I 3 0(as 0. For I 2 we use the fact that W Lip loc, i.e. (H2, together with the inequality u + v C and the definition of z to get the estimate W (z W (u u v W (u+w(v dx C dx + dx. es S i0 \ es Assumptions (4.18 and (4.19 imply that this vanishes as 0. In order to estimate I 1, note that z u = ϕ(u v+(1 ϕ[ (v u] and z + u = ϕ(u v+ u + v ϕ[ (v u], so we estimate ( z 2 u 2 S i0 (4.22 [ C 1 2 u v L 2 ( S e 2 u v 1 2 ( u + v + L 2 (S i u v L 2 (S i0 L 2 ( e S 1 2 ( u + v L 2 (S i0 ]. 17

20 The bound u + v C allows to estimate the L 2 norm by the L 1 norm, therefore the first term in (4.22 vanishes by (4.18, the second by (4.18 and (4.20, and the third by (4.20. Lemma 4.7. Under the assumptions of Proposition 4.4 we can find sets S i0, S and a function ϕ which fulfill the assumptions of Lemma 4.6. Proof. Since U U, we can assume U, U R. Consider then the family of S i defined by (4.16. Let us denote by k be the largest integer for which S i and note that k = O( α. As the functional is increasing on sets in R, the bound on the energy (4.15 allows to assume that F (u,s+f (v,s C. Since the functional is additive on disoint sets in R (see Prop and k i=0 S i S, weget k i=0 {F (u,s i +F (v,s i } F (u,s+f (v,s C. As all terms in the sum are nonnegative, we get that for 2/3 of the indices i F (u,s i +F (v,s i 3 C = C α. ( k Such strips satisfy (4.17. The argument used above will be referred to as averaging argument. This averaging argument shows in addition that for 2/3 of the indices i u v C α u v. (4.24 S i S Hence we can find at least one strip S i0 which fulfills both (4.23 and (4.24. There exists a constant C 1 such that this strip is the disoint union of at least C 1 α 1 strips of the form (4.25. So another averaging argument yields a strip S S i0 of the form S = { x U : ( 1 dist(x, U i0 } V for some N, (4.25 in which we have ( u v C 1 1 α C α es S u v = C u v. (4.26 S As u v L 1 (S 0, equations (4.24 and (4.26 imply (4.18. Furthermore (4.3, (4.23 and S i0 C α imply (4.19. Moreover using the fact that the renormalization is negative, (4.2 together with (4.23 give: { u 2 + v 2 } 0 S i0 which implies (4.20. Finally from the definition of S given in (4.26, it is also possible to construct a function ϕ satisfying (

21 Using both previous Lemmata, we can now prove Proposition 4.4. Proof. Let i 0, S i0 and ϕ be as in Lemma 4.6 and Lemma 4.7. Since the functionals F are additive and setting z := ϕu +(1 ϕv we have F (z,u V = F (z, (U V U i0 +F (z, (U V (R N \ U i F (z, (U V (U i0 +1 \ U i0 = F (u, (U V U i0 +F (v, (U V (R N \ U i F (z, (U V (U i0 +1 \ U i0 F (u, U +F (v, V +F (z,s i0. By Lemma 4.6, F (z,s i0 0as 0, whenever (4.15 holds. In the following, we provide some estimates from above and from below for the Γ-limit, which are useful in order to represent the limit as an integral functional. Proposition 4.8. Assume that (H1 to (H5 hold and that g is as in Proposition 3.7. Then, there exists a constant C 3 > 0 such that Γ lim inf F (χ E, Ω C 3 P (E,Ω Ω A, E Ω. (4.27 Proof. Let n 0andletu n χ E in L 1 (Ω. We may assume that lim inf F n (u n, Ω <, hence there exists a subsequence such that A := lim F n (u n, Ω = lim inf F n (u n, Ω <, and u n χ E L 1 0. Now (4.10 implies that there exists a C>0such that χ E Clim inf F n (u n, Ω for a further subsequence. However by construction Ω C lim inf F n (u n, Ω = lim F n (u n, Ω = lim inf F n (u n, Ω, which proves the claim. Proposition 4.9. Assume that (H1 to (H5 hold. Then, there exists a constant C 2 > 0 such that for any Ω Awith Lipschitz boundary and for any E Ω, we have Γ lim sup F (χ E, Ω C 2 P (E,Ω. (4.28 Proof. By approximating E with regular sets E k such that P (E k, Ω converges to P (E,Ω, we can assume that E Ω is a smooth hypersurface. To prove (4.28 it 19

22 is enough to choose n 0 and construct a sequence of functions u n H 1 (Ω such that: u n χ E in L 1 (Ω and lim sup F n (u n, Ω C 2 P (E,Ω. n Let R n R n be such that F n (v, Ω = F n (v, R n for all v H 1 (Ω. By Prop. 3.11, this is the maximal R R n which is contained in Ω. The renormalization is given by R n c. Define A 0 n := {z Z N : α n(q + z R n, dist ( α n(q + z, E < 2 α n}, A ± n := {z Z N : ±d E ( α z 0, dist ( α n (Q + z, E 2α n }, Σ n := α (z + Q, R n ± := α (z + Q. z A 0 n z A ± n Consider the positive, periodic minimizer u + of G 1 α(,q on the unit cube. 1 α Assumption (H4 implies that the positive and the negative global minimizer differ by the constant 2. We extend u + periodically to R N and denote the extended 1 α function by u + as well. Consider an even cut-off function Φ C (R, increasing 1 α on [0, andsuchthatφ(r =0if r < 1, and Φ(r =1if r > 2. We denote by γ the unique strictly increasing function, asymptotic at ± to the two stable zeroes ±1 ofw, and satisfying (1.4 with γ(0 = 0. Let δ 3beafixed natural number such that, if we let x := δ log, thenγ(±x =±1+O( 2δ and γ (±x =O( 2δ. Following [2], we consider a function γ C 1,1 (R C (R \{±x, ±2x }which coincides with γ on [ x,x ] and assumes the asymptotic values ±1 outside the interval ( 2x, 2x. Then, the sequence ( u n (x :=γ n d ( E(x de (x +Φ n α (u + ( n ( x 1 α n α 1 (4.29 n satisfies u n = u ± ( n (x/ α nonr ± 1 α n, if (H4 holds. E is regular, so there exists a constant C = C(N such that lim sup n Σ n α n CP(E,Ω. Let v + n (x :=u + ( n 1 α ( x α n. Then the renormalization is given by G n (v + n,r n. It follows (cf that there exists a constant C(W > 0 such that G n (v n, Σ n CP(E,Ω 1 α n + ω n, where ω n is such that lim n ω n α 1 n = 0. As the periodic minimizer u + is bounded 1 α in L, we may assume that u n 2. Then we get F n (u n, Ω = G n (u n,r n G n (v n,r n =G n (u n, Σ n G n (v n, Σ n ( n u n 2 + W (u n dx + 1 Σ n n α g( x n Σ n α u n dx + C 1 α n n ( n u n 2 + W (u n dx + C g P (E,Ω, n Σ n 20

23 where C is a constant depending only on N. Therefore, recalling [13, 14] we get lim sup F n (u n (c W + C g P (E,Ω. n Remark Notice that if we drop (H4, we can still show that Proposition 4.9 holds whenever α<2/3. Indeed, thanks to (H2, (H3 and Proposition 3.7 we get G 1 α(u + 1 α,q G 1 α(u 1 α,q C 2(1 α, for some C > 0, which implies that there exits a constant c with 0 c and lim sup α c < such that min G = R H 1 (R G α 1 α(u ±,Q= R (c 1 α + C 2 3α = R c + o(1. (4.30 Hence we can conclude as above. On the other hand, if α>2/3we cannot in general drop (H4 in order to avoid a Γ-limit which is always in {0, + }. Indeed, if W C 3 (R, the asymptotic expansion for u ± shows that u (x u + (x = 2(1 α W (1 (W (1 g 2 (x+o ( 2(1 α, hence estimate (4.30 is sharp for a general smooth 3 potential. Once we have both the fundamental estimate and the estimates from above and below, we can reason as in [1, Theorem 3.3] and get the following result. Proposition Assume (H1 to (H5. Then, there exists a local functional F 0 : L 1 loc (RN A [0, + ], and a subsequence of functionals F n (, Ω which Γ-converge to F 0 (, Ω for any Ω Awith Lipschitz boundary. Moreover, for any u BV loc (R N ; {0, 1}, F 0 (u, is the restriction to A of a regular Borel measure. 5 Representation Theorem and properties of the Γ-limit In this section we derive further properties of the Γ-limit. Throughout this section we shall always assume that (H1-(H5 hold, and that g L N c 0 with c 0 as in Proposition 3.7. Let us first introduce the following notation. Definition 5.1. Let u ± be the periodic extensions of the minimizers of G (,Q, let Φ and γ be as in the proof of Proposition 4.9, and let Q ν be a unit cube centered at the origin with two of its faces orthogonal to ν. Weset H(ν, x :={y R N : y x, ν 0}, χ ν,x := χ H(ν,x, Q ν,x ρ := x + ρq ν, ( u ν,x dh(ν,x (x :=γ +Φ ( ( d H(ν,x (u + (x 1, u ν,x,α(x :=u ν,x x 1 α α. 21

24 Observe that χ ν,x is the characteristic function of a half-space orthogonal to ν and centered at x, andu ν,x (y is an interpolation between the two absolute minimizers across the hyperplane orthogonal to ν. Recalling [3, Theorem 3] (see also [1, Theorem 3.5], we obtain a representation result for the functional F 0. Theorem 5.2. There exists a function ϕ : R N S N 1 (0, + such that ϕ(x, ν E (xdh N 1 if χ E BV (Ω, F 0 (χ E,B= E B + otherwise, for any Ω Awith Lipschitz boundary and any Borel set B Ω. Moreover the function ϕ satisfies C 3 ϕ(x, ν C 2, ϕ(x, ν = lim sup ρ 1 N m(ρ, x, ν, (5.1 ρ 0 + where C 2,C 3 > 0 are as in Propositions 4.9 and 4.8, while m(ρ, x, ν is defined by ( } m(ρ, x, ν :=min {F 0 u, Q ν,x ρ : u = χ ν,x in R N \ Q ν,x ρ. (5.2 Relation (5.1 looks slightly different from the formula in [3], but, because of the choice of closed cubes, (5.1 is implied by the result in [3]. More information on ϕ can be extract from the representation formula (5.1, like x-independence, convexity and a more explicit representation. To this end, we need two lemmas which allow us to neglect boundary effects. Let us choose a function u ν,x ρ problem defined by (5.2, namely: F 0 ( u ν,x ρ which solves the minimizing, Q ν,x ρ = m(ρ, x, ν. (5.3 Lemma 5.3. Given x R N, there exists a countable set E x R such that, for any ρ>0 with ρ/ E x,thereexistsasequenceη n ρ, η n <ρ, such that ( F 0 u ν,x ρ, Qν,x ρ = lim F ( 0 u ν,x n η n, int(q ν,x ρ. Proof. Fix (ν, x S N 1 R N and fix R>0. To simplify notation, we set Q ρ := Q ν,x ρ and u ρ := u ν,x ρ for all ρ>0. Let g R : (0,R [0,, η F 0 (u η,q R. Then g R is a decreasing function on the interval (0,R, hence it has a countable set of discontinuities E R. Notice that for R 1 R 2 the two functions g R1 and g R2 differ by a constant on (0,R 1. Hence E R1 E R2, whenever R 1 R 2. So E x = 0<R E R is countable, and the claim follows. 22

25 Lemma 5.4. Let u ν,x ρ be as in (5.3. For all x R N and ρ>0, ρ/ E x,thereexist u ν,x ρ in L 1 (Q ν,x ρ asequenceη ρ, withη <ρ, and a sequence of functions u ν,x such that u Hloc 1 (RN, u = u ν,x,α on R N \ Q ν,x,and F 0 ( u ν,x ρ ρ+η 2, ( Qν,x ρ = lim F u,q ν,x ρ. (5.4 Proof. As in the proof of the previous lemma, we simplify the notation by dropping the dependence of sets and functions on x and ν. By Lemma 5.3 we can find a sequence η k ρ, η k <ρ, such that F 0 (u ρ, Q ρ = lim k F 0(u ηk,q ρ, where u ηk = χ ν,x on R N \ Q ηk. For any k, we consider a Γ-realizing sequence w k, u ηk such that F 0 (u ηk,q ρ = lim F (w k,,q ρ. By Lemma 4.4, applied with U = Q ηk,u = Q ρ+η k,v = Q ρ \ Q ηk and u = w k,, 2 v = u ν,x,α, there exists a cut-off function ϕ between U and U. Letting u k, := ϕu +(1 ϕv, from the energy estimate (4.14 and Proposition 4.9 we obtain lim F (u k,,q ρ lim F (w k,,qρ+η k + lim F (u x,ν 2,α,Q ρ \ Q ηk lim F (w k,,q ρ +C 2 (ρ N 1 η N 1 k = F 0 (u ηk,q ρ +C 2 (ρ N 1 η N 1 k. Then, a diagonalization argument proves the claim. Remark 5.5. Notice that, in Lemma 5.4, we can choose η ρ independently of 0; in particular we can assume that for any k N there exists a 0 N such that η <ρ k α for any 0. In the following proposition, we want to show that the Γ-limit is homogeneous, i.e. the integrand function ϕ does not depend on x R N. Proposition 5.6. The function ϕ given by Theorem 5.2 does not depend on x, moreover its one homogeneous extension ϕ as defined in (2.5 is convex. Proof. Let us fix ν S N 1 and x, y R N, x y. We have to show that ϕ(x, ν =ϕ(y,ν. (5.5 Let u x,ν ρ be as in (5.3. For simplicity we write u x ρ := ux,ν ρ. Lemma 5.4 asserts the existence of a sequence u which equals u ν,x,α on a tubular neighborhood of the boundary of Q ρ and such that (5.4 holds. To simplify notation, 23

26 we drop the dependence of functions and cubes on the direction ν, which is fixed throughout this proof. Let τ Z N be defined as [ ] yi x i (τ i := and v (z :=u (z τ. Here [r] denotes the largest integer smaller or equal than r. Notice that τ (y x andv ( v( :=u x ρ ( y + x. For any r>1, we have F 0 (v, Q y ρ F 0 (v, Q y rρ lim inf F (v,q y rρ ( = lim inf F (v, τ + Q x ρ +F (v,q y rρ \ ( τ + Q x ρ = lim inf (F (u,q x ρ +F (u x ( τ,q y rρ \ ( τ + Q x ρ = lim F (u,q x ρ + lim F (u x ( τ,q y rρ \ ( τ + Q x ρ F 0 (u x ρ, Q x ρ+c 2 ρ N 1 (r N 1 1. Letting r 1, we then get F 0 (v, Q y ρ F 0 (u x ρ, Qx ρ. The choice of u x ρ then implies m(ρ, y, ν m(ρ, x, ν, where m(ρ, x, ν is defined in (5.2. By exchanging x and y, we obtain the equality for any ρ/ E x E y. Then, observing that we can rewrite (5.1 in the form ϕ(x, ν = lim sup ρ 0 +,ρ/ (E x E y ρ 1 N m(ρ, x, ν, we finally get (5.5. Once x-independence is established, the fact that the extension of ϕ is a convex function follows by standard semicontinuity results (see for example [7], [5]. Remark 5.7. Note that if ϕ is independent of x, then by dilating the variable x we see that m(ρ, ν =ρ N 1 m(1,ν=ρ N 1 ϕ(ν. In particular the set of discontinuities E x is empty for any x R N. Moreover, by the convexity of ϕ, the minimizers u η of m are always characteristic functions of a half-space. We want to prove that the Γ-limit is independent of the subsequence. In order to do so, it is convenient to work with blow-up sequences and the functional G as in Definition 3.1. We begin by showing that we can choose a suitable minimizing sequence which coincides, far from the interface, with the absolute minimizers on the cube. First let us introduce some notation. u ± denotes the periodic extension to R N of the minimizers of G (,Q. Let λ>0, ν S N 1,andset Q := Q ν,0 and [λ Q] := (z + Q. {z Z N : Q z+λ bq} 24

27 Lemma 5.8. There exist constants 0 <δ<1/3, 0 > 0, λ 0 > 0 and γ 1 > 0, such that for any sequence u with boundary values u (x =u ν,0 (x on R N \ [λ Q], which is uniformly bounded in L and satisfies the energy bound ( Cλ N 1 G (u, [λ Q] G (u ±, [λ Q], (5.6 there exists a sequence ũ with ũ (x =u (x on R N \ [λ Q], andsetss, which are unions of unit cubes, such that for any < 0 and λ>λ 0 the following holds: a ũ = u + or ũ = u on [λ Q] \ S ; b G (ũ, [λ Q] G (u, [λ Q] + Cλ N 1 γ 1 ; c S [λ Q] δ Cλ N 1. Proof. In the following we will consider u as a function on R N, extended by u ν,0 R N \ [λ Q]. Given a constant 0 <γ<1/3, we set Z γ := {z Z N : G (u,z+ Q G (u ±,z+ Q γ}, S γ := (z + Q. From the upper bound (5.6 we have S γ [λ Q] Cλ N 1 γ. Fix now a constant γ 1 <γ/[n(n 1] and let Z := { z Z N : dist(z + Q, S γ 2 γ 1 }, S := z Z γ z Z (z + Q. From the boundary conditions we know that S [λ Q] > 0. Possibly reducing γ 1, we can also choose 0 <δ<1/3 such that γ + Nγ 1 <δ.since we do not have any information on H N 1 ( S γ, the best available upper bound on S is S [λ Q] Cλ N 1 γ ( γ 1 N = Cλ N 1 (γ+nγ 1 <Cλ N 1 δ, (5.7 and condition c is satisfied. We call a cube positive, if {x Q + z : u (x > 0} 1 2, i.e. if [Hu ( / α ] = 1 on the cube, where [Hu] is defined in (4.11, and negative else. For x R N \ S γ,we define v (x by 2v (x :=([Hu ( / α ]( α x+1u + (x+([hu ( / α ]( α x 1u (x. We want to give an estimate of u v L 1 ((S [λq]\s b γ. First we show that there cannot be positive cubes in (S [λ Q] \ S γ which touch negative cubes on one facet. Indeed, let us assume that we can find two adacent cubes, say Q 1 and Q 2, contained in S \S γ, such that u is mostly positive in Q 1 and mostly negative in Q 2. Note that the energy scales with N 1 α under the change of variables y = α x, so (4.13 implies that there exists a constant Ĉ(W, g > 0such that G (u, int(q 1 Q 2 Ĉ. on 25

28 Therefore at least one of the cubes must be in S γ, and v is a well-defined H 1 -function on [λ Q] \ S γ. From (4.5 we get for Q 1, Q 2 as above {u < 1/2} Q 1 C γ N N 1, {u > 1/2} Q 2 C γ N N 1. (5.8 By assumptions (H2 and (H3 there is a constant c such that { c(u 1 W (u 2 if u 1 2, c(u +1 2 if u 1 2. Recall that G (u +,Q 0and u W (u 0. Using (5.8, we have for sufficiently small on a positive cube, which we call for simplicity Q, γ > G (u,q G (u +,Q 1 [W (u +g(u 1] dx g(u 1 dx u 1 <3/2 u < g L γ N { N c u 1 2 g u 1 } dx u 1 <3/2 2 g L γ N { N (1/2 u 1 2} dx 2 Q g 2 L, 2 g u 1 <3/2 hence Q {2 g u(x 1 <3/2} u 1 2 dx C 1+γ. (5.9 From (5.8, (5.9, the L -bound on u and v and since γ<1/3 weget u v L 1 (Q = u 1 (v 1 L 1 (Q C [ + γ N 1+γ ] N C γ N N 1, and the same holds for negative cubes as well. Since γ 1 <γ/[n(n 1] we have τ := γ/(n 1 Nγ 1 > 0, so summing over the cubes (see 5.7 we get u v L 1 ((S [λ b Q]\S γ CλN 1 τ. (5.10 In what follows we mimic the proof of the fundamental estimate, with the important difference that the sets are not given, but depend on. For i N, i dist([λ Q] \ S,S γ, we define the sets U i as follows U 0 := S γ, U i+1 := {z Z N, (z+q S, dist(z+q,u i =0} (z + Q. Let also S i := {U i+1 \ U i }. By the previous L 1 -estimate (5.10 we get gu + gv Cλ N 1 τ. (5.11 (S [λ b Q]\S γ (S [λ b Q]\S γ 26

29 (Note that A g 1=0ifA is a union of cubes. This allows us to estimate the nonnegative parts of the functional separately. The idea is to use the upper bound (5.6 and follow the proof of Lemma 4.4. Indeed, (5.11 and (5.6 imply { ( u 2 + v (W (u +W(v } dx<cλ N 1. (S [λ b Q]\S γ Since there are at least γ 1 strips S i contained in S \S γ, by an averaging argument we can find 0 1, such that { ( u 2 + v (W (u +W(v } dx<cλ N 1 γ 1. (5.12 S 0 [λ b Q] Notice that 0 1, i.e. the chosen strip does not touch the set S 0. Averaging again, we can also assume u v L 1 (S 0 [λ bq] CλN 1 τ+γ 1. (5.13 Let us now divide the strip S 0 into smaller strips Σ of width, andletϕ (x bea smooth cut-off function such that 0 ϕ 1, ϕ 1onV,ϕ 0on[λ Q] \ V +1, where V 0 = U i, V +1 = {x U i+1 : dist(x, V ( +1} and Σ := V +1 \ V. Since the boundary of the cubic set S γ is uniformly Lipschitz, we can also assume ϕ C 1 for some C independent of. We want to choose an index such that the function satisfies condition b. Notice first that ũ := (1 ϕ u + ϕ v G(ũ, [λ Q] G(u, [λ Q] G(ũ,S 0 G(u,S 0 (5.14 { = ( ( } ũ 2 u 2 W (ũ W (u + + g(ũ u dx S 0 [λq] b { ( ( } ũ 2 u 2 W (ũ W (u + dx + Cλ N 1 τ+γ 1, S 0 [λ b Q] since using (5.13 we have g(ũ u dx g L u v L 1 (S 0 [λq] b CλN 1 τ+γ 1. S 0 [λ b Q] Hence, it remains to prove { ( ũ 2 u 2 ( } W (ũ W (u + S 0 [λ bq] dx Cλ N 1 γ 1. (

30 Since the number of the smaller strips in S 0 is of order 1, by a further averaging argument and using (5.13, we can find an index such that u v dx Cλ N 1 τ+γ 1. (5.16 Σ [λ bq] Recalling (5.12 and reasoning as in Lemma 4.4, estimate (4.22, we obtain { ( ũ 2 u 2 + W (ũ } W (u dx S 0 [λq] b { ( v 2 + u 2 + W (u } +W (v dx S 0 [λq] b { u v W (ũ } W (v dx Σ [λq] b Cλ N 1 γ 1 u v + C dx Cλ N 1 γ 1, Σ [λ b Q] where we denote by C a general positive constant. By (5.14, this implies G(ũ, [λ Q] G(u, [λ Q] + Cλ N 1 γ 1, which is condition b. It remains to prove that ũ coincides with u outside of [λ Q]. Note that by construction of v and the fact that u = u ν,0 on R N \ [λ Q], any cube in R N \ [λ Q] such that u v must be contained in S 0 U 0. As 0 1, we obtain ũ = u on R N \ [λ Q]. We show now that the Γ limit does not depend on the particular subsequence and on the parameter α. In order to do this, we characterize the limit function ϕ(ν. For any Borel set A R N, we define Fg A (E :=c W P (E,A+ g(xχ E (x dx. Theorem 5.9. We have the following representation for the function ϕ(ν: 1 { ϕ(ν = ψ(ν := lim inf λ + λ N 1 min F [λqν,0 ] g (E : (5.17 } E R N such that χ E = χ ν,0 on R N \ [λq ν,0 ]. In particular, the Γ limit does not depend on the subsequence and on the parameter α (0, 1. Proof. Fix ν S N 1,set Q := Q ν,0 and let [λ Q] be as in Lemma 5.8. We divide the proof into two steps. A 28

31 Step 1. Let us prove ϕ ψ. We recall from Lemma 5.4, applied with x =0andρ =1,that F 0 (χ ν,0, Q = ϕ(ν dh N 1 = lim F (u, Q, H(ν,0 bq where u Hloc 1 (RN aresuchthatu = u ν,0,α on R N \ Q ν,0 η 1, η < 1. Notice that we can assume η < 1 4 α gives Q ν,0 1+η 2 {x Q : dist(x, R N \ [λ Q] 1} [λ Q]. 1+η 2 Let now λ be the biggest integer less than or equal to α u (x/λ. Since we have F (u, Q C for some C>0, it follows ( C F (u, Q (N 1α G 1 α(v 1 α, [λ Q] G 1 α Since λ α λ + 1, from (5.18 it follows G 1 α (v 1 α, [λ Q] G 1 α (u ±, [λ 1 α Q] Cλ N 1,,forsome 0and (see Remark 5.5, which,andsetv 1 α(x := (u ±, [λ 1 α Q], (5.18 possibly considering a bigger constant C. Set := 1 α. Then the conditions of Lemma 5.8 are satisfied, and we may assume that v e = u ± e outside S e, for some set S e such that S e [λ Q] ( δ Cλ N 1, for some 0 <δ<1/3. Let us fix ρ>0 such that δ<ρ< 1 3. As the renormalization is nonnegative, we obtain from the Co-Area formula Cλ N 1 G e (v e,s e [λ Q] Ge (u ± e,s e [λ Q] ( Ce ρ W (sp({ve >s},s e [λ Q] ds + dx 1+Ce ρ 1 Ce ρ 1+Ce ρ S e [λ b Q] gv e W (sp({ve >s},s e [λ Q] ds 2 g δ λ N 1. Again from Lemma 5.8 we know that {v e >s} int(s e for any s [ 1 + C ρ, 1 C ρ ], hence P({v e >s},s e [λ Q] = P({ve >s}, [λ Q]. Let now t := arg min 1+Ce ρ s 1 Ceρ P ({v e >s}, [λ Q] and let E ({v := e >t } [λ Q] Then we have 1 Ce ρ W (sp({ve >s}, [λ Q] ds (cw C ρ P(E, [λ Q]. 1+Ce ρ 29