Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

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The Main Result
We consider the family of functionals F ε (u, ) := ε|∇ 2 u| 2 + 1 ε 1 − |∇u| 2 2 dx, (1.1) where ⊂ R 2 is a C 2 bounded open set, ε > 0 and u ∈ W 2,2 0 ( ). These functionals were introduced in [4] and proposed as a model for blistering in [27]. In these cases we are interested in the minimizers u ε of F ε in the space where n denotes the outer normal to . The final goal is the understanding of the behavior of u ε as ε → 0. In [27] (and more explicitly in [5]) it is conjectured that u ε →ū := dist(·, ∂ ), (1.2) at least for convex domains . A first partial result in this direction was obtained in [16,Theorem 5.1], where the authors proved that if is an ellipse, then lim ε→0 min F ε (·, ) = F 0 (ū, ), (1.3) where F 0 is the candidate asymptotic functional that we are going to introduce in (1.4).
The main result of this paper is the proof of (1.2) in the same setting as in [16], namely Theorem 1.1. Let ⊂ R 2 be an ellipse and, for every ε > 0, let u ε be a minimizer of F ε (·, ). Then This result is obtained as a corollary after showing thatū is the unique minimizer of a suitable asymptotic problem for F ε (·, ) as ε → 0. In order to rigorously introduce it, we recall some previous results (see also the introduction of [10] for a presentation of the history of the problem).

Previous Results
In what follows, denotes a C 2 bounded open subset of R 2 . Independently from the validity of 1.2, it is conjectured already in [4] that (1) if u ε is such that lim sup ε→0 F ε (u ε , ) < ∞, then u ε converges up to subsequences to a Lipschitz solution u of the eikonal equation |∇u| = 1; (2) if u ε is a sequence of minimizers of F ε (·, ), then any limit u of u ε minimizes the functional among the solutions of the eikonal equation. Here, J ∇v denotes the jump set of ∇v and ∇v ± the corresponding traces.
A positive answer to the first point was obtained independently in [12] and [2]. A fundamental notion in this analysis and in particular in [12] is the one of entropy, borrowed from the field of conservation laws. Definition 1.2. We say that ∈ C ∞ c (R 2 ; R 2 ) is an entropy if for every open set ⊂ R 2 and every smooth m : → R 2 it holds that div m = 0 and |m| 2 = 1 ⇒ div( (m)) = 0. (1.5) We will denote by E the set of entropies.
We will consider the following family of entropies introduced first in [6,16]: there (α 1 , α 2 ) is an orthonormal system in R 2 .
Collecting the results of [12] and [2] we get the following statement: Theorem 1.3. Let ε k → 0 and u k ∈ W 2,2 0 ( ) be such that lim sup k→∞ F ε k (u k , ) < ∞. Then m k := ∇ ⊥ u k is pre-compact in L 1 ( ). Moreover if m k converges to m in L 1 ( ), then |m| = 1 a.e. in , for every entropy ∈ E it holds that namely μ is a locally finite Radon measure on . We moreover set Finally we denote by The functionalF 0 (·, ) coincides with F 0 (·, ) in the subspace of 0 ( ) whose elements have gradient in BV loc ( ) (see [2]) and it is the natural candidate to be the -limit of the functionals F ε (·, ) as ε → 0 + .
Although A( ) ⊂ BV loc ( ), elements of A( ) share with BV functions most of their fine properties.
The analogy with the structure of elements in A( )∩BV loc ( ) is not complete: for these functions properties (1) and (3) can be improved to (1.6) In order to prove (3') from (3) one should show that μ is concentrated on J . This is considered as a fundamental step towards the solution of the -limit conjecture and it remains open. Notice moreover that by means of Theorem 1.5 we can give a meaning to the definition of the functional F 0 (·, ) even for solutions u to the eikonal equation with ∇ ⊥ u ∈ A( ) \ BV loc ( ); Property (3') would imply that F 0 coincides withF 0 on the whole 0 ( ). A fundamental tool in the study of fine properties of elements of A( ) is the kinetic formulation [18] (see also [23] in the framework of scalar conservation laws). Here we use a more recent version obtained in [13].
where χ : × R/2π Z is defined by We observe that if σ solves (1.7), then σ + μ ⊗ L 1 also solves (1.7) for every μ ∈ M loc ( ). This ambiguity is resolved in [13] by considering the unique σ 0 solving (1.7) such that The above kinetic formulation encodes the entropy production of the family of entropies Condition (1.5) is equivalent to d ds (e is ) · e is = 0 for every s ∈ R/2π Z, therefore for every ∈ E we can define ψ : Notice that ∈ E π if and only if ψ is π -periodic. Rephrasing the construction in [13], we have the following identity: for every ∈ E π and every ζ ∈ C 1 A possibly weaker version of (3') is the following: (3") Eq. (1.6) holds for every ∈ E π . This is equivalent to require that ν 0 := ( p x ) |σ 0 | ∈ M loc ( ) is concentrated on J and moreover it would be sufficient to establish the equality F 0 =F 0 . The following proposition is a partial result in this direction for general m ∈ A( ); we remark here that a key step of the proof of Theorem 1.1 is to establish (3") for a class of m including the limits of ∇ ⊥ u ε , where u ε is a minimizer of F ε (·, ) and is an ellipse.
. Then for ν 0 -a.e. x ∈ \ J there existss =s(x) ∈ R/2π Z such that Among other results, the same expression for σ 0,x has been obtained very recently in [22] under the additional assumption that div (m) ∈ L p ( ) for every ∈ E. As the authors point out, it is still not known if this additional assumption is sufficient to establish that indeed σ 0 vanishes.

The Asymptotic Problem
Adapting the argument in [30] for scalar conservation laws to this context, it is possible to prove that the elements of A( ) with finite energy have strong traces in L 1 at the boundary of . However, the conditions do not guarantee that ∂u ∂n = −1 on ∂ ; in other words we can have boundary layers. In order to take them into account we slightly reformulate the minimum problem for F ε (·, ): given δ > 0 we define Being of class C 2 , we can take δ > 0 sufficiently small so that the function − dist(x, ∂ ) belongs to W 2,2 (S δ ). We therefore consider the minimum problems for the functionals F ε (·, δ ) on the space Notice that for every u ∈ ( ) the function u δ : δ → R defined by belongs to δ ( ) and Similarly the restriction to of any function in δ ( ) belongs to ( ), so that the two minimum problems are equivalent. We will also denote by We will prove the following result: We show now that Theorem 1.1 is a corollary of Theorem 1.8 and the previous mentioned results: indeed let ε k → 0 as k → ∞ and for any k let u ε k be a minimizer of F ε k (·, ) on ( ). By Theorem 1.3 and (1.3) we have that every limit point u 0 of u ε k belongs to 0 ( ) and moreover it holds Sinceū δ is the only minimizer ofF 0 (·, δ ) in 0 δ ( ), then u δ 0 =ū δ , namely u 0 =ū.

Zero-Energy States
The only case in which the behavior of minimizers of F ε (·, ) as ε → 0 is completely understood is when lim ε→0 min F ε (·, ) = 0. All the sets admitting sequences with vanishing energy were characterized in [17] and with the appropriate boundary conditions the limit function is in these casesū = dist(·, ∂ ). A quantitative version of this result is proven in [20] (see also [19]). In a different direction, it was shown in [21] that the vanishing of the two entropy defect measures div e 1 ,e 2 (m) and div ε 1 ,ε 2 (m) is sufficient to establish div (m) = 0 for every ∈ E. Here we denoted by (e 1 , e 2 ) the standard orthonormal system in R 2 and by the orthonormal system obtained by performing a rotation of (e 1 , e 2 ) by π/4.

States with a Vanishing Entropy Defect Measure
The case when is an ellipse is special since we know a priori that there exists an orthonormal system (α 1 , α 2 ) in R 2 for which the minimizers u δ in A( δ ) of the asymptotic problem F 0 (·, δ ) satisfy This situation has been considered more extensively in [14,15], where in particular the authors proved the minimizing property of the viscosity solution (1.3) for more general domains and functionals. In this direction we only mention here that the same arguments of this paper allow to prove Theorem 1.1 also in the case where is a stadium, namely a domain of the form We finally mention that under the additional assumption (1.11) we can prove Property (3").

A Micromagnetics Model
A family of functionals E ε strictly related to (1.1) was introduced in [28,29] in the context of micro-magnetics. An analogous result to Theorem 1.1 was proved in [8] even for general smooth domains , while the -limit conjecture is still open also in this setting. Although Theorem 1.5 has a perfect analogue for the elements in the asymptotic domain of E ε (see [7]), the main difficulty seems to be a still not complete understanding of the fine properties of these elements. In this direction we notice that the method used here to establish Proposition 1.7 gives the analogue in this setting of the concentration property (3') (see [25]).

Lagrangian Representation of Elements in A( )
The Lagrangian representation is an extension of the classical method of characteristics to the non-smooth setting: it was introduced in the framework of scalar conservation laws in [9,24] building on the kinetic formulation from [23]. This approach is strongly inspired by the decomposition in elementary solutions of non-negative measure valued solutions of the linear transport equation, called superposition principle (see [1]). Indeed by Theorem 1.6, the vector fields m ∈ A( ) are represented by the solution χ of the linear transport equation (1.7). The main difficulty in this case is due to the source term which is merely a derivative of a measure. This issue is reflected in the lack of regularity of the characteristics detected by our Lagrangian representation, which have bounded variation but they are in general not continuous. A fundamental feature for our analysis is that we can decompose the kinetic measure σ in (1.7) along the characteristics.

Lagrangian Representation
We introduce the following space of curves: given T > 0, we let We will always consider the right-continuous representative of the component γ s . Moreover we will adopt the notation from [3] for the decomposition of the measure Dv where v ∈ BV(I ; R) for some interval I ⊂ R, whereDv denotes the sum of the absolutely continuous part and the Cantor part of Dv and D j v denotes the jump part of Dv. We will need to consider alsoDv for functions v ∈ BV(I ; R/2π Z). In this caseDv =Dw where w is any function in BV(I ; R) such that for every z ∈ I the value w(z) belongs to the class v(z) in R/2π Z. For every t ∈ (0, T ) we consider the section and we denote Definition 2.1. Let m ∈ A( ) and be a W 2,∞ -open set compactly contained in We say that a finite non-negative Radon measure ω ∈ M( ) is a Lagrangian representation of m in if the following conditions hold: (1) for every t ∈ (0, T ) it holds that For every curve γ ∈ we define the measure σ γ ∈ M((0, T ) × × R/2π Z) by (2.4) Notice that since R/2π Z is not ordered, given s 1 = s 2 ∈ R/2π Z the condition s 1 < s 2 is not defined. Nevertheless we use the notation s ∈ (s 1 , s 2 ) or s 1 < s < s 2 to indicate the following condition (depending only on the orientation of R/2π Z): if t 1 , t 2 ∈ R are such t 1 < t 2 < t 1 + 2π , e it 1 = e is 1 and e it 2 = e is 2 then there exists t ∈ (t 1 , t 2 ) such that e it = e is .

Lemma 2.2. Let ω be a Lagrangian representation of m ∈ A( ) on . Let us denote by
Proof. We show that (2.5) holds when tested with every function of the form By the chain-rule for functions with bounded variation we have the following equality between measures: where J γ denotes the jump set of γ . Therefore, proceeding in the chain (2.6), we have therefore in order to establish e is · ∇ xχ , φ = ∂ s σ ω , φ it suffices to prove that By Point (4) in Definition 2.1 for ω-a.e. γ ∈ it holds that ϕ(γ (t − γ +)) = ϕ(γ (t + γ −)) = 0, and, in particular, where we used (2.1) in the second equality and thatχ does not depend on t in the last equality. This concludes the proof.
is a minimal kinetic measure if it satisfies (1.7) and for every σ solving (1.7) it holds that We moreover say that ω is a a minimal Lagrangian representation of m if it is a Lagrangian representation of m according to Def. 2.1 and The existence of a minimal kinetic measure is proven in the following lemma: Proof. Since ∂ s σ is uniquely determined by (1.7), we have that a kinetic measure σ is minimal if and only if for ν σ -a.e. x ∈ the disintegration σ x satisfies the following inequality: Therefore all minimal kinetic measures are of the form where α : → R is a measurable function such that for ν σ 0 -a.e. x ∈ it holds that The existence of such an α is trivial and in particular it holds that In Sect. 3 we will show that for every m ∈ A( ) there exists a unique minimal kinetic measure σ min , namely that for ν min -a.e. x ∈ there exists a unique α(x) such that (2.8) holds.
The main result of this section is The existence of a Lagrangian representation for weak solutions with finite entropy production to general conservation laws on the whole (0, T ) × R d has been proved in [24]. The case of bounded domains when is a ball was considered in [25] for the class of solutions to the eikonal equation arising in [29]. The extension to the case where is a W 2,∞ open set does not cause any significant difficulty. In particular the argument proposed in [25] applies here with trivial modifications and leads to the following partial result: Lemma 2.6. In the setting of Proposition 2.5, let σ ∈ M loc ( × R/2π Z) be a locally finite measure satisfying (1.7). Then there exists a Lagrangian representation ω of m on such that We now prove Proposition 2.5 relying on Lemma 2.6.
Proof of Proposition 2.5. Let m ∈ A( ) and letσ be a minimal kinetic measure. By Lemma 2.6, there exists a Lagrangian representation ω of m such that By definition of σ ω it holds that By Lemma 2.2, the measure σ ω satisfies (2.5); beingσ a minimal kinetic measure for m, it follows that T σ ≤ σ ω . In particular the inequalities in (2.10) and (2.11) are equalities and (2.9) follows.
The following lemma is a simple application of Tonelli theorem and (2.1); since it is already proven in [26], we refer to it for the details.
We denote by g the set of curves γ ∈ such that the two properties above hold.

Structure of the Kinetic Measure
The main goal of this section is to prove Proposition 1.7. As a corollary we will obtain the concentration property (3") presented in the introduction for solutions m ∈ A( ) with a vanishing entropy defect measure. The key step is the following regularity result (the strategy of the proof is borrowed from [25], where an analogous statement was proved for the solutions to the eikonal equation arising in the micromagnetics model mentioned in the introduction, and we finally observe that in that situation this result is sufficient to establish the concentration property (3'), while it is not the case here): , and setx :=γ x (t) ands :=γ s (t+). Then there exists c > 0 such that for every δ ∈ (0, 1/2) we have at least one of the following: (1) the lower density estimate holds true: (2) the following lower bound holds true: The same statement holds by settings :=γ s (t−).
In particular In the remaining part of the proof we assume that L 1 (E − (r )) > r/2, being the case L 1 (E + (r )) > r/2 analogous. Given ε > 0, we consider the strip be the nontrivial interiors of the connected components of Notice that we have the estimate For every i ∈ N we consider and the measurable restriction map We finally consider the measurẽ We observe thatω ∈ M + ( ), since, for every N > 0, by Point (3) in Definition 2.1. The advantage of the measureω is that it is concentrated on curves whose x-components are transversal toγ x on the whole domain of definition. This property allows us to prove the following claim: Claim 1. There exists an absolute constantc > 0 such that forω-a.e. (γ , t − γ , t + γ ) ∈ it holds Proof of Claim 1. It follows from (3.1) and the characteristic equation (2.2) that there exists a Lipschitz function fγ : R → R such that Similarly forω-a.e. (γ , t − γ , t + γ ) ∈ there exists a Lipschitz function f γ such that for L 1 -a.e. z ∈ R. By the definitions of S r,ε in (3.2) and of fγ in (3.3), it easily follows that By construction ofω, forω-a.e. (γ , t − γ , t + γ ) ∈ and L 1 -a.e. t ∈ (t − γ , t + γ ) it holds that On the other hand Therefore, by (3.5) and (3.6), we have for some universalc > 0. This concludes the proof of the claim.

By construction we have
for every t ∈ (0, T ). Therefore On the other hand, sinceγ ∈ g and L 1 (E − (r )) > r/2 there existsε > 0 such that for every ε ∈ (0,ε) it holds that We consider the split = > ∪ < , where We will prove the following claim, from which the lemma follows immediately: There exists an absolute constant c 1 > 0 such that the two following implications hold true: (2) ifω( < ) ≥ r δ 2 T 10c , then Proof of (1). By definition of > and the assumption in (1) we have Proof of (2). Forω-a.e. (γ , t − γ , t + γ ) ∈ < , the image of γ x is contained in B 2r (x) and Tot.Var.(γ s ) ≥ δ 5 . Since ω is a minimal Lagrangian representation, this implies that
Proof. In particular let σ be a minimal kinetic measure; since σ is π -periodic in the variable s, it follows from Proposition 3.2 that for ν min -a.e. x ∈ \ J it holds that for somes ∈ R/2π Z and some c ∈ R depending on x. The necessary and sufficient condition (2.7) for minimality trivially implies c = 0. By Theorem 1.5 and (1.9) it holds that The following identity was obtained in Sect. 4.2 of [13]: for every β ∈ [0, π/2] it holds that where g β : R/2π Z → R is a π -periodic defined by Observe that the constraint div m = 0 implies that for H 1 -a.e. x ∈ J it holds m + · n = m − · n. Therefore, with the notation introduced in the statement, we have n = ±e is . We prove (3.10) first in the case β ∈ [0, π/2]. Choosing˜ such that ψ˜ (s) = ψ (s +s), we deduce from (3.11) that This shows that for ν min -a.e. x ∈ J with β ∈ (0, π/2) there exist two constants c 1 > 0 and c 2 ∈ R such that σ x = c 1 (g β (· −s) + c 2 )L 1 . It is a straightforward computation to check that the choice in (3.10) is the unique that satisfies the constraint in (2.7). In particular σ x is uniquely determined for ν min -a.e. x ∈ J such that β ∈ (0, π/2). The case β ∈ (π/2, π), can be reduced to the previous case exchanging m + with m − , and therefore changing the sign of n and replacings withs + π . Since ∂ s ψ and g β for β ∈ (0, π/2] are π -periodic, then the same computations as above leads to Similarly the choice in (3.10) is the unique that satisfies the constraint (2.7). σ x being uniquely determined for ν min -a.e. x ∈ , the measure σ min is unique.

Solutions with a Single Vanishing Entropy
The goal of this section is to prove the following result about solutions with vanishing entropy production: For ν min -a.e. x ∈ J it holds n = ±e is , therefore in order to show that J is contained in a countable union of horizontal and vertical segments, it is sufficient to observe that for every β ∈ (0, π) it holds that This can be proven directly by using the explicit expression of g β in (3.10). Alternatively, we refer to [6,Lemma 2.4], where the authors show that for m ∈ A( ) ∩ BV ( ) it holds that where α ∈ R/2π Z is such that n = ±e i(α+ π 4 ) . Theorem 1.5 implies that the same computation is valid for every m ∈ A( ). Since cos(2α) = 0 ⇒ α ∈ π 4 + π 2 Z, then div ε 1 ,ε 2 (m) = 0 implies that n = e is withs ∈ π 2 Z a.e. with respect to the measure ν min J . Now we prove that ν min is concentrated on J : by Corollary 3.4, for ν min -a.e.
Remark 3.8. The same argument shows that, in order to prove that ν min is concentrated on J , the assumption div ε 1 ,ε 2 (m) = 0 can be replaced with div (m) = 0 for any ∈ E π such that {s : ∂ s ψ (s) = 0} is at most countable.

Uniqueness of Minimizers on Ellipses
The goal of this section is to prove Theorem 1.8. Since the functionalF 0 is invariant by rotations, then we will assume without loss of generality that the major axis of the ellipse is parallel to x-axis in the plane.
The next result is essentially contained in [16] (see also [15]); for completeness, we give the proof here. Proof. In [2], the authors noticed that for every u ∈ A( δ ) it holds that Let us denote bym := ∇ ⊥ūδ . Since for every u ∈ δ ( ) it holds ∇ ⊥ u =m in S δ , then it follows from (4.1) that = div e 1 ,e 2 (m)( δ ) =F 0 (ū δ , δ ), where in the last equality we used div ε 1 ,ε 2 (m) = 0 and div e 1 ,e 2 (m) ≥ 0. This shows, in particular, thatū δ is a minimizer ofF 0 (·, δ ) in 0 δ ( ). Moreover for every minimizer u ofF 0 (·, δ ) in 0 δ ( ), the inequality in (4.2) is an equality and this completes the proof. Proof. The proof is divided into three steps: in Step 1 we link the assumptions in (4.3) with the sign of ∂ s σ min relying on Corollary 3.4 and Proposition 3.6. Then we will prove in Step 2 that the entropy defect measures of every m as in the statement are concentrated on the axis of the ellipse. We finally prove in Step 3 that this last condition forces m to satisfy (4.4).
Step 1. Let m ∈ A δ ( ) be as in the statement and σ min be its minimal kinetic measure. Then, for every φ ∈ C 1 c ( δ × R/2π Z) such that φ ≥ 0 and it holds that Proof of Step 1. Since div ε 1 ,ε 2 (m) = 0, it follows from Proposition 3.6 that for ν min -a.e. x ∈ J the normal to J at x is n(x) = e is(x) for some s(x) ∈ π 2 Z. Up to exchange m + and m − , we can therefore assume without loss of generality that n(x) = (1, 0) or n(x) = (0, 1) for ν min -a.e. x ∈ J . We denote by J h ⊂ J the points for which n = (0, 1) and J v ⊂ J the points with n(x) = (1, 0). We consider these two cases separately.
A similar argument excludes that ν min ({x ∈ J h : x 2 = b}) > 0 if b < 0 and that ν min ({x ∈ J v : x 1 = a}) > 0 if a = 0; see Fig. 2 which illustrates the sets E that need to be considered in these cases.
Step 3. We prove that the unique m ∈ A δ ( ) for which ν min is concentrated on the axis of the ellipse satisfies (4.4). In particular we show that m =m oñ δ = {x ∈ δ : x 1 < 0, x 2 > 0}, this being the argument for the other analogous quadrants.
Letx ∈ be a Lebesgue point of m and lets(x) ∈ (π/2, π) be such that e is(x) = −∇ dist(x, ∂ ). y s2 x 1 x 2 Fig. 3. The picture represents the points y s 1 , ys (x) , y s 2 , while the arrows represent the values ofm at these points For every s ∈ (π/2, π) let t s > 0 be the unique value such that y s :=x + t s e is ∈ ∂ δ/2 ∩˜ δ .
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