On a minimization problem with a mass constraint in dimension two

On a minimization problem with a mass constraint in dimension two Nelly Andr´e∗ and Itai Shafrir† August 20, 2013

Abstract We continue our study initialized in [1] of a singular perturbation type minimization problem with a mass constraint, involving a potential vanishing on two curves in the plane. In the case of a two dimensional nonconvex domain (and under some additional assumptions) we are able to prove a convergence result for the minimizers and characterize the limit as a solution of a mixed Dirichlet-Neumann boundary condition problem with a mass constraint. Mathematics Subject Classification. Primary 35J20; Secondary 35B25, 35J60, 58E50

1

Introduction

The following problem was introduced and studied by Sternberg [10]. Let Γ1 and Γ2 be two disjoint, smooth and simple closed curves in R2 of lengths l1 = l(Γ1 ) and l2 = l(Γ2 ), respectively, such that Γ1 lies inside Γ2 and the origin 0 lies inside Γ1 . Let W : R2 → [0, ∞) be a smooth function (i.e., at least of class C 4 ) satisfying W > 0 on R2 \ (Γ1 ∪ Γ2 ) and W = 0 on Γ1 ∪ Γ2 .

(H1 )

Since W attains its minimal value zero on Γ1 ∪ Γ2 , the normal derivative Wn equals zero on Γ1 ∪ Γ2 . We make the generic assumption that Wnn > 0 on Γ1 ∪ Γ2 . ∗ †

D´epartement de Math´ematiques, Universit´e de Tours, 37200 Tours, France Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

1

(H2 )

We shall also assume a coercivity condition on W at infinity: there exist constants R0 > 0 and c˜0 > 0 such that W (y) ≥ c˜0 |y| for |y| ≥ R0 .

(H3 )

Let G be a bounded smooth domain in RN . In order to simplify notation, we shall assume in the sequel, without loss of generality, that G has unit volume: (1.1)

µ(G) = 1 . For each ε > 0 consider the energy functional ∫ W (u) Eε (u) = |∇u|2 + over u ∈ H 1 (G, R2 ) . 2 ε G

(1.2)

Let Rc be a positive number such that the circle {|x| = Rc } lies between the two curves Γ1 and Γ2 . The number Rc represents the constraint in the following minimization problem: ∫ min{Eε (u) : u ∈ H 1 (G, R2 ),

|u| = Rc } .

(Pε )

G

Denoting by uε a minimizer in (Pε ), we are interested in the asymptotic behavior of the minimizers {uε } and their energies Eε (uε ), as ε goes to 0. In [1] it was shown that the asymptotic behavior of the minimizers depends in a crucial way on the geometry of G through a certain problem, involving the isoperimetric profile of G. Recall that the isoperimetric profile of G satisfying (1.1) is the function I = IG : (0, 1) → R defined by I(t) = min{PerG Ω : Ω ⊂ G s.t. χΩ ∈ BV (G) and |Ω| = t} .

(1.3)

Clearly, I(t) is a symmetric function w.r.t. the middle point 1/2 and we refer the reader to [9] and the references therein for more information on it. As in [1] we define the interval I0 = [β1 , β2 ] by m 2 − Rc M 2 − Rc I0 := [ , ], (1.4) m2 − m1 M2 − M1 where (1.5) mj = min |x| and Mj = max |x| (j = 1, 2) . x∈Γj

x∈Γj

Finally, the geometric problem relevant to our study which was introduced in [1] is min{I(t) : t ∈ I0 } . 2

(1.6)

In [1] it was shown that when the minimum in (1.6) is attained (only) at one of the end points, βj of I0 (and under more technical assumptions), there exists a subsequence s.t. uεn → u∗ in L1 (G), where u∗ has the simple form  x(1) x ∈ G , 1 u∗ (x) = (1.7) x(2) x ∈ G \ G1 , with G1 a set of minimal perimeter I(βj ) among all sets of volume βj . It was noted in [1] that the above is always the case for a convex G. On the other hand, for a nonconvex G, the minimum in (1.6) may be attained only at an interior point α ∈ (β1 , β2 ), and then the behavior of the minimizers is expected to be quite different. This latter case will be characterized by “property (NC)”: Definition 1. We shall say that the pair G and I0 have property (NC) if the minimum in (1.6) is attained at a unique interior point α of the interval I0 . The domain drawn in Figure 1 (with an appropriate I0 ) is an example where property (NC) holds. Another, more explicit example, consists of a domain of unit area, symmetric with respect to both the x and y axes, given by G = {(x, y) : −f (x) < y < f (x), x ∈ (−a, a)}, such that 0 < m := min[−a,a] f = f (0) < a (0 is the unique minimum of f ), with 21 lying in the interior of I0 . We remark that the uniqueness of α is assumed only for the sake of simplicity. The difference between the two cases (the “convex” and “nonconvex”) is manifested already by the following energy estimate, [1, Theorem 3] that we recall below: Theorem 1. If property (NC) holds and W satisfies (H1 ) − (H3 ), then there exists a constant C, independent of ε, such that Eε (uε ) − 2D I(α) ≤ C . ε

(1.8)

Above, the constant D, representing a “distance” between the two curves Γ1 and Γ2 , is defined, following Sternberg [10], by ∫ 1 ( )1/2 0 W (γ(t)) |γ (t)| dt . (1.9) D := inf 2 γ∈Lip([0,1],R ), γ(0)∈Γ1 ,γ(1)∈Γ2

0

3

More generally, for any pair of points x, y ∈ R2 we define dW (x, y) =

∫

where L(γ) =

1

(

inf

γ∈Lip([0,1],R2 ), γ(0)=x,γ(1)=y

L(γ) ,

)1/2 0 W (γ(t)) |γ (t)| dt .

(1.10)

(1.11)

0

Note that thanks to scaling invariance we may replace the interval [0, 1] in (1.10)–(1.11) by any interval [0, A], A > 0. We define the corresponding distance functions to the curves Γ1 , Γ2 as follows: Ψj (ζ) = dW (ζ, Γj ) := inf dW (ζ, x) , j = 1, 2 , ζ ∈ R2 .

(1.12)

e = min(Ψ1 , Ψ2 ) . Ψ

(1.13)

x∈Γj

We also set It is well known (c.f. [10, 6]) that for j = 1, 2, Ψj ∈ Lip(R2 ) is a solution of the eikonal-type equation |∇Ψj (ζ)|2 = W (ζ) a.e. on R2 . (1.14) It was further shown in [2] that Ψj is regular in a neighborhood of Γj , i.e., ∃d0 > 0 s.t. Ψj is of class C 2 in {x : Ψj (x) < d0 } , j = 1, 2 .

(1.15)

Moreover, we have Ψj (x) ∼ W (x) ∼ dist2 (x, Γj ) on {Ψj (x) < d0 } ,

(1.16)

where “dist” stands for the euclidean distance. As in [1], we denote the set of geodesics w.r.t. the degenerate metric dW ((1.10)), realizing the infimum in (1.9), by G = {γ (i) : i ∈ I} ,

(1.17)

where I is a set of indices. The main result of this paper is a convergence result that characterizes the limit of {uε }, when ε goes to zero. Unfortunately, we were able to establish this result only in dimension two, although we conjecture that it should be valid in any dimension. Therefore, we shall assume in the sequel that N = 2. In addition, we shall also make

4

the simplifying assumption that G is simply connected. When N = 2 it is well known that if G1 is a set realizing the minimum in (1.3) for t = α, then the separation surface Σ := ∂G1 ∩ G ,

(1.18)

consists of a finite union of segments (see Figure 1), Σ=

J ∪

Σ(j) ,

(1.19)

j=1

that are all orthogonal to the boundary ∂G.

G1

Σ

(1)

G2

Σ

(2)

G1

Figure 1: In this example Σ is composed of two segments We denote for each λ > 0, Σλ = {x ∈ G : dist(x, Σ) ≤ λ} :=

J ∪

(j)

Σλ .

(1.20)

j=1

Our main result is the following: Theorem 2. Assume the hypotheses of Theorem 1, and in addition that N = 2, G is simply connected and that W satisfies a1 W (y) ≤ ∇W (y) · y ≤ a2 W (y) ,

|y| ≥ R1 ,

(H30 )

for some positive constants a1 , a2 and R1 (we may take R1 > R0 ). Suppose (for simplicity) that there is a unique minimizing α ∈ I0 in (1.6). Assume also that the set G consists of a single geodesic γ with endpoints p1 ∈ Γ1 and p2 ∈ Γ2 . Then, there is a set G1 realizing the minimum for I(α) in (1.3) such that for every λ > 0 we have uε Gj → Uj in H 1 (Gj \ Σλ ) ∩ C(Gj \ Σλ ) , j = 1, 2 , (1.21) 5

¯ 1 , and the pair (U1 , U2 ) ∈ H 1 (G1 , Γ1 ) × H 1 (G2 , Γ2 ) is a minimizer for where G2 = G \ G the following problem: ∫ {∫ 2 E0 := min |∇v1 | + |∇v2 |2 : vj ∈ H 1 (Gj , Γj ) , Tr vj ∂Gj ∩G = pj , j = 1, 2 , G1 G2 ∫ ∫ } |v1 | + |v2 | = Rc . (1.22) G1

G2

Furthermore, the following more precise formula than (1.8) for the energy holds: Eε (uε ) =

2D I(α) + E0 + o(1) . ε

(1.23)

Remark 1.1. Existence of a minimizer (U1 , U2 ) ∈ H 1 (G1 , Γ1 ) × H 1 (G2 , Γ2 ) to the problem (1.22) follows easily by the direct method of the calculus of variations. It will be ek = g −1 ◦ Uk where gk is convenient to associate with Uk (k = 1, 2) the S 1 -valued map U k dgk (eiφ ) lk 1 a diffeomorphism of S onto Γk satisfying | dφ | = 2π , ∀φ ∈ [0, 2π), k = 1, 2. For each (j)

(j)

Σ(j) let Gk denote the component of Gk for which Σ(j) ⊆ ∂Gk ∩ G, k = 1, 2. Since ek ∈ H 1 (G(j) , S 1 ) and G(j) is simply connected there exists a lifting φ(j) ∈ H 1 (G(j) , R) U k k k k ek = eiφk(j) on G(j) (see [4, 5]). Define a function hk (φ) by hk (φ) = |gk (eiφ )|. such that U k (j) The function φk is a minimizer for the problem ∫ ∫ (j) 2 1 min{ |∇φ| : φ ∈ H (Gk ) , hk (φ) = αj,k } , (j)

Gk

for some constant αj,k (αj,k =

(j)

Gk

∫ (j)

Gk

|Uk |). Therefore, it solves the equation

−∆φk = aj,k h0k (φk ) , (j)

(j)

(1.24) (j)

for some constant aj,k , with zero Neumann condition on ∂Gk ∩ ∂G and (constant) (j) (j) (j) Dirichlet condition on ∂Gk ∩ G. It follows that φk is a smooth function in Gk . ¯ k holds as Therefore, U1 and U2 are smooth maps away from Σ (continuity of Uk in G well as one can see by a reflection argument near the corners). Remark 1.2. The uniqueness assumptions on α and G in Theorem 2 are not essential and are made only for the sake of simplicity of the statement of the theorem. Remark 1.3. After completing our work we learned about a recent interesting article by Lin, Pan and Wang [8]. They study a different, but related problem: the Dirichlet problem for a similar type of energy (for very special and suitably chosen boundary data {gε }). Their framework is more general than ours since they allow for arbitrary 6

dimension both in the domain and the target (we allow arbitrary dimension for the domain only in the energy estimate of Theorem 1, proven in [1]). We mention that our two dimensional result provides convergence in stronger norms, namely in H 1 and uniform convergence (that probably can be improved to C k -convergence) away from Σ. It is interesting to note that a nontrivial limit as we found in our Theorem 2 is obtained under simple and natural mass constraint (compare with Remark 1.2(5) in [8]). The paper is organized as follows. In Section 2 we prove some preliminary estimates needed for the proof of Theorem 2 (the proof of two technical results is postponed to the Appendix). Finally, the proof of Theorem 2 is completed in Section 3. Acknowledgment. The second author (I.S.) acknowledges the support by the Israel Science Foundation (grant no. 1279/08).

2

Preliminary results

Let {uε } be a family of minimizers for (Pε ), assuming that the assumptions of Theorem 2 hold. The next lemma shows that the sets were uε takes values close to Γ1 and to Γ2 , have measures close to α and 1 − α, respectively. Lemma 2.1. We have lim |µ(Ψ1 (uε ) ≤ ε1/2 ) − α| + |µ(Ψ2 (uε ) ≤ ε1/2 ) − (1 − α)| = 0 .

ε→0

(2.1)

Furthermore, for a subsequence εn → 0 we have for every t ∈ (0, d0 ], χ{Ψ1 (uεn ) 0, T (r) = sup{s ∈ (0, d1 ) : I(α + s) − I(α) ≤ r} .

(3.3)

For each Ck0 as in Lemma 3.1, we define dM = dkM0 = min

max

1≤j≤J x∈∂Ck0 ∩G2

12

dist(x, Σ(j) ) .

(3.4)

Proposition 3.1. There exists a constant c3 such that, for Ck0 as in Lemma 3.1, we have ( 1 ) dM ≤ δ(ε) := c3 max ε 2 , T (c2 ε) . (3.5) Proof. Assume that dM is attained at x0 for j = j0 and let T denote the closure of the component of ∂Ck0 ∩ G2 containing x0 . Set dm = min dist(x, Σ(j0 ) ) and η = x∈T

dm . dM

We distinguish two possibilities: (i) η ≤ 12 . (ii) η > 12 . Case (i): Let dm be attained at the point y0 . It may happen that y0 ∈ Σ(j0 ) , i.e., η = 0, as shown in Figure 2. Suppose an arclength parametrization of the part of the curve T joining x0 to y0 is given by γ on the interval [a, b], with γ(a) = x0 and γ(b) = y0 . Denote by γ1 the projection of γ on Σ(j0 ) and by γ2 the projection in the orthogonal direction. From Lemma 3.1 it follows that ∫ b ∫ ∫ b ∫ b b 0 |γ20 (t)|2 0 0 c2 ε ≥ dt |γ (t)| dt − γ1 (t) dt ≥ (1 − |γ1 (t)|) dt = 0 a a a a 1 + |γ1 (t)| ∫ b 1 (dM − dm )2 ≥ | γ20 (t)|2 = , 2(b − a) a 2(b − a) implying that

√ dM ≤ dM − dm = O( ε) . (3.6) 2 Case (ii): The assumption η > 12 implies in particular that T = ∂Ck0 ∩G2 , dist(T , Σ) > 0 and dM l(Σ(j0 ) ) , (3.7) µ(Ck0 ∪ G1 ) − µ(G1 ) = µ(Ck0 ∩ G2 ) ≥ 2 where l(Σ(j0 ) ) denotes the length of the component (segment) Σ(j0 ) , see Figure 3. Applying Lemma 3.1 yields I(µ(Ck0 ∪ G1 )) − I(α) ≤ PerG (Ck0 ∪ G1 ) − I(α) = l(T ) − l(Σ(j0 ) ) ≤ c2 ε .

(3.8)

By (3.7)–(3.8) and the definition (3.3) of T (r) we obtain dM l(Σ(j0 ) ) ≤ T (c2 ε) . 2 Combining (3.6) and (3.9) yields (3.5). 13

(3.9)

Σ

Ck

G1

0

G2

x0

y0

Figure 2: An example of Case (i) with dm = 0, hence η = 0

Σ

Ck

G1

y0

0

x0

G2

Figure 3: An example of Case (ii)

14

Obviously, an analogous statement holds for the components Dl , l ∈ L0 . Before moving to the next proposition we introduce some more notation. We denote by Ω the domain lying between Γ1 and Γ2 . First we define the functions θ1 , θ2 by θ1 (x) = dW (x, Γ2 ) − (D − d0 ), for x ∈ Ω with Ψ1 (x) = d0 , θ2 (x) = dW (x, Γ1 ) − (D − d0 ), for x ∈ Ω with Ψ2 (x) = d0 . Then we define the functions θ˜1 , θ˜2 on Γ1 and Γ2 , respectively, by θ˜j (x) = min{θj (y) : y ∈ Ω, Ψj (y) = d0 , dΓj (˜ sj (y), pj ) ≥ dΓj (x, pj )}, j = 1, 2 , (3.10) where dΓj stands for the geodesic distance on Γj . Consecutively, we define the functions Qj (t) = sup{dΓj (x, pj ) : x ∈ Γj , θ˜j (x) ≤ t} , j = 1, 2.

(3.11)

Finally, put ˜ = δ(ε) + Q1 ( 1 l2 ε 12 ) + Q2 ( 1 l2 ε 12 ) + ε 21 . δ(ε) (3.12) 41 42 The motivation for the above definitions will become clearer in the course of the proof of Proposition 3.2 below. In the sequel we shall use in a neighborhood of each component Σ(j) (j = 1, . . . , J) of Σ the system of coordinates (σ, τ ) = (σj , τj ) obtained by projecting the point x on Σ(j) . More precisely, τj = τj (x) is the signed distance from x to Σ(j) (with the convention that points with τ < 0 belong to G1 ) and σj = σj (x) ∈ [0, l(Σ(j) )] is the arclength parameter on the segment Σ(j) corresponding to the nearest point projection of x on Σ(j) . Notice that both σj and τj are affine functions of the original coordinates 1 of (x1 , x2 ). We next prove a lower bound for the energy in the neighborhood Σ2δ(ε)+4ε ˜ 2 Σ (see (1.20)). We start with a lemma. ˜ ˜ + 4ε 21 ) such that 2δ(ε) Lemma 3.2. There exist c4 > 0 and d(ε) ∈ (δ(ε), J ∫ ∑ j=1

{τj =d(ε)}∪{τj =−d(ε)}

|∇uε |2 +

c4 W (uε ) ≤ . 2 ε d(ε)

(3.13)

Proof. By Proposition 3.1 and Lemma 2.3, if x ∈ Mε and dist(x, Σ) > δ(ε), then x belongs to one of the components Ci , i ∈ K2 , or Di , i ∈ L2 . Since the total perimeter ˜ ˜ + ε 21 ) such that δ(ε) of these components is at most O(ε), we may find δ1 (ε) ∈ (δ(ε), (j) (j) both l1 = {τj = δ1 (ε)} and l2 = {τj = −δ1 (ε)} do not intersect Mε and such that

15

∫

1

(j) (j) l1 ∪l2

and

∫

W (uε ) ≤ Cε 2 , for j = 1, . . . , J. By (1.16) it follows that also

(j)

l2

1

∫

1

(j)

l1

Ψ1 (uε ) ≤ Cε 2

Ψ2 (uε ) ≤ Cε 2 . Therefore, for each j = 1, . . . , J,

2 Eε (uε ; {|τj | ≤ δ1 (ε)}) ≥ ε

∫ {|τj |≤δ1 (ε),σj ∈[0,l(Σ(j) )]}

|∇(Ψ1 (uε ))|

∫ 2 ∇(Ψ1 (uε )) · ∇τ ≥ ε {|τj |≤δ1 (ε),σj ∈[0,l(Σ(j) )]} (3.14) ∫ l(Σ(j) ) ( ) 2 = Ψ1 (uε (σj , δ1 (ε))) − Ψ1 (uε (σj , −δ1 (ε))) dσ ε 0 1 2 ≥ (D − cε 2 )l(Σ(j) ) ε

Summing (3.14) for j = 1, . . . , J yields Eε (uε ; Σδ1 (ε) ) ≥ By (1.8) and (3.15) we conclude that

1 2DI(α) − Cε− 2 . ε

∫

1

(3.15) 3

G\Σδ1 (ε)

W (uε ) ≤ Cε 2 . Therefore, there exists

δ2 (ε) ∈ (δ1 (ε), δ1 (ε) + ε 2 ) such that ∫ {τj =−δ2 (ε)}∪{τj =δ2 (ε)}

W (uε ) ≤ Cε , j = 1, . . . , J .

(3.16)

We can now repeat the argument used above in (3.14)–(3.15), with δ2 (ε) replacing δ1 (ε), to obtain 2DI(α) Eε (uε ; Σδ2 (ε) ) ≥ −C. (3.17) ε By (3.17) and (1.8) it follows that Eε (uε ; G \ Σδ2 (ε) ) ≤ C, hence there exists d(ε) ∈ (δ2 (ε), 2δ2 (ε)) satisfying (3.13). Proposition 3.2. There exists c5 > 0 such that 2DI(α) Eε (uε ; Σd(ε) ) − ≥ ε (j) J ∫ ) c5 ∑ l(Σ ) ( 2 2 dΓ1 (˜ s1 (uε (s, −d(ε)), p1 ) + dΓ2 (˜ s2 (uε (s, d(ε)), p2 ) ds + o(1) . (3.18) d(ε) j=1 0 Proof. We claim: there exists c5 > 0 such that for every j = 1, . . . , J and every σj ∈

16

(0, l(Σ(j) )) we have ∫

( ∂u W (uε (σj , τ )) ) ε | (σj , τ )|2 + dτ ∂τ ε2 −d(ε) ) 2D 2 ( ≥ − Ψ1 (uε (σj , −d(ε))) + Ψ2 (uε (σj , d(ε))) ε ε ) c5 ( 2 + dΓ1 (˜ s1 (uε (σj , −d(ε)), p1 ) + d2Γ2 (˜ s2 (uε (σj , d(ε)), p2 ) + o(1) . (3.19) d(ε) d(ε)

To prove the claim it suffices to show that ∫

( ∂u W (uε (σj , τ )) ) ε | (σj , τ )|2 + dτ ∂τ ε2 −d(ε) ) ( )) 2D 2 ( ( − Ψ1 uε (σj , −d(ε)) + Ψ2 uε (σj , d(ε)) ≥ ε ε ) c5 2 ( dΓ1 s˜1 (uε (σj , −d(ε))), p1 + o(1) , (3.20) + d(ε) d(ε)

since then an analogous estimate to (3.20) holds when we replace s˜1 and p1 by s˜2 and p2 , respectively. Finally (3.19) would follow by adding the two estimates and dividing by 2. To prove (3.20) we begin by setting η1 = inf{τ ∈ (−d(ε), d(ε)) : Ψ1 (uε (σj , τ )) = d0 } , y0 = uε (σj , −d(ε)) , y1 = uε (σj , η1 ) . We first consider the contribution to the integral on the l.h.s. of (3.20) from the interval (η1 , d(ε)). We have ∫

d(ε)

η1

∫ ( ∂u )1/2 ∂u W (uε (σj , τ )) ) 2 d(ε) ( ε 2 ε (σ, τ ) dτ | (σj , τ )| + dτ ≥ W (u (σ , τ )) ε j 2 ∂τ ε ε η1 ∂τ ( ) 2 2 ≥ dW (y1 , uε (σj , d(ε))) ≥ dW (y1 , Γ2 ) − Ψ2 (uε (σj , d(ε))) ε ε ) 2( = θ1 (y1 ) + D − d0 − Ψ2 (uε (σj , d(ε))) . (3.21) ε

On the interval (−d(ε), η1 ), uε (σj , ·) takes values close to Γ1 (since Ψ1 (uε (σj , τ )) ≤ d0 ). This enables us to get a more precise estimate, taking into account also the contribution of the component of the gradient which is “tangential” to Γ1 (a similar argument was used in the proof of [2, Lemma 3.3]). At each point (σj , τ ) with τ ∈ (−d(ε), η1 ) we write ˜ |∂τ uε |2 = |∂τ(t) uε |2 + |∂τ(˜s) uε |2 , 17

where ˜ ∂τ(t) uε

( ) ( ) ∂τ Ψ1 (uε ) ∂τ s˜1 (uε ) ∇Ψ1 (uε ) ∇˜ s(uε ) (˜ s) = = ∂τ uε · and ∂τ uε = = ∂τ uε · . |∇Ψ1 (uε )| |∇Ψ1 (uε )| |∇˜ s(uε )| |∇˜ s(uε )|

Since for some β > 0, |∇˜ s1 (y)| ≤

√1 β

whenever Ψ1 (y) ≤ d0 , we conclude that

( ) |∂τ(˜s) uε |2 ≥ β|∂τ s˜1 (uε ) |2 , for τ ∈ (−d(ε), η1 ) . From the above we conclude that ∫ η1 ( ∫ η1 ) ( ) ∂uε |∂τ (Ψ1 (uε ))|2 W (uε ) 2 W (uε (σj , τ )) | (σj , τ )| + dτ ≥ + +β|∂τ s˜1 (uε ) |2 2 2 ∂τ ε W (uε ) ε −d(ε) −d(ε) ∫ d2 (˜ s1 (y0 ), s˜1 (y1 )) 2 η1 ∂τ (Ψ1 (uε )) + β Γ1 ≥ ε −d(ε) d(ε) + η1 d2Γ1 (˜ s1 (y0 ), s˜1 (y1 )) 2 . (3.22) = (d0 − Ψ1 (y0 )) + β ε d(ε) + η1 Adding together (3.21) and (3.22) gives ∫

( ∂u d2Γ1 (˜ s1 (y0 ), s˜1 (y1 )) W (uε (σ, τ )) ) ε 2 | (σ, τ )| + dτ ≥ β ∂τ ε2 d(ε) + η1 −d(ε) ( ) 2 D − Ψ1 (uε (−d(ε), σ)) − Ψ2 (uε (d(ε), σ)) + θ1 (y1 ) . (3.23) + ε d(ε)

We consider two cases: (i) dΓ1 (˜ s1 (y0 ), s˜1 (y1 )) > (ii) dΓ1 (˜ s1 (y0 ), s˜1 (y1 )) ≤

dΓ1 (˜ s1 (y0 ),p1 ) 2

,

dΓ1 (˜ s1 (y0 ),p1 ) 2

.

In case (i), (3.20) clearly follows from (3.23), so it remains to consider case (ii). Denote by z1 a point on Γ1 satisfying 1 dΓ1 (z1 , p1 ) = dΓ1 (z1 , s˜1 (y0 )) = dΓ1 (p1 , s˜1 (y0 )) . 2 s1 (y0 )) d2 (p1 ,˜

˜

We distinguish two possibilities. If θ1 (zε 1 ) > Γ1 d(ε) then (3.20) follows again from (3.23). Indeed, this follows from (3.10) by the inequality θ1 (y1 ) ≥ θ˜1 (z1 ) that holds since by (ii) 1 s1 (y0 ), p1 ) = dΓ1 (z1 , p1 ) . dΓ1 (˜ s1 (y1 ), p1 ) ≥ dΓ1 (˜ s1 (y0 ), p1 ) − dΓ1 (˜ s1 (y1 ), s˜1 (y0 )) ≥ dΓ1 (˜ 2 18

We assume then that

d2Γ1 (p1 , s˜1 (y0 )) θ˜1 (z1 ) ≤ . ε d(ε)

(3.24)

We claim that the above inequality implies that d2Γ1 (˜ s1 (y0 ), p1 ) = o(1) . d(ε)

(3.25)

Clearly, combining (3.25) with (3.23) yields (3.20). It suffices then to prove (3.25). First, 2 (Γ )ε 1 1 note that (3.24) implies that θ˜1 (z1 ) ≤ l 4d(ε) ≤ 14 l2 (Γ1 )ε 2 . By the definition (3.11) of Q1 it follows that 1 1 dΓ1 (˜ s1 (y0 ), p1 ) = 2dΓ1 (z1 , p1 ) ≤ 2Q1 ( l2 (Γ1 )ε 2 ) . 4 1

Since d(ε) ≥ Q1 ( 14 l2 (Γ1 )ε 2 ), we obtain that d2Γ1 (˜ s1 (y0 ), p1 ) 1 1 ≤ 4Q1 ( l2 (Γ1 )ε 2 ) , d(ε) 4 and (3.25) follows. The validity of the claim (3.19) is now established. Finally, we note that by (3.13) and (1.16) we have for each j ∈ {1, . . . , J} ∫

∫

l(Σ(j) )

Ψ1 (uε (σj , −d(ε)) + Ψ2 (uε (σj , d(ε)) ≤ C 0

l(Σ(j) )

W (uε ) ≤ 0

Cε2 ≤ Cε3/2 , d(ε)

so integrating (3.19) for σj ∈ (0, l(Σ(j) )) and summing over j = 1, . . . , J yields (3.18). Thanks to Proposition 3.2 we can now conclude the convergence of {uεn } away from Σ. Proposition 3.3. For a subsequence we have uεn * u∗ in H 1 (G \ Σλ ), ∀λ > 0 , where u∗ : G → Γ1 ∪ Γ2 satisfies ∫ |u∗ | = Rc , u∗ |Gj ∈ H 1 (Gj , Γj ) and Tr(u∗ , ∂Gj ∩ G) = pj , j = 1, 2. G

Proof. For each λ > 0 we have by Proposition 3.2 and the upper-bound, Eε (uε , G \ Σλ ) ≤ C ,

19

(3.26)

hence, passing to a diagonal subsequence, we may extract a subsequence satisfying 1 uεn * u∗ in Hloc (G \ Σ) , with u∗ |Gj ∈ H 1 (Gj , Γj ), j = 1, 2 .

We shall next verify that u∗ satisfies both the constraint and the boundary conditions in (3.26). ∫ |u∗ | = Rc . Claim 1: G

First, notice that for any λ > 0, ∫ ∫ Rc = |uεn | = G

∫ |uεn | +

|uεn | .

Σλ

Since uεn → u∗ in L1 (G \ Σλ ) we have

∫

G\Σλ

|u∗ | = G\Σλ

∫

(λ)

G\Σλ

|uεn | + oεn (1). Here and in the

(λ)

sequel oεn (1) denotes a quantity that goes to zero with εn , for every fixed λ > 0. By (H3 ) and (1.8), ∫

∫ |uεn | =

∫

Σλ ∩{|uεn |>R0 }

Σλ

|uεn | +

Σλ ∩{|uεn |≤R0 }

|uεn |

∫

≤ c˜0

Σλ ∩{|uεn |>R0 }

W (uεn ) + CλR0 ≤ C(εn + λR0 ) .

Therefore, ∫

∫

∫

|u∗ | = G

|u∗ | + G\Σλ

∫

∫

|u∗ | = Σλ

|u∗ | + oλ (1) =

|uεn | + o(λ) εn (1) + oλ (1)

G\Σλ ∫G\Σλ (λ) = Rc − |uεn | + o(λ) εn (1) + oλ (1) = Rc + oεn (1) + oλ (1) . Σλ

Letting εn → 0 and then λ → 0 yields the claim. Claim 1: Tr(u∗ , ∂Gj ∩ G) = pj , j = 1, 2. In the sequel we shall write again for the sake of simplicity uε instead of uεn . Since the geodesic distance on Γj is equivalent to the euclidean distance, we get from Lemma 3.2 and Proposition 3.2 that J ∫ ∑ j=1

{τj =±d(ε)}

|∇uε |2 +

s1 (uε ) − p1 |2 C W (uε ) |˜ + ≤ . 2 2 ε d (ε) d(ε)

20

(3.27)

By (3.27) and (1.16) we have for every j, ∫ ∫ 1 2 2 |uε − p1 | ≤ |uε − s˜1 (uε )|2 + |˜ s1 (uε ) − p1 |2 d(ε) {τj =−d(ε)} d(ε) {τj =−d(ε)} C ≤ d(ε)

∫ {τj =−d(ε)}

W (uε ) + C ≤

Cε2 + C ≤ C . (3.28) d2 (ε)

We now define a new map wε on G1 by  u (x) x ∈ G1 \ Σd(ε) , ε wε (x) = τ τ (1 − j )p1 + j uε (σj , d(ε)) x ∈ Σ(j) , j = 1, . . . , J . d(ε) d(ε) d(ε) Using (1.8),(3.18) and (3.27)–(3.28) we get by a direct computation that ∫ |∇wε |2 + |wε |2 ≤ C , Σd(ε)

so we have wε * u∗ in H 1 (G1 ), implying that Tr(u∗ , ∂G1 ∩G) = p1 . The same argument shows that also Tr(u∗ , ∂G2 ∩ G) = p2 . Next, we prove a lower bound for the energy. ∫ Proposition 3.4. Eε (uε ) ≥ 2DI(α) + |∇u∗ |2 + o(1) . ε G Proof. It suffices to consider a subsequence uεn as in Proposition 3.3. Fix any λ > 0. By Proposition 3.2, 2DI(α) Eεn (uεn ; Σλ ) ≥ + o(1) . (3.29) εn By Proposition 3.3, ∫ lim inf Eεn (uεn ; G \ Σλ ) ≥ |∇u∗ |2 . (3.30) εn →0

G\Σλ

Since λ is arbitrary, the result follows by combining (3.29) and (3.30). In order to identify the limit u∗ we shall need the following upper-bound which improves the estimate of (1.8). 1 2 Proposition 3.5. ∫ There exists a family of functions {vε } ⊂ H (G, R ), each satisfying the constraint |vε | = Rc , such that G

Eε (vε ) ≤

2DI(α) + E0 + o(1) , ε

where E0 is defined in (1.22). 21

(3.31)

Proof. We shall need some notation associated with the geodesic γ that were used in [1]. We associate with γ an arclength parametrization on [0, L] and define z(s) as the solution of the ODE dz √ = W (γ(z(s))) , z(0) = L/2 , (3.32) ds which is defined on the whole real line and satisfies lim z(s) = 0 and

s→−∞

Since

and

lim z(s) = L .

s→∞

√ W (γ(z(s))) ∼ |γ(z(s)) − p1 | ∼ z(s) √ W (γ(z(s))) ∼ |γ(z(s)) − p2 | ∼ L − z(s)

as s → −∞ as s → ∞ ,

(3.33)

(3.34)

we have, for some positive constants c˜1 , c˜2 , 0 ≤ z(s) ≤ Cec˜1 s

for s < 0 ,

(3.35)

0 ≤ L − z(s) ≤ Ce−˜c2 s

for s > 0 .

(3.36)

Recall that (U1 , U2 ) denote minimizers for the problem in (1.22). It will be convenient to denote by u0 the map defined on the whole domain G \ Σ by u0 |Gk = Uk , k = 1, 2. (T ) Our construction of a test function vε will depend on a real parameter T = Tε , whose value will be determined later by the constraint. (T ) (T ) Fix a small λ > 0. In G \ Σλ we set vε = u0 , so it remains to define vε in each (j) Σλ (see (1.20)). Consider first the case T ≥ 0. We set Sε = T ε ln

1 1 1 1 and Kε = Sε + 2( + )ε ln . ε c˜1 c˜2 ε (T )

(j)

For each j = 1, . . . , J we next define vε for x ∈ Σλ , using the coordinates (σj , τj ) = (j) (σj (x), τj (x)) as follows. Recall the definition of φk in Remark 1.1 and let π2 be the (j) (j) constant which is the trace of φ2 on ∂G2 ∩ G (so that g2 (eiπ2 ) = p2 ). Then define,

22

writing for short g˜2 (φ) := g2 (eiφ ),    u0 (x)      p1       affine func. of τj joining p1 to γ(z(−2 lnc˜1/ε ))   1  ln 1/ε τ −ε−Sε −2ε c˜ 1 vε(T ) (x) = γ(z( j )) ε      affine func. of τj joining γ(z(2 lnc˜1/ε )) to p2  2  ( ) ) ((  ) (  (j) τj −(Kε +2ε) ) τj −(Kε +2ε)  φk (x) + 1 − π2 g˜2   ε ln 1ε ε ln 1ε    u0 (x)

τj < 0 τj ∈ [0, Sε ] τj ∈ [Sε , Sε + ε] τj ∈ (Sε + ε, Kε + ε) τj ∈ [Kε + ε, Kε + 2ε] τj − Kε − 2ε ∈ (0, ε ln 1ε ]

τj > Kε + 2ε + ε ln 1ε . (3.37) (j) Since µ(Σλ ) = λl(Σ(j) ) + O(λ2 ), we find by a direct computation, J ∫ ∫ ∑ (T ) (|vε | − |u0 |) − G

j=1

Σ

(j)

1 (|p1 | − |u0 |) ≤ Cε ln , ε 1 ∩{τj >0}

(3.38)

T ε ln ε

for some constant C independent of T (provided that T ε ln 1ε 0, ε + ε (t) u = uε + t(φ − α− uε ) + o(t) t < 0, where

1 α± = Rc

(∫ {uε 6=0}

∫

uε ·φ± |uε |

{uε =0}

(A.3)

) |φ| .

Using (A.3) yields ∫

t Eε (u ) − Eε (uε ) = 2t ∇uε · ∇(φ − α± uε ) + 2 ε G

∫ ∇W (uε ) · (φ − α± uε ) + o(t) . (A.4)

(t)

G

Taking the two limits t → 0+ and t → 0− in (A.4) and using the minimality property of uε we obtain that ∫ ∫ } ∫ uε 1 λε { ·φ+ |φ| ≤ 2∇uε · ∇φ + 2 ∇W (uε ) · φ Rc ε {uε 6=0} |uε | {uε =0} G ∫ ∫ { } λε uε ≤ ·φ− |φ| , (A.5) Rc {uε 6=0} |uε | {uε =0} ∫

where

2|∇uε |2 +

λε = G

24

1 ∇W (uε ) · uε . ε2

From (A.5) we deduce that −2∆uε + ε12 ∇W (uε ), viewed as a distribution in H −1 (G), is actually a function λε fε ∈ L∞ (G, R2 ) satisfying (A.2). Finally, the Neumann boundary condition in (A.1) follows by preforming the above computation with φ ∈ C ∞ (G, R2 ). The next lemma provides an L∞ bound for uε whose proof was shown to us by Petru Mironescu. Note that such an estimate was proved by Gurtin and Matano in [7] in the scalar case, but their argument does not seem to apply here. Lemma A.2 (Mironescu). Let R1 be given in (H30 ). Then, there exists an ε0 such that kuε kL∞ (G) ≤ R1 , ε < ε0 .

(A.6)

Proof. We fix a function Φ ∈ C ∞ [0, ∞) satisfying: (i) Φ(t) = 0 on [0, R1 ]. c (ii) 0 < Φ0 (t) ≤ 2 on (R1 , ∞). Put t +1 φ = Φ(|uε |)uε . Using φ as a test function in (A.1) and the identity uε ·

∂uε 1 ∂|uε |2 ∂|uε | = = |uε | , ∂xi 2 ∂xi ∂xi

yields ∫

∫ 2 Φ(|uε |) 2 Φ(|uε |)|∇uε | + Φ (|uε |)|uε | ∇|uε | + ∇W (uε ) · uε ε2 G G ∫ )∫ 1 ( 1 2 = 2|∇uε | + 2 ∇W (uε ) · uε Φ(|uε |)|uε | . (A.7) Rc G ε G 2

0

Since the first integral on the L.H.S. of (A.7) is nonnegative, using the assumptions (H30 ) and (H3 ) yields ∫ ∫ )∫ a1 c˜0 1 ( 1 2 Φ(|uε |)|uε | ≤ 2|∇uε | + 2 ∇W (uε ) · uε Φ(|uε |)|uε | . ε2 {|uε |>R1 } Rc G ε {|uε |>R1 } (A.8) Next we claim that ∫ ( 1 ) 1 (A.9) 2|∇uε |2 + 2 ∇W (uε ) · uε = O 3/2 . ε ε G 25

First, by the energy estimate (1.8) we have ∫ (1) |∇uε |2 = O . ε G

(A.10)

√ Next, we note that the assumption (H2 ) implies that |∇W | ≤ c W in some neighborhood of Γ1 ∪ Γ2 , hence the same estimate holds in BR1 (0) (for a different c) and we deduce, using again (1.8), that ∫ ∫ )1/2 ( 1 ) CR12 ( 1 |∇W (u ) · u | ≤ W (u ) = O . (A.11) ε ε ε ε2 {|uε |≤R1 } ε2 ε3/2 {|uε |≤R1 } Finally, using (H30 ) with (1.8) yields ∫ ∫ (1) 1 a2 |∇W (u ) · u | ≤ W (u ) = O . ε ε ε ε2 {|uε |>R1 } ε2 {|uε |>R1 } ε

(A.12)

Combining (A.10) with (A.11)–(A.12) we are led to (A.9). Plugging (A.9) in (A.8) implies that for ε < ε0 , ∫ Φ(|uε |)|uε | = 0 , {|uε |>R1 }

i.e., the set {|uε | > R1 } is empty. Proof of Proposition 2.1. For each x0 ∈ G with dist(x0 , ∂G) ≥ ε define the rescaled function u˜ε (x) = uε (x0 + εx) on B1 (0). Using (A.1) we conclude that u˜ε satisfies the equation ∫ ) 1 1 ε2 ( ∆˜ uε = ∇W (˜ uε ) − 2|∇uε |2 + 2 ∇W (uε ) · uε fε (x0 + εx) in B1 (0) . (A.13) 2 2 ε G From (A.6) and (A.9) we deduce that the R.H.S. of (A.13) is bounded in L∞ (B1 (0)), uniformly in ε < ε0 . By standard elliptic estimates it follows that ∇˜ uε is bounded in L∞ (B1/2 (0)). Rescaling back we deduce the desired bound for ∇uε in L∞ loc (G). The estimate at points x0 with dist(x0 , ∂G) < ε follows similarly from elliptic estimates for a Neumann boundary problem on the domain obtained by rescaling of the domain B2ε (x0 ) ∩ G

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