A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

Yoshikazu Giga; Jun Okamoto; Masaaki Uesaka

doi:10.1515/acv-2020-0113

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A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

Yoshikazu Giga , Jun Okamoto and Masaaki Uesaka

Published/Copyright: June 23, 2021

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From the journal Advances in Calculus of Variations Volume 16 Issue 1

Abstract

An explicit representation of the Gamma limit of a single-well Modica–Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi–Warren–Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness are useful to characterize the limit of minimizers of the Kobayashi–Warren–Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problems is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

Keywords: Gamma convergence; Modica–Mortola functional; Kobayashi–Warren–Carter energy

MSC 2010: 49Q20; 74A50; 74N05

1 Introduction

In this paper, we are interested in a singular limit called the Gamma limit of a single-well Modica–Mortola functional under the graph convergence, the convergence with respect to the Hausdorff distance of graphs, which is finer than conventional L 1 -convergence or convergence in measure. A single-well Modica–Mortola functional was introduced by Ambrosio and Tortorelli [1, 2] to approximate the Mumford–Shah functional [29]. A typical explicit form of their functional now called the Ambrosio–Tortorelli functional is

ℰ ε ( u , v ) : = σ ∫ Ω v 2 | ∇ u | 2 d x + λ ∫ Ω ( u - g ) 2 d x + E ε ( v ) ,

with small parameter ε > 0 , where E ε is a single-well Modica–Mortola functional of the form

E ε ( v ) : = 1 2 ⁢ ε ∫ Ω ( v - 1 ) 2 d x + ε 2 ∫ Ω | ∇ v | 2 d x .

Here g is a given function defined in a bounded domain Ω in 𝐑 n and σ ≥ 0 , λ ≥ 0 are given parameters. The potential energy part ( v - 1 ) 2 is a single-well potential. If it is replaced by a double-well potential like ( v 2 - 1 ) 2 , the corresponding energy E ε well approximates (a constant multiple of) the surface area of the interface, and this observation goes back to Modica and Mortola [24, 25]. Even for the single-well potential, if v is close to zero around some interface, then it is expected that E ε still approximates the surface area of the interface. This observation enables us to prove that, for σ > 0 , the Gamma limit of ℰ ε ⁢ ( u , v ) in the convergence in measure is a Mumford–Shah functional; see [1, 2, 10].

If E ε ⁢ ( v ε ) is bounded for small ε > 0 , then it is rather clear that v ε → 1 in L 1 as ε → 0 , so that v ε → 1 almost everywhere by taking a suitable subsequence. Therefore, it seems natural to consider the Gamma convergence in L 1 -sense. However, if one considers

(1.1) E b ε ⁢ ( v ) = E ε ⁢ ( v ) + b ⁢ v ⁢ ( 0 ) 2

for b > 0 , where Ω = ( - 1 , 1 ) , then we see L 1 -convergence is too weak because, in the limit stage, the effect of the term involving b is invisible, but this should be counted.

To illustrate the point, we calculate the unique minimizer w ε of E b ε ⁢ ( v ) , that is,

E b ε ⁢ ( w ε ) = min ⁡ { E b ε ⁢ ( v ) ∣ v ∈ H 1 ⁢ ( - 1 , 1 ) } .

This is a strict convex problem so that the minimizer exists and is unique. Moreover, its Euler–Lagrange equation is linear. A simple manipulation shows that the minimizer of E b ε with the Neumann boundary conditions w ε ′ ⁢ ( ± 1 ) = 0 is given by

w ε ⁢ ( x ) = 1 + b ⁢ ( - e - 2 ε - 1 ) 1 - e - 4 ε + b ⁢ ( 1 + e - 2 ε ) 2 ⁢ e - | x | ε + b ⁢ ( - e - 2 ε - e - 4 ε ) 1 - e - 4 ε + b ⁢ ( 1 + e - 2 ε ) 2 ⁢ e | x | ε .

It converges to 1 locally uniformly outside zero, but

lim ε → 0 ⁡ w ε ⁢ ( 0 ) = 1 1 + b > 0 and lim ε → 0 ⁡ E b ε ⁢ ( w ε ) = b 1 + b < b .

Since E b ε ⁢ ( 1 ) = b for any ε > 0 , the information that w ε ⁢ ( x ) → 1 almost everywhere is insufficient to identify the behavior of minimizers w ε .

$Figure 1 The graphs of w ε {w_{\varepsilon}} as the minimizers of E b ε {E^{\varepsilon}_{b}} defined by (1.1) when b = 1 {b=1} and ε = 10 - 1 , 10 - 2 , 10 - 3 {\varepsilon=10^{-1},10^{-2},10^{-3}} .$

Figure 1

The graphs of w ε as the minimizers of E b ε defined by (1.1) when b = 1 and ε = 10 - 1 , 10 - 2 , 10 - 3 .

We show the graph of w ε for several ε > 0 in Figure 1. We see that the graph of w ε is dropping sharply at x = 0 , and its sharpness increases as ε → 0 . Hence it is natural to consider the graph convergence of w ε , and its limit is a set-valued function Ξ so that Ξ ⁢ ( x ) = { 1 } for x ≠ 0 and Ξ ⁢ ( 0 ) = [ 1 1 + b , 1 ] .

Our first goal is to give an explicit representation formula for the Gamma limit of E b ε under the graph convergence as well as compactness. We discuss such problems only in a one-dimensional domain since the problem is already complicated. The graph convergence enables us to characterize the limit of above w ε as a minimizer of the Gamma limit of E b ε .

Our second goal is to give an explicit representation formula for the Gamma limit of the Kobayashi–Warren–Carter energy. A typical form of the energy is

E KWC ε ⁢ ( u , v ) = σ ⁢ ∫ Ω v 2 ⁢ | ∇ ⁡ u | + E ε ⁢ ( v ) ,

where ∫ v 2 ⁢ | ∇ ⁡ u | denotes a weighted total variation of a Radon measure ∇ ⁡ u ; see Section 5 for a precise definition. Here and hereafter, we suppress d ⁢ x unless ∇ ⁡ u is absolutely continuous with respect to the Lebesgue measure. This energy was first proposed by [18, 20] to model motion of multi-phase problems in materials sciences. This energy looks similar to the Ambrosio–Tortorelli functional ℰ ε . It is obtained by inhomogenizing Dirichlet energy ∫ | ∇ ⁡ u | 2 ⁢ d ⁢ x by putting weights ∫ v 2 ⁢ | ∇ ⁡ u | 2 ⁢ d ⁢ x with a single-well Modica–Mortola functional. By this observation, we call ℰ ε an Ambrosio–Tortorelli inhomogenization of the Dirichlet energy when λ = 0 . From this point of view, the Kobayashi–Warren–Carter energy is interpreted as an Ambrosio–Tortorelli inhomogenization of the total variation. It turns out that natural topology for studying the limit of functionals as ε → 0 is quite different.

For the Ambrosio–Tortorelli functional, it is enough to consider the L 1 × L 1 convergence since v ε ⁢ ( x ) → 1 except at most finitely many points where lim ⁢ inf ∗ ⁡ v ε ⁢ ( x ) = 0 if one assumes that ℰ ε ⁢ ( u ε , v ε ) is bounded and u ε → u , v ε → v in L 1 (see [1, 2, 10]). Here lim ⁢ inf ∗ denotes the relaxed liminf, and we shall give its definition in Section 2. For the Kobayashi–Warren–Carter energy, however, the situation is quite different. Indeed, if one considers

u ⁢ ( x ) = { 1 , 0 < x < 1 , 0 , - 1 < x < 0 ,

then E KWC ε ⁢ ( u , v ) = E σ ε ⁢ ( v ) with Ω = ( - 1 , 1 ) . Thus the natural convergence for v must be in the graph convergence as we discussed before. Note that, in our problem, v ε → 1 except at most countably many points where lim ⁢ inf ∗ ⁡ v ε may be nonzero. One merit of the graph convergence is that it is very strong, so when we consider the Gamma limit problem, we do not need to restrict ourselves to the space of special BV functions as for the Ambrosio–Tortorelli functional.

Our first main result is a characterization of the Gamma limit of E b ε in the graph convergence (Theorem 2.1). To show the Gamma convergence, we need to prove the two types of inequalities often called liminf and limsup inequalities. To show the liminf inequality, a key point is to study a general behavior near the set Σ of all exceptional points of the limit set-valued function Ξ; here we say a point x is exceptional if Ξ ⁢ ( x ) is not a singleton. To describe the behavior near Σ, a conventional method is to find a suitable accumulating sequence as in [10, proof of Proposition 3.3]. However, unfortunately, it seems that this argument does not apply to our setting since Σ can be a countably infinite set. Thus we are forced to introduce a new method to show the liminf inequality. When we study an absolutely continuous function u ε on a bounded interval I, that is, u ε ∈ W 1 , 1 ⁢ ( I ) , we associate its unfolding U ε by replacing the variable by the arc-length parameter of the graph. Namely, we set U ε ⁢ ( s ) = u ε ⁢ ( x ε ⁢ ( s ) ) , s ∈ J ε = s ε ⁢ ( I ) , where x ε = x ε ⁢ ( s ) is the inverse function of the arc-length parameter

s ε ⁢ ( x ) = ∫ 0 x ( 1 + ( u x ε ⁢ ( z ) ) 2 ) 1 2 ⁢ d ⁢ z .

If the total variation of u ε is bounded, then the length of J ε is bounded as ε → 0 . The unfolding U ε has several merits compared with the original one. First, { U ε } and { x ε } are uniformly Lipschitz with constant 1. Second, the total variation of U ε and u ε are the same as expected. It is easy to study the convergence as ε → 0 of unfolding U ε compared with the original u ε . Among other results, we are able to characterize the relaxed limits lim ⁢ inf ∗ ⁡ u ε , lim ⁢ sup ∗ ⁡ u ε by the limit of U ε and x ε . We use this unfolding for 1 2 ⁢ ( v ε - 1 ) 2 in the case of E b ε to show liminf inequalities, where { v ε } is a given sequence with a bound for E b ε ⁢ ( v ε ) . The proof for limsup inequalities is not difficult, although one has to be careful that there are countably many points where the limit of v ε is not equal to one.

We also established a compactness under the graph convergence with a bound for E b ε (Theorem 2.2). This can be easily proved by use of unfoldings.

Based on results on E b ε , we are able to prove the Gamma convergence of the Kobayashi–Warren–Carter energy E KWC ε under the graph convergence (Theorem 2.3). If u is a piecewise constant function with countably many jump points { a ℓ } ℓ = 1 ∞ ⊂ Ω with positive jump { b ℓ } ℓ = 1 ∞ , we see that

E KWC ε ⁢ ( u , v ) = E ε ⁢ ( v ) + σ ⁢ ∑ ℓ = 1 ∞ b ℓ ⁢ v 2 ⁢ ( a ℓ ) .

The Gamma limit for such fixed u is easily reduced to the results of E b ε . However, to establish the liminf inequality for E KWC ε for both u ε and v ε , we have to establish some lower estimate for a sequence ∫ Ω v ε 2 ⁢ | ∇ ⁡ u ε | ⁢ d ⁢ x as ε → 0 , which is an additional difficulty. However, we still do not need to use SBV space here.

The Gamma convergence problem of the Modica–Mortola functional, which is the sum of Dirichlet type energy and potential energy, was first studied by [24]. Since then, there is a large number of works discussing the Gamma convergence. However, the topology is either L 1 or convergence in measure. In our Gamma limit, the topology is the graph convergence, which is finer than in previous studies. In [25], the L 1 Gamma limit of a double-well Modica–Mortola functional is characterized as a number of transition points in one-dimensional setting. Later, in [23, 32], it was extended to the multi-dimensional setting, and the limit is a constant multiple of the surface area of the transition interface. This type of the Gamma convergence results as well as compactness is important to establish the convergence of local minimizer [21] as well as the global minimizer. However, the convergence of critical points are not in the framework of a general theory, and a special treatment is necessary [15]. The double-well Modica–Mortola functional is by now well studied, even in the level of gradient flow, called the Allen–Cahn equation. The limit ε → 0 is often called the sharp interface limit, and the resulting flow is known as the mean curvature flow. For an early stage of development of the theory, see [6, 7, 8, 9].

A single-well Modica–Mortola functional was first used in [1] to approximate the Mumford–Shah functional. The Gamma limit of the Ambrosio–Tortorelli functional is by now well studied [1, 2, 10]. However, convergence of critical points is studied only in one dimension [11]. The Ambrosio–Tortorelli type approximation is now used in various problems. In [12], the Ambrosio–Tortorelli type approximation is introduced to describe brittle fractures. Its evolution is also described in [13]. For the Steiner problem, such approximation was also proposed [22], and its Gamma limit is established [4]. However, all these problems are closer to the Ambrosio–Tortorelli inhomogenization of the Dirichlet energy, not of the total variation.

For the Kobayashi–Warren–Carter energy, its gradient flow for fixed ε is somewhat studied. Note that the well-posedness itself is non-trivial because, even if one assumes v ≡ 1 , the gradient flow of E KWC ε is the total variation flow and the definition of a solution itself is non-trivial; see [17] for example. Apparently, there is no well-posedness result for the original system proposed by [20, 18, 19]. According to [18], its explicit form is

τ 1 ⁢ v t = s ⁢ Δ ⁢ v + ( 1 - v ) - 2 ⁢ s ⁢ v ⁢ | ∇ ⁡ u | ,

(1.2) τ 0 ⁢ v 2 ⁢ u t = s ⁢ div ⁡ ( v 2 ⁢ ∇ ⁡ u | ∇ ⁡ u | ) ,

where τ 0 , τ 1 and s are positive parameters. This system is regarded as the gradient flow of E KWC ε with F ⁢ ( v ) = ( v - 1 ) 2 , ε = 1 , σ = s with respect to a kind of weighted L 2 norm whose weight depends on the solution. If one replaces (1.2) by

τ 0 ⁢ ( v 2 + δ ) ⁢ u t = s ⁢ div ⁡ ( ( v 2 + δ ′ ) ⁢ ∇ ⁡ u | ∇ ⁡ u | + ν ⁢ ∇ ⁡ u )

with δ > 0 , δ ′ ≥ 0 and ν ≥ 0 satisfying δ ′ + ν > 0 , then the studies of existence and large-time behavior of solutions are developed in [16, 26, 27, 33, 30, 31], under homogeneous settings of boundary conditions. However, the uniqueness question is almost open, and there is just one result [16, Theorem 2.2] for the one-dimensional solution, under ν > 0 . Meanwhile, the line of previous results can be extended to the studies of non-homogeneous cases of boundary conditions. For instance, if we impose the non-homogeneous Dirichlet boundary condition for (1.2), then we can further observe various structural patterns of steady-state solutions, under one-dimensional setting, two-dimensional radially-symmetric setting, and so on (cf. [28]).

This paper is organized as follows. In Section 2, we recall the notion of the graph convergence and state our main Gamma convergence results as well as compactness. In Section 3, we introduce the notion of unfoldings. Section 4 is devoted to the proof of the Gamma convergence of E b ε as well as the compactness in the graph convergence. Section 5 is devoted to the proof of the Gamma convergence of the Kobayashi–Warren–Carter energy.

2 Singular limit under graph convergence

We first recall the basic notion of set-valued functions; see [3] for example. Let ( M , d M ) be a compact metric space. We consider a set-valued function Γ defined in M such that Γ ⁢ ( x ) is a compact set in 𝐑 for each x ∈ M . If its graph ⁡ Γ defined by

graph Γ : = { ( x , y ) ∈ M × 𝐑 ∣ y ∈ Γ ( x ) , x ∈ M }

is closed, we say that Γ is upper semicontinuous. Let ℬ denote the totality of a bounded, upper semicontinuous set-valued function. In other words,

ℬ : = { Γ ∣ graph Γ is compact in M × 𝐑 } .

For Γ 1 , Γ 2 ∈ ℬ , we set

d g ( Γ 1 , Γ 2 ) : = d H ( graph Γ 1 , graph Γ 2 ) ,

where d H denotes the Hausdorff distance of two sets in M × 𝐑 . The Hausdorff distance d H is defined as usual,

d H ( A , B ) : = max { sup z ∈ A dist ( z , B ) , sup w ∈ B dist ( w , A ) }

for A , B ⊂ M × 𝐑 , where

dist ( z , B ) : = inf w ∈ B dist ( z , w ) , dist ( z , w ) : = ( d M ( z 1 , w 1 ) 2 + | z 2 - w 2 | 2 ) 1 2

for z = ( z 1 , z 2 ) and w = ( w 1 , w 2 ) . It is easy to see that ( ℬ , d g ) is a complete metric space. The convergence with respect to d g is called the graph convergence.

We next recall the notion of semi-convergent limit for sets. For a family of closed subsets { Z ε } 0 < ε < 1 in M × 𝐑 , we set

lim sup ε → 0 Z ε : = ⋂ ε > 0 cl ( ⋃ 0 < δ < ε Z δ ) , lim inf ε → 0 Z ε : = cl ( ⋃ ε > 0 ⋂ 0 < δ < ε Z δ ) ,

where cl denotes the closure in M × 𝐑 . These semi-limits can be defined for sequences like { Z j } j = 1 ∞ with trivial modification.

Lemma 2.1.

A sequence { Γ j } j = 1 ∞ ⊂ B converges to Γ in the sense of the graph convergence if and only if

lim sup j → ∞ ⁡ graph ⁡ Γ j = lim inf j → ∞ ⁡ graph ⁡ Γ j = graph ⁡ Γ .

Proof.

Note that the Hausdorff convergence to A for a sequence { A j } j = 1 ∞ of compact sets is equivalent to saying that,

for any z ∈ A , there is a sequence z j ∈ A j such that z j → z ( j → ∞ ) and
if w j ∈ A j converges to w, then w ∈ A .

Since (i) and (ii) are equivalent to

lim inf j → ∞ ⁡ A j ⊃ A , lim sup j → ∞ ⁡ A j ⊂ A ,

respectively, the Hausdorff convergence is equivalent to saying that

A = lim inf j → ∞ ⁡ A j = lim sup j → ∞ ⁡ A j .

Thus the proof is complete. ∎

We next recall relaxed convergent limits of functions. Let { g j } be a sequence of real-valued functions on M. For x ∈ M , we set

lim ⁢ sup ∗ j → ∞ g j ( x ) : = lim j → ∞ sup { g k ( y ) | | y - x | < 1 j , k ≥ j } ,

lim ⁢ inf ∗ j → ∞ g j ( x ) : = lim j → ∞ inf { g k ( y ) | | y - x | < 1 j , k ≥ j } ;

see [14, Chapter 2] for more detail. By definition, the lim ⁢ sup ∗ ⁡ w j is upper semicontinuous and lim ⁢ inf ∗ ⁡ w j is lower semicontinuous.

Let C ⁢ ( M ) be the Banach space of all continuous real-valued functions on M equipped with the norm ∥ f ∥ ∞ = sup x ∈ M ⁡ | f ⁢ ( x ) | , f ∈ C ⁢ ( M ) . For g ∈ C ⁢ ( M ) , we associate a set-valued function Γ g such that Γ g ⁢ ( x ) = { g ⁢ ( x ) } for x ∈ M . Clearly, Γ g ∈ ℬ .

Lemma 2.2.

Let { g j } j = 1 ∞ ⊂ C ⁢ ( M ) be a bounded sequence. Then the semi-limit Γ + = lim sup j → ∞ ⁡ Γ g j still belongs to B . Let K be the set-valued function of the form

K ( x ) : = { y ∈ 𝐑 ∣ lim ⁢ inf ∗ j → ∞ g j ( x ) ≤ y ≤ lim ⁢ sup ∗ j → ∞ g j ( x ) } .

Then Γ + ⁢ ( x ) ⊂ K ⁢ ( x ) for all x ∈ M .

Proof.

The first statement is trivial. To prove Γ + ⊂ K , it suffices to prove that the limit y = lim j → ∞ ⁡ y j , y j ∈ Γ g j ⁢ ( x j ) belongs to K ⁢ ( x ) if x j → x . Since y j = g j ⁢ ( x j ) , by definition of relaxed limits lim ⁢ sup ∗ and lim ⁢ inf ∗ , it is easy to see that lim ⁢ inf ∗ j → ∞ ⁡ g j ⁢ ( x ) ≤ y ≤ lim ⁢ sup ∗ j → ∞ ⁡ g j ⁢ ( x ) . Thus Γ + ⁢ ( x ) ⊂ K ⁢ ( x ) . ∎

We next discuss an equivalent condition of the graph convergence.

Lemma 2.3.

The following statements hold.

Let { g j } j = 1 ∞ ⊂ C ⁢ ( M ) be a bounded sequence. Then the semi-limit Γ - = lim inf j → ∞ ⁡ Γ g j belongs to ℬ .
Assume that M is locally arcwise connected. For each x ∈ M , the following three conditions are equivalent.
1. Γ - ⁢ ( x ) contains both lim ⁢ inf ∗ ⁡ g j ⁢ ( x ) and lim ⁢ sup ∗ ⁡ g j ⁢ ( x ) .
2. K ⁢ ( x ) ⊂ Γ - ⁢ ( x ) .
3. Γ - ⁢ ( x ) = Γ + ⁢ ( x ) = K ⁢ ( x ) .
In particular, Γ g j converges to K in the graph sense if and only if one of (a), (b), (c) holds for all x ∈ M .

Proof.

(1) follows from the definition, and we focus on the proof of (2). Assume (a) so that Γ - ⁢ ( x ) contains ∂ ⁡ ( K ⁢ ( x ) ) . Then there exist x j ∈ M , y j = g j ⁢ ( x j ) such that x j → x , y j → y ^ for y ^ = lim ⁢ inf ∗ ⁡ g j ⁢ ( x ) , and there exist x ¯ j ∈ M , y ¯ j ∈ g j ⁢ ( x ¯ j ) such that x ¯ j → x , y ¯ j → y ¯ for y ¯ = lim ⁢ sup ∗ ⁡ g j ⁢ ( x ) .

By assumption, for any δ > 0 , there exists an arc γ j connecting x j to x ¯ j , lying in a δ-neighborhood B δ of x provided that j is sufficiently large. Since g j is continuous on γ j ⊂ B δ , the intermediate value theorem implies that [ y j , y ¯ j ] ⊂ g j ⁢ ( B δ ) . Thus K ⁢ ( x ) ⊂ Γ - ⁢ ( x ) . Thus (b) follows.

Assume (b). By Lemma 2.2, we know Γ + ⁢ ( x ) ⊂ K ⁢ ( x ) . By definition of Γ - , we see Γ - ⊂ Γ + . Thus we conclude (c). It is easy to see that (c) implies (a). The proof is now complete. ∎

We next consider an important subclass of ℬ . Let 𝒜 be the family of Γ ∈ ℬ such that Γ ⁢ ( x ) is a closed interval for all x ∈ M . Let 𝒜 0 be the subfamily of 𝒜 such that Γ ⁢ ( x ) is the singleton { 1 } with at most countably many exceptions of x ∈ M . Such Γ is uniquely determined by { x i } i = 1 ∞ , where Γ ⁢ ( x i ) = [ ξ i - , ξ i + ] with ξ i - < ξ i + contains 1 and Γ ⁢ ( x ) = { 1 } if x ∉ { x i } i = 1 ∞ . We call such a point x i an exceptional point of Ξ ∈ 𝒜 0 , so that Σ is the set of all exceptional points of Ξ.

We next study the compactness in the graph convergence.

Lemma 2.4.

Let { g j } j = 1 ∞ ⊂ C ⁢ ( M ) be a bounded sequence. Assume that

η - ⁢ ( x ) < η + ⁢ ( x ) 𝑓𝑜𝑟 ⁢ x ∈ S ,

η - ⁢ ( x ) = η + ⁢ ( x ) = 1 𝑓𝑜𝑟 ⁢ x ∈ M ∖ S ,

where S is a countable set and

η - ⁢ ( x ) = lim ⁢ inf ∗ j → ∞ ⁡ g j ⁢ ( x ) , η + ⁢ ( x ) = lim ⁢ sup ∗ j → ∞ ⁡ g j ⁢ ( x ) .

If 1 ∈ [ η - ⁢ ( x ) , η + ⁢ ( x ) ] , then there is a subsequence { g j k } such that Γ g j k converges to some Γ 0 ∈ A 0 in the graph sense.

Proof.

We write S = { x i } i = 1 ∞ . By definition, there is a subsequence { g 1 , j - } of { g j } such that

η - ⁢ ( x 1 ) = lim j → ∞ ⁡ g 1 , j - ⁢ ( y 1 , j )

with some { y 1 , j } converging to x 1 . We set

η 1 + ( x 1 ) : = lim ⁢ sup ∗ j → ∞ g 1 , j - ( x 1 ) ≤ η + ( x 1 ) .

Since η - = η + = 1 outside S, we see η + ⁢ ( x 1 ) ≥ 1 . We take a further subsequence { g 1 , j } of { g 1 , j - } so that

η 1 + ⁢ ( x 1 ) = lim j → ∞ ⁡ g 1 , j ⁢ ( z 1 , j )

with some { z 1 , j } converging to x 1 . We repeat this procedure for x 2 , x 3 , … and find a subsequence { g ℓ , j } j = 1 ∞ so that

lim j → ∞ ⁡ g ℓ , j ⁢ ( y ℓ , j ) = lim ⁢ inf ∗ j → ∞ ⁡ g ℓ , j ⁢ ( x ℓ ) ≤ 1 , lim j → ∞ ⁡ g ℓ , j ⁢ ( z ℓ , j ) = lim ⁢ sup ∗ j → ∞ ⁡ g ℓ , j ⁢ ( x ℓ ) ≥ 1

with some { y ℓ , j } , { z ℓ , j } converging to x ℓ for ℓ = 1 , 2 , … , k . By a diagonal argument, we see that { g k , k } k = 1 ∞ has the property that

ξ - ( x ) : = lim ⁢ inf ∗ k → ∞ g k , k ( x ) , ξ + ( x ) : = lim ⁢ sup ∗ k → ∞ g k , k ( x )

belong to Γ - ⁢ ( x ) = lim inf k → ∞ ⁡ Γ g k , k for x ∈ M . We now apply Lemma 2.3 (2) to conclude that Γ g k , k converges to Γ with Γ ⁢ ( x ) = [ ξ - ⁢ ( x ) , ξ + ⁢ ( x ) ] , x ∈ M . By construction, Γ ⁢ ( x ) = { 1 } for x ∈ M ∖ S and ξ - ⁢ ( x ) ≤ 1 ≤ ξ + ⁢ ( x ) for x ∈ S . Thus Γ ∈ 𝒜 0 , so the proof is now complete. ∎

We now define several functionals when M = I ¯ or 𝐓 = 𝐑 / 𝐙 , where I is a bounded open interval in 𝐑 and I ¯ = cl ⁡ I . For a real-valued function v on M and ε > 0 , a single-well Modica–Mortola functional is defined by

E sMM ε ( v ) : = ε 2 ∫ M | d ⁢ v d ⁢ x | 2 d x + 1 2 ⁢ ε ∫ M F ( v ) d x .

Here the potential energy F is a single-well potential. We shall assume that

F ∈ C ⁢ ( 𝐑 ) is nonnegative and F ⁢ ( v ) = 0 if and only if v = 1 ,
lim inf | v | → ∞ ⁡ F ⁢ ( v ) > 0 ,
(growth condition) there are positive constants c 0 , c 1 such that F ⁢ ( v ) ≥ c 0 ⁢ | v | 2 - c 1 for all v ∈ 𝐑 .

Remark 2.1.

Obviously, (F2 ′ ) implies (F2).

We are interested in a Gamma limit of E sMM ε not in usual L 1 -convergence, but the graph convergence, which is of course finer than L 1 topology. As usual, we set

G ⁢ ( v ) = | ∫ 1 v F ⁢ ( τ ) ⁢ d ⁢ τ | .

A typical example of F ⁢ ( v ) is F ⁢ ( v ) = ( v - 1 ) 2 . In this case, G ⁢ ( v ) = 1 2 ⁢ ( v - 1 ) 2 . To write the limit energy for Ξ ∈ 𝒜 0 , let Σ = { x i } i = 1 ∞ denote the totality of points where Ξ ⁢ ( x i ) is a nontrivial closed interval [ ξ i - , ξ i + ] such that ξ i - ≤ 1 ≤ ξ i + . This set can be a finite set. By definition, Ξ ⁢ ( x ) = { 1 } if x ∉ { x i } i = 1 ∞ . In the case that M = 𝐓 , we define

E sMM 0 ( Ξ , 𝐓 ) : = { 2 ⁢ ∑ i = 1 ∞ { G ⁢ ( ξ i - ) + G ⁢ ( ξ i + ) } for ⁢ Ξ ∈ 𝒜 0 , ∞ otherwise .

In the case that M = I ¯ , one has to modify the value when x i is the end point of I ¯ . The energy is defined by

E sMM 0 ( Ξ , I ¯ ) : = { ∑ i = 1 ∞ { 2 ⁢ ( G ⁢ ( ξ i - ) + G ⁢ ( ξ i + ) ) - κ i ⁢ max ⁡ ( G ⁢ ( ξ i - ) , G ⁢ ( ξ i + ) ) } for ⁢ Ξ ∈ 𝒜 0 , ∞ otherwise ,

where κ i = 0 if x i ∈ I and κ i = 1 if x i ∈ ∂ ⁡ I . For brevity, we simply write v j → 𝑔 Ξ by the abuse of notation if v j ∈ C ⁢ ( M ) is a sequence such that v j → Ξ ( j → ∞ ) in the sense of the graph convergence. We also use v ε → 𝑔 Ξ as ε → 0 if ε is a continuous parameter.

We shall state that the Gamma limit of E sMM ε is E sMM 0 as ε → 0 under the graph convergence. For later applications, it is convenient to consider a slightly more general functional of the form

E sMM ε , b ( v ) : = E sMM ε ( v ) + b ( v ( a ) ) 2 ,

where a ∈ M ̊ = int ⁡ M and b ≥ 0 . The corresponding limit functional is

E sMM 0 , b ( Ξ , M ) : = E sMM 0 ( Ξ , M ) + b ( min Ξ ( a ) ) 2 .

Theorem 2.1 (Gamma limit under graph convergence).

Assume the following conditions:

M = I ¯ or 𝐓 ;
F satisfies (F1) and (F2);
a ∈ M ̊ = int ⁡ M and b ≥ 0 .

Then the following inequalities hold.

(liminf inequality) Let { v ε } 0 < ε < 1 be in H 1 ⁢ ( M ) ⊂ C ⁢ ( M ) . If v ε → 𝑔 Ξ ∈ ℬ , then

E sMM 0 , b ⁢ ( Ξ , M ) ≤ lim inf ε → 0 ⁡ E sMM ε , b ⁢ ( v ε ) .

In particular, if the right-hand side is finite, then Ξ ∈ 𝒜 0 .
(limsup inequality) For any Ξ ∈ 𝒜 0 , there is { w ε } 0 < ε < 1 ⊂ H 1 ⁢ ( M ) ⊂ C ⁢ ( M ) such that w ε → 𝑔 Ξ and

E sMM 0 , b ⁢ ( Ξ , M ) = lim ε → 0 ⁡ E sMM ε , b ⁢ ( w ε ) .

We also have a compactness result.

Theorem 2.2 (Compactness).

Assume that M = I ¯ or T . Assume that F satisfies (F1) and (F2 ′ ). Let { v ε j } j = 1 ∞ be in H 1 ⁢ ( M ) ⊂ C ⁢ ( M ) . Assume that sup j ⁡ E sMM ε j ⁢ ( v ε j ) < ∞ for ε j → 0 as j → ∞ . Then there exists a subsequence { v ε k ′ } such that v ε k ′ → 𝑔 Ξ with some Ξ ∈ A 0 .

By combining the Gamma convergence result and the compactness, a general theory yields the convergence of a minimizer of E sMM ε , b ; see [5, Theorem 1.21] for example. Note that, in the case of b = 0 , the minimum of E sMM ε ⁢ ( v ) is zero and is attained only at constant function v = 1 , so the convergence of minimizers is trivial.

Corollary 2.1.

Assume the same hypotheses of Theorem 2.1 and (F2 ′ ). Let v ε be a minimizer of E sMM ε , b on H 1 ⁢ ( M ) . Then there is a subsequence { v ε k } k = 1 ∞ such that v ε k → 𝑔 Ξ 0 with some Ξ 0 ∈ A 0 . Moreover, Ξ 0 is a minimizer of E sMM 0 , b . Furthermore, Ξ 0 ⁢ ( x ) = { 1 } if x ≠ a and Ξ 0 ⁢ ( a ) = [ p 0 , 1 ] , where p 0 is a minimizer of 2 ⁢ G ⁢ ( p ) + b ⁢ p 2 with p ∈ [ 0 , 1 ] .

Remark 2.2.

If F ′ ⁢ ( v ) ⁢ ( v - 1 ) ≥ 0 , then G is convex so that 2 ⁢ G ⁢ ( p ) + b ⁢ p 2 is strictly convex for b > 0 . In this case, the minimizer is unique. If F ⁢ ( v ) = ( v - 1 ) 2 so that G ⁢ ( v ) = 1 2 ⁢ ( v - 1 ) 2 , then 2 ⁢ G ⁢ ( p ) + b ⁢ p 2 = ( p - 1 ) 2 + b ⁢ p 2 and its minimizer is 1 b + 1 and its minimal value is E sMM 0 , b ⁢ ( Ξ 0 , M ) = b b + 1 .

Our theory has an application to the Kobayashi–Warren–Carter energy [20, 18, 19] which can be interpreted as an Ambrosio–Tortorelli inhomogenization of the total variation energy. Its typical form is

E KWC ε ( u , v ) : = σ ∫ M ̊ v 2 | d ⁢ u d ⁢ x | + E sMM ε ( v )

for σ ≥ 0 . The first integral denotes the total variation of u with weight v 2 . See Section 5 for a more rigorous definition. Note that if u x = 0 outside a and u jumps at a with jump 1, then E KWC ε ⁢ ( u , v ) = E sMM ε , σ ⁢ ( v ) , so our E sMM ε , σ ⁢ ( v ) is considered a special value of E KWC ε ⁢ ( u , v ) by fixing such u. For Ξ ∈ 𝒜 0 , let Σ = { x i } i = 1 ∞ be the set of all exceptional points of Ξ. (Note that the set Σ can be finite.) Let ξ i - = min ⁡ Ξ ⁢ ( x i ) for x i ∈ Σ . For u ∈ BV ⁢ ( M ̊ ) , let J u denote the set of jump discontinuities of u, i.e.,

J u : = { x ∈ M ̊ ∣ d ( x ) = | u ( x + 0 ) - u ( x - 0 ) | > 0 } ,

where u ⁢ ( x + 0 ) (resp. u ⁢ ( x - 0 ) ) denotes the trace from right (resp. left). For ( u , Ξ ) ∈ L 1 ⁢ ( M ̊ ) × ℬ , we set

E KWC 0 ( u , Ξ , M ) : = { σ ⁢ ∫ M ̊ ∖ ( J u ∩ Σ ) | d ⁢ u d ⁢ x | + σ ⁢ ∑ i = 1 ∞ d i ⁢ ( ξ i - ) 2 + E sMM 0 ⁢ ( Ξ , M ) for ⁢ u ∈ BV ⁢ ( M ̊ ) ⁢ and ⁢ Ξ ∈ 𝒜 0 , ∞ otherwise ,

Theorem 2.3 (Gamma limit).

Assume that the same hypotheses of Theorem 2.1 concerning M and F hold.

(liminf inequality) Let { v ε } 0 < ε < 1 be in H 1 ⁢ ( M ) ⊂ C ⁢ ( M ) . Assume that v ε → 𝑔 Ξ ∈ ℬ as ε → 0 . Let { u ε } ⊂ L 1 ⁢ ( M ) satisfy u ε → u in L 1 ⁢ ( M ̊ ) as ε → 0 . Then

E KWC 0 ⁢ ( u , Ξ , M ) ≤ lim inf ε → 0 ⁡ E KWC ε ⁢ ( u ε , v ε ) .
(limsup inequality) For any Ξ ∈ 𝒜 0 and u ∈ BV ⁢ ( M ̊ ) , there exist

{ w ε } 0 < ε < 1 ⊂ H 1 ⁢ ( M ̊ ) ⊂ C ⁢ ( M ) 𝑎𝑛𝑑 { u ε } 0 < ε < 1 ⊂ L 1 ⁢ ( M )

such that w ε → 𝑔 Ξ and u ε → u in L 1 satisfying

E KWC 0 ⁢ ( u , Ξ , M ) = lim ε → ∞ ⁡ E KWC ε ⁢ ( u ε , w ε ) .

Remark 2.3.

(i) From the proof of Lemma 2.1, it suffices to assume u ε → u in L loc 1 ⁢ ( M ̊ ∖ Σ 0 ) , where

Σ 0 = { x ∈ M ∣ min ⁡ Ξ ⁢ ( x ) = 0 }

in the statement of Theorem 2.3 (i). Since Ξ must be in 𝒜 0 and E sMM 0 ⁢ ( Ξ , M ) < ∞ , this set Σ 0 must be a finite set.

(ii) We may add a fidelity term λ ⁢ ∥ u - g ∥ L 2 ⁢ ( M ̊ ) 2 to energies E KWC ε , E KWC 0 for λ > 0 with given g ∈ L 2 ⁢ ( M ̊ ) like the Ambrosio–Tortorelli functional ℰ ε and the Mumford–Shah functional. More precisely, the statement of Theorem 2.3 is still valid for

E KWC ε , λ ( u , v ) : = E KWC ε ( u , v ) + λ ∫ M ̊ | u - g | 2 d x ,

E KWC 0 , λ ( u , Ξ , M ) : = E KWC 0 ( u , Ξ , M ) + λ ∫ M ̊ | u - g | 2 d x .

The next compactness result easily follows from the compactness (Theorem 2.2) in ℬ and L 1 -compactness of BV ⁢ ( Ω ) , where Ω is an open set such that Ω ¯ ⊂ M ̊ ∖ Σ 0 .

Theorem 2.4 (Compactness).

Assume the same hypothesis of Theorem 2.2 concerning M and F. Let λ > 0 be fixed. Let { v ε j } j = 1 ∞ be in H 1 ⁢ ( M ) ⊂ C ⁢ ( M ) and { u ε j } ⊂ L 2 ⁢ ( M ̊ ) . Assume that

sup j ⁡ E KWC ε j , λ ⁢ ( u ε j , v ε j ) < ∞ 𝑓𝑜𝑟 ⁢ ε j → 0 .

Then there exists a subsequence { ( u ε k ′ , v ε k ′ ) } such that u ε k ′ → u in L loc 1 ⁢ ( M ̊ ∖ Σ 0 ) with some u ∈ L loc 1 ⁢ ( M ̊ ∖ Σ 0 ) and that v ε k ′ → 𝑔 Ξ with some Ξ ∈ A 0 . Here Σ 0 = { x ∈ M ∣ min ⁡ Ξ ⁢ ( x ) = 0 } .

By combining the Gamma convergence result and the compactness, a general theory yields the convergence of a minimizer of E KWC ε , λ ; see [5, Theorem 1.21] for example.

Corollary 2.2.

Assume the same hypothesis of Theorem 2.1. Let ( u ε , v ε ) be a minimizer of E KWC ε , λ . Then there is a subsequence { ( u ε k , v ε k ) } k = 1 ∞ such that v ε k → 𝑔 Ξ , u ε k → u in L loc 1 ⁢ ( M ̊ ∖ Σ 0 ) and that the limit ( u , Ξ 0 ) be a minimizer of E KWC ε , λ . Here Σ 0 = { x ∈ M ∣ min ⁡ Ξ 0 ⁢ ( x ) = 0 } .

3 Unfolding by arc-length parameters

For a bounded open interval I, let u be a real-valued C 1 function on I ¯ , that is, u ∈ C 1 ⁢ ( I ¯ ) . To simplify notation, we set I = ( 0 , r ) . Then the arc-length parameter s of the graph curve y = u ⁢ ( x ) is defined as

s ( x ) = s u ( x ) : = ∫ 0 x ( 1 + u x 2 ( z ) ) 1 2 d z .

One is able to extend this definition for general u ∈ BV ⁢ ( I ) . By definition, s ⁢ ( ⋅ ) is strictly monotone increasing. It is easy to see that s ⁢ ( ⋅ ) is continuous if and only if the derivative u x has no point mass, that is, u has no jump, which is equivalent to u ∈ C ⁢ ( I ¯ ) . The inverse function x = x ⁢ ( s ) of s = s ⁢ ( x ) is always Lipschitz with Lipschitz constant 1, that is, Lip ⁡ ( x ) ≤ 1 . Indeed, since d ⁢ x d ⁢ s = ( 1 + u x 2 ) - 1 2 , the inequality | d ⁢ x d ⁢ s | ≤ 1 always holds. For u ∈ C ⁢ ( I ¯ ) ∩ BV ⁢ ( I ) , we define an unfolding U by arc-length parameter of the form U ⁢ ( s ) = u ⁢ ( x ⁢ ( s ) ) . The function U is defined on J ¯ with J = ( 0 , L ) , where L is the length of the graph u on I ¯ . Note that L ≥ r , the length of I.

We begin with several basic properties of the unfoldings.

Lemma 3.1.

Assume that u ∈ W 1 , 1 ⁢ ( I ¯ ) .

U is Lipschitz continuous on J . More precisely, Lip ⁢ ( U ) ≤ 1 .
The total variation of U on J equals that of u in I , that is, TV ⁢ ( u ) = TV ⁢ ( U ) .

Proof.

(i) Since

U s = u x ( 1 + u x 2 ) 1 2 ,

Lip ⁢ ( U ) ≤ 1 is rather clear.

(ii) By definition,

TV ⁢ ( U ) = ∫ 0 L | U s | ⁢ d ⁢ s = ∫ 0 r | u x | ⁢ d ⁢ x = TV ⁢ ( u ) . ∎

Since q = p ( 1 + p 2 ) 1 2 is equivalent to p = q ( 1 - q 2 ) 1 2 , we see that

d ⁢ x d ⁢ s = 1 ( 1 + u x 2 ) 1 2 = ( 1 - U s 2 ) 1 2 .

We next discuss compactness for unfoldings and the lower semicontinuity of TV ⁢ ( ⋅ ) .

Lemma 3.2.

Assume that { u ε } 0 < ε < 1 ⊂ W 1 , 1 ⁢ ( I ) with a bound for TV ⁢ ( u ε ) and ∥ u ε ∥ ∞ . Then there is a subsequence such that U ε tends to some function V with Lip ⁢ ( V ) ≤ 1 uniformly in a domain of definition of V. Moreover, TV ⁢ ( V ) ≤ lim inf ε → 0 ⁡ TV ⁢ ( u ε ) .

Proof.

Since TV ⁢ ( u ε ) is bounded, so is the length L ε of the graph of u ε . The existence of a convergent subsequence follows from the Ascoli–Arzela theorem. A basic lower semicontinuity of TV ⁢ ( ⋅ ) yields

TV ⁢ ( V ) ≤ lim inf ε → 0 ⁡ TV ⁢ ( U ε ) .

The right-hand side equals TV ⁢ ( u ε ) as proved in Lemma 3.1 (ii), so the proof is now complete. ∎

We raise the question whether or not a Lipschitz function V on J with Lip ⁢ ( V ) ≤ 1 can be written as u ⁢ ( x ⁢ ( s ) ) . This is in general not true if there is a non-trivial interval such that U s = 1 (or U s = - 1 ). Indeed, if U s = ± 1 , then x ⁢ ( s ) is not invertible.

In spite of this lack of the correspondence, however, the following lemma states that the limit of the unfolding contains the information on the pointwise behavior of u ε k . (See also Figure 2.)

Theorem 3.1.

Assume that { u ε k } k = 1 ∞ ⊂ W 1 , 1 ⁢ ( I ) is a sequence uniformly bounded in TV and its unfolding U ε k converges uniformly to V in the domain J of definition of V.

If x ε k , the inverse of the arc-length parameter of u ε k , converges uniformly to a limit x ¯ in J, then

( lim ⁢ sup ∗ k → ∞ ⁡ u ε k ) ⁢ ( x ) = max ⁡ { V ⁢ ( s ) ∣ x ¯ ⁢ ( s ) = x } ,

( lim ⁢ inf ∗ k → ∞ ⁡ u ε k ) ⁢ ( x ) = min ⁡ { V ⁢ ( s ) ∣ x ¯ ⁢ ( s ) = x } , x ∈ I ¯ .

Remark 3.1.

The length of J is bounded by a bound of TV ⁢ ( u ε ) plus the length of I.

$Figure 2 The visual example of Theorem 3.1. The sequence of functions u ε {u^{\varepsilon}} is unfolded to U ε {U^{\varepsilon}} (the upper right figure). When U ε {U^{\varepsilon}} converges uniformly to V, the corresponding limit of u ε {u^{\varepsilon}} can be no longer captured as a single-valued function but is possibly multi-valued. The red part of the graph of V (the lower right image), however, corresponds to the multi-valued part (the red part in the lower left image), and its maximum and minimum coincide with the upper and lower relaxed limit of u ε {u^{\varepsilon}} , respectively.$

Figure 2

The visual example of Theorem 3.1. The sequence of functions u ε is unfolded to U ε (the upper right figure). When U ε converges uniformly to V, the corresponding limit of u ε can be no longer captured as a single-valued function but is possibly multi-valued. The red part of the graph of V (the lower right image), however, corresponds to the multi-valued part (the red part in the lower left image), and its maximum and minimum coincide with the upper and lower relaxed limit of u ε , respectively.

Proof.

Since the proof is symmetric, we only give a proof for lim ⁢ sup ∗ . Let J x = { s ∈ J ∣ x ¯ ⁢ ( s ) = x } . We take s * ∈ J x such that V ⁢ ( s * ) = max J x ⁡ V . Since V is the limit of U ε , we have

V ⁢ ( s * ) = lim k → ∞ ⁡ u ε k ⁢ ( x ε k ⁢ ( s * ) ) ≤ ( lim ⁢ sup ∗ k → ∞ ⁡ u ε k ) ⁢ ( x ) .

To prove the converse inequality, we set J x σ = { s ∈ J ∣ | x ¯ ⁢ ( s ) - x | ≤ σ } . Since x ε k converges to x ¯ uniformly in J, for sufficiently large k, say k > k 0 ⁢ ( σ ) , x ε k ⁢ ( J x 2 ⁢ σ ) ⊃ { y ∈ I ∣ | y - x | ≤ σ } ; here k 0 ⁢ ( σ ) can be taken so that k 0 ⁢ ( σ ) → ∞ as σ → ∞ and k 0 ⁢ ( σ ) > 1 σ . We thus observe that

sup | y - x | ≤ σ ⁡ u ε k ⁢ ( y ) ≤ sup ⁡ { u ε k ⁢ ( y ) ∣ y ∈ x ε k ⁢ ( J x 2 ⁢ σ ) } = sup ⁡ { U ε k ⁢ ( s ) ∣ s ∈ J x 2 ⁢ σ }

for k > k 0 ⁢ ( σ ) . Sending σ → 0 , we observe that

lim σ ↓ 0 ⁡ sup | y - x | ≤ σ k > k 0 ⁢ ( σ ) ⁡ u ε k ⁢ ( y ) ≤ max s ∈ J x ⁡ V ⁢ ( s ) .

The left-hand side agrees with lim ⁢ sup ∗ k → ∞ ⁡ u ε k since

{ ( y , k ) | | y - x | < 1 k 0 ⁢ ( σ ) , k > k 0 ( σ ) } ⊂ { ( y , k ) | | y - x | < σ , k > k 0 ( σ ) } ⊂ { ( y , k ) | | y - x | < σ , k > 1 σ } .

We thus conclude that

( lim ⁢ sup ∗ k → ∞ ⁡ u ε k ) ⁢ ( x ) ≤ max ⁡ { V ⁢ ( s ) ∣ s ∈ J x } .

The proof is now complete. ∎

We next prove the inequality connecting the total variation and the relaxed limit in terms of the unfolding (Figure 3).

$Figure 3 The visual explanation of Theorem 3.2. If the graph of { u ε k } {\{u^{\varepsilon_{k}}\}} converges to the graph as the top, its unfolding converges to V, whose graph is like the bottom. Then J x i = { s ∈ J ∣ x ¯ ⁢ ( s ) = x i } {J_{x_{i}}=\{s\in J\mid\overline{x}(s)=x_{i}\}} can be decomposed as the union of { J x i , j } j = 1 , 2 , … {\{J_{x_{i},j}\}_{j=1,2,\dots}} by labelling the disjoint intervals where V does not vanish.$

Figure 3

The visual explanation of Theorem 3.2. If the graph of { u ε k } converges to the graph as the top, its unfolding converges to V, whose graph is like the bottom. Then J x i = { s ∈ J ∣ x ¯ ⁢ ( s ) = x i } can be decomposed as the union of { J x i , j } j = 1 , 2 , … by labelling the disjoint intervals where V does not vanish.

Theorem 3.2.

Assume the same hypothesis of Theorem 3.1. Then the set Σ of points x where

lim ⁢ sup ∗ k → ∞ ⁡ u ε k ⁢ ( x ) > lim ⁢ inf ∗ k → ∞ ⁡ u ε k ⁢ ( x )

has at most countable cardinality. Assume furthermore that, outside Σ, the limit must be zero and

lim inf k → ∞ ⁡ u ε k * ⁢ ( x ) = 0

for all x ∈ I . Then

lim inf k → ∞ ⁡ TV ⁢ ( u ε k ) ≥ ∑ x ∈ Σ 2 ⁢ χ ⁢ ( x ) ⁢ lim ⁢ sup ∗ k → ∞ ⁡ u ε k ⁢ ( x ) ,

where χ ⁢ ( x ) = 1 for x ∈ I and χ ⁢ ( x ) = 1 2 for x ∈ ∂ ⁡ I .

Proof.

If # ⁢ Σ is uncountable, then there is an infinite number of intervals J x i such that max ⁡ V - min ⁡ V > c 0 with some c 0 > 0 . This is impossible by Theorem 3.1 since TV ⁢ ( V ) < ∞ . Thus Σ is at most a countable set.

We write Σ = { x i } i = 1 ∞ and J i = J x i . We set ρ i = max J i ⁡ V . The cases are divided into two cases depending on whether or not J i ¯ contains a boundary point of J. The total variation is estimated as

TV ⁢ ( V ) ≥ ∑ i = 1 ∞ 2 ⁢ χ i ⁢ ρ i ,

where χ i = χ ⁢ ( x i ) . Thus Theorem 3.1 and Lemma 3.2 yield the desired result. ∎

We decompose J i by

J i : = cl ( ⋃ j = 1 ∞ J x i , j ) ,

where V > 0 in an open interval J x i , j and V = 0 on ∂ ⁡ J x i , j . The union can be finite.

We introduce χ ¯ on subsets of J x i which reflects behavior finer than that of χ on the boundary. We set, for x = x i ∈ Σ ,

χ ¯ ⁢ ( J x i , j ) = { 1 if ⁢ J x i , j ∩ ∂ ⁡ I = ∅ , 1 2 otherwise .

By definition,

TV ⁢ ( V ) ≥ ∑ i = 1 ∞ ∑ j = 1 ∞ 2 ⁢ χ ¯ ⁢ ( J x i , j ) ⁢ ρ x i , j with ⁢ ρ x i , j = max J x i , j ⁡ V .

Similarly, to obtain Theorem 3.2, we are able to prove a stronger result.

Theorem 3.3.

Assume the same hypothesis of Theorem 3.1. Then

lim inf k → ∞ ⁡ TV ⁢ ( u ε k ) ≥ ∑ x ∈ Σ ∑ j = 1 ∞ 2 ⁢ χ ¯ ⁢ ( J x , j ) ⁢ ρ x , j ,

where ρ x , j and χ ¯ are determined from V as above.

4 Proof of convergence of functional and compactness

We shall prove the characterization of the Gamma limit of the single-well Modica–Mortola functional by the results of the previous section on unfoldings.

Proof of Theorem 2.1.

(i) (liminf inequality) We discuss the case M = I ¯ . We may assume I = ( 0 , r ) . Assume that v ε → 𝑔 Ξ ∈ ℬ with v ε ∈ H 1 ⁢ ( M ) . By the Modica–Mortola inequality, which follows from α 2 + β 2 ≥ 2 ⁢ α ⁢ β for numbers, we have

E sMM ε ⁢ ( v ε ) ≥ ∫ M | d ⁢ v ε d ⁢ x | ⁢ F ⁢ ( v ε ) ⁢ d ⁢ x = ∫ M | G ⁢ ( v ε ) x | ⁢ d ⁢ x .

The right-hand is equal to TV ⁢ ( u ε ) if one sets u ε = G ⁢ ( v ε ) ≥ 0 . We may assume that E sMM ε ⁢ ( v ε ) is bounded for ε ∈ ( 0 , 1 ) so that TV ⁢ ( u ε ) is bounded for ε ∈ ( 0 , 1 ) and that ∫ M F ⁢ ( v ε ) ⁢ d ⁢ x → 0 as ε → 0 . By (F2), the latter convergence implies that v ε → 1 in measure. By taking a subsequence, we see that v ε ′ → 1 a.e. so that u ε ′ → 0 a.e. This implies that

lim ⁢ inf ∗ ε → 0 ⁡ u ε ⁢ ( x ) = 0 for all ⁢ x ∈ M .

By taking a subsequence, we may assume that the inverse function x ε k of the arc-length parameter of u ε k converges to some x ¯ . Applying Theorem 3.3, we see that

lim inf k → ∞ ⁡ E sMM ε k ⁢ ( v ε k ) ≥ ∑ x ∈ Σ ∑ j = 1 ∞ 2 ⁢ χ ¯ ⁢ ( J x , j ) ⁢ ρ x , j ,

where Σ is the set where lim ⁢ sup ∗ ⁡ u ε k ⁢ ( x ) > 0 and ρ x , j is determined by limit V of u ε . Note that Σ is at most countable. If v ε → 𝑔 Ξ , then u ε → 𝑔 Θ and Θ ⁢ ( x ) = { 0 } if x ∉ Σ and

Θ ⁢ ( x i ) = [ 0 , max ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) ] for ⁢ x i ∈ Σ

by Lemma 2.3. By Theorem 3.1, at least one of ρ x i , j should equal max ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) . However, if ξ i - < 1 < ξ i + , then v ε - 1 is sign-changing near x i . In this case, one of ρ x i , j must be equal to min ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) . Thus we observe that

∑ j = 1 ∞ χ ¯ ⁢ ( J x i , j ) ⁢ ρ x i , j ≥ G ⁢ ( ξ i + ) + G ⁢ ( ξ i - ) , x i ∈ Σ ∩ I .

If x ∈ Σ ∩ ∂ ⁡ I , one has to be more careful. For x i ∈ Σ ∩ ∂ ⁡ I , we see that

(4.1) ∑ j = 1 ∞ χ ¯ ⁢ ( J x i , j ) ⁢ ρ x , j ≥ min ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) + 1 2 ⁢ max ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) .

Indeed, without loss of generality, we assume that G ⁢ ( ξ i - ) < G ⁢ ( ξ i + ) . When G ⁢ ( ξ i - ) = 0 , (4.1) is rather easy to prove since the right-hand side is equal to 1 2 ⁢ G ⁢ ( ξ i + ) , and then we may assume that G ⁢ ( ξ i - ) > 0 . Then there are at least two indices, denoted by j = 1 , 2 , such that

χ ¯ ⁢ ( J x i , 1 ) = 1 2 , χ ¯ ⁢ ( J x i , 2 ) = 1 and { ρ x i , 1 , ρ x i , 2 } = { G ⁢ ( ξ i - ) , G ⁢ ( ξ i + ) } .

The left-hand side is dominated from below by

χ ¯ ⁢ ( J x i , 1 ) ⁢ ρ x i , 1 + χ ¯ ⁢ ( J x i , 2 ) ⁢ ρ x i , 2 = 1 2 ⁢ ρ x i , 1 + ρ x i , 2 .

The right-hand side is minimized in the case that ρ x i , 1 = G ⁢ ( ξ i + ) , ρ x i , 2 = G ⁢ ( ξ i - ) . We thus obtain inequality (4.1).

We now conclude that

lim inf k → ∞ ⁡ E sMM ε k ⁢ ( v ε k ) ≥ ∑ i = 1 ∞ 2 ⁢ ( 1 - κ i ) ⁢ ( G ⁢ ( ξ i + ) + G ⁢ ( ξ i - ) ) + κ i ⁢ [ 2 ⁢ min ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) + max ⁡ ( G ⁢ ( ξ i + ) , G ⁢ ( ξ i - ) ) ] = E sMM 0 ⁢ ( Ξ , I ¯ ) ,

which is the desired liminf inequality for b = 0 . Since v ε → 𝑔 Ξ , we see that lim inf ⁡ b ⁢ ( v ε ⁢ ( a ) ) 2 ≥ b ⁢ ( min ⁡ Ξ ⁢ ( a ) ) 2 . Thus the desired liminf inequality follows for b > 0 . The case M = 𝐓 is easier since there is no boundary point.

(ii) (limsup inequality) This follows from explicit construction of function w ε as for the standard double-well Modica–Mortola functional. For ξ ≠ 1 ( ξ ∈ 𝐑 ) and x > 0 , let v ⁢ ( x , ξ ) be a function determined by

| ∫ ξ v ( 1 F ⁢ ( ρ ) ) ⁢ d ⁢ ρ | = x .

This equation is uniquely solvable by (F1) for all x ∈ [ 0 , x * ) with

x * : = | ∫ ξ 1 ( 1 F ⁢ ( ρ ) ) d ρ | .

Note that v solves the initial value problem

{ d ⁢ v d ⁢ x = F ⁢ ( v ) , x ∈ ( 0 , x * ) , v ⁢ ( 0 , ξ ) = ξ ,

although this problem may admit many solutions. We also note that v is monotone and that lim x → x * ⁡ v ⁢ ( x , ξ ) = 1 including the case x * = ∞ . We consider the even extension of v, still denoted by v, that is, v ⁢ ( x , ξ ) = v ⁢ ( - x , ξ ) for x ∈ ( - x * , 0 ] . We next translate and rescale v. Let v ε be of the form

v ε ( x , z , ξ ) : = v ( x - z ε , ξ ) , x ∈ 𝐑 .

By the equality case of the Modica–Mortola functional, we see that

E sMM ε ⁢ ( v ε ) = ∫ M | d ⁢ v ε d ⁢ x | ⁢ F ⁢ ( v ε ) ⁢ d ⁢ x = ∫ M | G ⁢ ( v ε ) x | ⁢ d ⁢ x .

The right-hand side is estimated from above by 2 ⁢ ( G ⁢ ( ξ ) - G ⁢ ( 1 ) ) = 2 ⁢ G ⁢ ( ξ ) , and if z is a boundary point of M, we may replace 2 ⁢ G ⁢ ( ξ ) by G ⁢ ( ξ ) .

In order to explain the main idea of the proof, we first study the case when all ξ i + = 1 although, logically, we need not distinguish this case from the general case. If all ξ i + = 1 , then it is easy to construct the desired w ε by setting w ε ⁢ ( x ) = min x i ∈ Σ ⁡ v ⁢ ( x , x i , ξ i - ) . Indeed, we still have

E sMM ε ⁢ ( w ε ) = ∫ M | G ⁢ ( w ε ) x | ⁢ d ⁢ x ,

and evidently, this total variation is dominated from above by ∑ i = 1 ∞ 2 ⁢ χ i ⁢ G ⁢ ( ξ i - ) . (The first identity can be proved by approximating w ε by a minimum of finitely many w ε .) We thus observe that E sMM ε ⁢ ( w ε ) ≤ E sMM 0 ⁢ ( Ξ , 𝐓 ) for all ε > 0 . The graph convergence w ε → 𝑔 Ξ is rather clear since w ε ⁢ ( x , 0 , ξ ) → 𝑔 Ξ 0 on any bounded closed interval as ε → 0 , where

Ξ 0 ⁢ ( x ) = { 1 , x ≠ 0 , [ ξ , 1 ] , x = 0 .

The proof for general ξ i ± is more involved. For δ > 0 , we cut off v by setting as follows: for ξ < 1 ,

v δ ⁢ ( x , ξ ) = { v ⁢ ( x , ξ ) if ⁢ v ⁢ ( x ) ≤ 1 - δ ⁢ β , β = | ξ - 1 | , ( | x | + c ) ∧ 1 if ⁢ v ⁢ ( x ) ≥ 1 - δ ⁢ β ,

and for ξ > 1 ,

v δ ⁢ ( x , ξ ) = { v ⁢ ( x , ξ ) if ⁢ v ⁢ ( x ) ≥ 1 + δ ⁢ β , β = | ξ - 1 | , ( - | x | + c ′ ) ∨ 1 if ⁢ v ⁢ ( x ) ≤ 1 + δ ⁢ β ,

where constants c , c ′ are taken so that v δ is (Lipschitz) continuous (Figure 4). We rescale and translate this v δ and set

v ε δ ( x , z , ξ ) : = v δ ( x - z ε , ξ ) x ∈ 𝐑 .

We consider the case when ξ < 1 . Since

d ⁢ v ε δ d ⁢ x = 1 ε ⁢ F ⁢ ( v ε δ ) for ⁢ v ε δ ≤ 1 - δ ,

we see that, for z ∈ M ,

(4.2) E sMM ε ⁢ ( v ε δ ) = ε 2 ⁢ ∫ M | d ⁢ v ε δ d ⁢ x | 2 ⁢ d ⁢ x + 1 2 ⁢ ε ⁢ ∫ M F ⁢ ( v ε δ ) ⁢ d ⁢ x ≤ ∫ v ε ⁢ ( x ) < 1 - δ | G ⁢ ( v ε ) x | ⁢ d ⁢ x + 2 ⁢ { ε 2 ⁢ ( 1 ε ) 2 ⋅ δ ⁢ β ⁢ ε + 1 2 ⁢ ε ⁢ max ⁡ { F ⁢ ( ρ ) ∣ 1 - δ ≤ ρ ≤ 1 } ⁢ δ ⁢ β ⁢ ε } ≤ 2 ⁢ G ⁢ ( ξ ) + 2 ⁢ β ⁢ δ

for sufficiently small δ, say δ < δ F , since F ⁢ ( ρ ) → 0 as ρ → 1 . This δ F depends only on F. A similar argument for ξ > 1 yields the same estimate (4.2).

$Figure 4 (Left) The construction of v δ ⁢ ( ⋅ , ξ ) {v^{\delta}(\,{\cdot}\,,\xi)} . In order to ensure the finiteness of the support, we take the cutoff by affine functions. (Right) The construction of v ε , i δ {v^{\delta}_{\varepsilon,{i}}} for general ξ i ± {\xi_{i}^{\pm}} . It is constructed by combining v ε δ ⁢ ( ⋅ , x i , ξ i - ) {v^{\delta}_{\varepsilon}(\,{\cdot}\,,x_{i},\xi_{i}^{-})} and v ε δ ⁢ ( ⋅ , x i , ξ i + ) {v^{\delta}_{\varepsilon}(\,{\cdot}\,,x_{i},\xi_{i}^{+})} with shift so that their supports touch at their endpoints.$

Figure 4

(Left) The construction of v δ ⁢ ( ⋅ , ξ ) . In order to ensure the finiteness of the support, we take the cutoff by affine functions. (Right) The construction of v ε , i δ for general ξ i ± . It is constructed by combining v ε δ ⁢ ( ⋅ , x i , ξ i - ) and v ε δ ⁢ ( ⋅ , x i , ξ i + ) with shift so that their supports touch at their endpoints.

We first consider the case M = 𝐓 . Let η = η ⁢ ( ε , δ , ξ ) be a number such that supp ⁡ ( v ε δ - 1 ) = [ z - η , z + η ] . For x i ∈ Σ , we set

v ε , i δ ( x ) : = { v ε δ ⁢ ( x , x i , ξ i - ) if ⁢ x ∈ ( x i - η i - , x i + η i - ) , v ε δ ⁢ ( x , x i + η i + + η i - , ξ i + ) otherwise ,

where η i - = η ⁢ ( ε , δ , η i - ) and η i + = η ⁢ ( ε , δ , η i + ) (Figure 4). This function is (Lipschitz) continuous and is strictly monotone from x i - η i + - η i - to x i . For v ε , i δ ⁢ ( x ) by (4.2), we see that

(4.3) E sMM ε ⁢ ( v ε , i δ ) ≤ 2 ⁢ ( G ⁢ ( ξ i + ) + G ⁢ ( ξ i - ) ) + 4 ⁢ β i ⁢ δ ,

where β i = max ⁡ ( | ξ i + - 1 | , | ξ i - - 1 | ) .

Our goal is to construct w ε such that w ε → 𝑔 Ξ , and for each μ > 0 , there is ε μ > 0 such that if ε < ε μ , then

(4.4) E sMM ε ⁢ ( w ε ) ≤ E sMM 0 ⁢ ( Ξ , 𝐓 ) + μ .

We order x i ∈ Σ so that β i is decreasing. We note that { β i } must converge to zero because

∑ i = 1 ∞ ( G ⁢ ( ξ i + ) + G ⁢ ( ξ i - ) ) < ∞ .

For each v ε , i δ , we set δ = δ i = δ i ⁢ ( μ ) such that ∑ i = 1 ∞ 4 ⁢ β i ⁢ δ i < μ ; this is, of course, possible for example by taking δ i = 2 - i - 2 ⁢ μ . Let j ⁢ ( μ , ε ) > 0 be the maximum number such that the support of { v ε , i δ i - 1 } i = 1 j ⁢ ( μ , ε ) is mutually disjoint. We set

w ε μ ( x ) : = 1 + ∑ i = 1 j ⁢ ( μ , ε ) ( v ε , i δ i ⁢ ( μ ) ( x ) - 1 )

and observe by (4.3) that

E sMM ε ⁢ ( w ε μ ) ≤ ∑ i = 1 j ⁢ ( μ , ε ) 2 ⁢ ( G ⁢ ( ξ i + ) + G ⁢ ( ξ i - ) ) + ∑ i = 1 j ⁢ ( μ , ε ) 4 ⁢ β i ⁢ δ i ≤ E sMM 0 ⁢ ( Ξ , 𝐓 ) + μ for all ⁢ ε > 0 .

Since j ⁢ ( μ , ε ) → ∞ as ε → 0 , we see that w ε μ → 𝑔 Ξ as ε → 0 for each μ > 0 . The desired w ε is obtained through a diagonal argument. Indeed, for a given ν > 0 , we take ε = ε ⁢ ( ν , μ ) such that d H ⁢ ( Γ w ε μ , Ξ ) < ν for ε ∈ ( 0 , ε ⁢ ( ν , μ ) ) . We may assume that ε ⁢ ( ν , μ ) is monotone in ν and μ, that is, ε ⁢ ( ν 2 , μ 2 ) ≤ ε ⁢ ( ν 1 , μ 1 ) if ν 1 ≥ ν 2 and μ 1 ≥ μ 2 . We then set

w ε : = w ε μ ℓ for ε ∈ [ ε ( ν ℓ + 1 , μ ℓ + 1 ) , ε ( ν ℓ , μ ℓ ) ) ,

where ν ℓ , μ ℓ ↓ 0 as ℓ → ∞ . We now observe that w ε → 𝑔 Ξ , and by (4.4), the desired estimate (4.4) holds for ε μ = ε ⁢ ( ν ℓ , μ ℓ ) for μ ℓ < μ .

We thus proved the limsup inequality for E sMM ε for M = 𝐓 . If b > 0 , we may assume that ξ i - = min ⁡ Ξ ⁢ ( a ) < 1 . It is easy to see that w ε ⁢ ( a ) = ξ - i for all ε > 0 by construction. Thus the limsup inequality for E sMM ε , b for b > 0 is obtained.

It remains to handle the case for M = I ¯ . Assume that x 1 ∈ Σ is the right end point of I ¯ . We first consider the case when G ⁢ ( ξ 1 - ) ≤ G ⁢ ( ξ 1 + ) . Instead of (4.2), we have

E sMM ε ⁢ ( v ε , 1 δ ) ≤ 2 ⁢ G ⁢ ( ξ 1 + ) + G ⁢ ( ξ 1 - ) + 3 ⁢ β 1 ⁢ δ .

If there is no other point of Σ on ∂ ⁡ I , arguing in the same way, we obtain the desired limsup inequality by the same construction of w ε . If G ⁢ ( ξ 1 - ) > G ⁢ ( ξ 1 + ) , then we modify the definition of v ε , 1 δ by

v ¯ ε , 1 δ ( x ) : = { v ε δ ⁢ ( x , x 1 , ξ i + ) if ⁢ x ∈ ( x 1 - η i , x 1 ] , v ε δ ⁢ ( x , x 1 + η 1 + + η 1 - , ξ i - ) otherwise .

The remaining argument is similar. A symmetric argument yields the limsup inequality in the case that Σ has the left end point of I ¯ . ∎

We next prove the compactness.

Proof of Theorem 2.2.

As in the proof of Theorem 2.1 (i), we see that

sup j ⁡ ∫ M | G ⁢ ( v ε j ) x | ⁢ d ⁢ x < ∞ .

By (F2), we see that

G ⁢ ( v ) ≤ F ⁢ ( v ) ⁢ | v - 1 | ≤ F ⁢ ( v ) 2 + ( v - 1 ) 2 2 , v ∈ 𝐑 .

By (F2 ′ ), we see F ⁢ ( v ) ≥ c 0 ⁢ ( v - 1 ) 2 - c 1 ′ , so G ⁢ ( v ) ≤ C ′ ⁢ F ⁢ ( v ) for v such that | v - 1 | is sufficiently large, v with some content C ′ > 0 . Since

1 ε j ⁢ ∫ M F ⁢ ( v ε j ) ⁢ d ⁢ x

is bounded, so is ∫ M G ⁢ ( v ε j ) ⁢ d ⁢ x . We set u ε j = G ⁢ ( v ε j ) and observe that TV ⁢ ( u ε j ) is bounded and ∥ u ε j ∥ L 1 is bounded. Since ∥ f - f av ∥ ∞ ≤ ∥ f x ∥ L 1 , where f av is the average of f over I, it follows that ∥ f ∥ ∞ ≤ ∥ f x ∥ L 1 + ∥ f x ∥ L 1 / | I | . This interpolation inequality yields a bound for ∥ u ε j ∥ ∞ . Applying Lemma 3.2, there is a subsequence U ε k converging to V uniformly, where U ε k is the unfolding of u ε k . Since we may assume that x ε k , the inverse of arc-length of u ε k , converges to x ¯ uniformly in M by taking a subsequence, applying Theorem 3.2 yields that

lim ⁢ sup ∗ k → ∞ ⁡ u ε k ⁢ ( x ) > lim ⁢ inf ∗ k → ∞ ⁡ u ε k ⁢ ( x ) , x ∈ Σ ,

for at most countably many x ∈ Σ . Since v ε k → 1 a.e. by taking a subsequence, we see that lim ⁢ inf ∗ ⁡ u ε k ⁢ ( x ) = 0 for all x ∈ M . This implies that v ε k satisfies all assumptions on a sequence { g j } of the compactness lemma (Lemma 2.4) with S = Σ . Then, by Lemma 2.4, we conclude that v ε k → 𝑔 Ξ with some Ξ ∈ 𝒜 0 . ∎

5 Singular limit of the Kobayashi–Warren–Carter energy

In this section, we shall study the Gamma limit of the Kobayashi–Warren–Carter energy.

We first derive an inequality for lower semicontinuity. Assume that M is either I ¯ or 𝐓 . Assume that

v ε → 𝑔 Ξ , u ε → u in L 1 ⁢ ( M ̊ ) as ε → 0 , where v ε ∈ C ⁢ ( M ) , u ε ∈ L 1 .

For the limits, we assume that

Ξ ∈ 𝒜 0 , that is, there is a countable set Σ = { x i } i = 1 ∞ ⊂ M such that Ξ ⁢ ( x ) = { 1 } for x ∉ Σ and

Σ ⁢ ( x i ) = [ ξ i - , ξ i + ] ∋ 1 with ⁢ ξ i - < ξ i + , for ⁢ x i ∈ Σ .

Moreover, ∑ i = 1 ∞ G ⁢ ( ξ i - ) < ∞ .
u ∈ BV ⁢ ( M ̊ ∖ Σ 0 ) , where Σ 0 = { x i ∈ Σ ∣ ξ i - = 0 } . (Since ∑ i = 1 ∞ G ⁢ ( ξ i - ) < ∞ , the set Σ 0 is a finite set.)

We define a weighted total variation

∫ M ̊ v ε 2 | d ⁢ u ε d ⁢ x | : = sup { ∫ M ̊ φ x u ε d x | | φ ( x ) | ≤ v ε 2 ( x ) , φ ∈ C c 1 ( M ̊ ) } ,

where C c 1 ⁢ ( M ) is the space of all C 1 functions in M ̊ with compact support in M ̊ . For u ∈ BV ⁢ ( M ̊ ∖ Σ 0 ) , let J u denote the set of jump discontinuities of u. In other words,

J u = { x ∈ M ̊ ∖ Σ 0 ∣ d ⁢ ( x ) = | u ⁢ ( x + 0 ) - u ⁢ ( x - 0 ) | > 0 } ,

where u ⁢ ( x ± 0 ) is the trace from right ( + ) and left ( - ). It is at most a countable set.

Lemma 5.1.

Assume conditions (C1)–(C3). Then

∫ M ̊ ∖ ( J u ∩ Σ ) | u x | + ∑ x ∈ Σ ′ d i ⁢ | ξ i - | 2 ≤ lim inf ε → 0 ⁡ ∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | ,

where d i = d ⁢ ( x i ) ≥ 0 , Σ ′ = Σ ∖ Σ 0 .

Proof.

It suffices to prove that, for any δ ∈ ( 0 , 1 ) ,

( 1 - δ ) 2 ⁢ ∫ M ̊ ∖ ( J u ∩ Σ 1 - δ ) | u x | + ∑ x i ∈ Σ 1 - δ ′ d i ⁢ | ξ i - | 2 ≤ lim inf ε → 0 ⁡ ∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | ,

where Σ 1 - δ = { x i ∈ Σ ∣ ξ i - < 1 - δ } , Σ 1 - δ ′ = Σ 1 - δ ∖ Σ 0 . By this notation, Σ 1 = Σ . Note that the set Σ 1 - δ is a finite set for δ > 0 since ∑ i = 1 ∞ G ⁢ ( ξ i - ) < ∞ .

Since Σ 0 is a finite set and

∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | ≥ ∫ M ̊ ∖ Σ 0 v ε 2 ⁢ | d ⁢ u ε d ⁢ x | ,

it suffices to prove that, for each interval { M j } j = 1 m , which is a connected component of M ̊ ∖ Σ 0 , the inequality

( 1 - δ ) 2 ⁢ ∫ M j ∖ ( J u ∩ Σ 1 - δ ) | u x | + ∑ x i ∈ Σ 1 - δ ′ x i ∈ M j d i ⁢ | ξ i - | 2 ≤ lim inf ε → 0 ⁡ ∫ M j v ε 2 ⁢ | d ⁢ u ε d ⁢ x | .

Thus we may assume that Σ 0 = ∅ .

We consider a δ 1 -open neighborhood of Σ 1 - δ , that is, X δ 1 = { x ∈ M ̊ ∣ dist ⁡ ( x , Σ 1 - δ ) < δ 1 } , and observe that

We may assume that X δ 1 consists of disjoint interval B δ 1 ⁢ ( x i ) = { x ∈ M ̊ ∣ | x - x i | < δ 1 } , x i ∈ Σ 1 - δ , by taking δ 1 small. Since v ε → 𝑔 Ξ , for sufficiently small ε, we observe that

v ε ≥ 1 - δ - δ 1 in ⁢ M ̊ ∖ X ¯ δ 1 ,

v ε ≥ ξ i - - δ 1 in ⁢ B δ 1 ⁢ ( x i ) , x i ∈ Σ 1 - δ .

We thus conclude that

lim inf ε → 0 ⁡ ∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | ≥ lim inf ε → 0 ⁡ { ( 1 - δ - δ 1 ) 2 ⁢ ∫ M ̊ ∖ X ¯ δ 1 | d ⁢ u ε d ⁢ x | + ∑ x i ∈ Σ 1 - δ ( ξ i - - δ 1 ) 2 ⁢ ∫ B δ 1 ⁢ ( x i ) | d ⁢ u ε d ⁢ x | } ≥ ( 1 - δ - δ 1 ) 2 ⁢ ∫ M ̊ ∖ X ¯ δ 1 | u x | + ∑ x i ∈ Σ 1 - δ ( ξ i - - δ 1 ) 2 ⁢ ∫ B δ 1 ⁢ ( x i ) | u x |

by lower semicontinuity of TV ⁢ ( ⋅ ) with respect to L 1 -convergence. The second term of the right-hand side is estimated from below by

∑ x i ∈ Σ 1 - δ ( ξ i - - δ 1 ) 2 ⁢ d i .

Note that δ 2 < δ 1 implies M ̊ ∖ X ¯ δ 1 ⊂ M ̊ ∖ X ¯ δ 2 . Sending δ 1 → 0 yields

( 1 - δ ) 2 ⁢ ∫ M ̊ ∖ Σ 1 - δ | u x | + ∑ x i ∈ Σ 1 - δ d i ⁢ | ξ i - | 2 ≤ lim inf ε → 0 ⁡ ∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | .

Replacement of M ̊ ∖ Σ 1 - δ by M ̊ ∖ ( J u ∩ Σ 1 - δ ) is rather trivial because, outside J u , the set Σ 1 - δ has measure zero with respect to the measure | u x | . ∎

We are now in position to give a proof for the Gamma limit of the Kobayashi–Warren–Carter energy.

Proof of Theorem 2.3.

(i) (liminf inequality) We may assume that lim inf ε → 0 ⁡ E KWC ε ⁢ ( u ε , v ε ) < ∞ . By Theorem 2.1 (i), we see that the limit Ξ satisfies (C2). Let Ω be an open set such that Ω ¯ is compact and contained in M ̊ ∖ Σ 0 . Assume that

∫ M ̊ v ε 2 ⁢ | d ⁢ u ε d ⁢ x |

is bounded. Since c : = min x ∈ Σ ′ Ξ ( x ) > 0 and v ε → 𝑔 Ξ , we set v ε ≥ c 2 > 0 on Ω ¯ for sufficiently small ε > 0 . Thus ∫ Ω | d ⁢ u ε d ⁢ x | is bounded. This implies that the limit u ∈ BV ⁢ ( Ω ) . We now conclude that u satisfies (C3).

Applying Theorem 2.1 for E sMM ε and Lemma 5.1 for ∫ v ε 2 ⁢ | d ⁢ u ε d ⁢ x | , we see that

σ ⁢ ∫ M ̊ ∖ ( J u ∩ Σ ) | u x | + σ ⁢ ∑ x ∈ Σ ′ d i ⁢ | ξ i - | 2 + E sMM 0 ⁢ ( Ξ , M ) ≤ lim inf ε → 0 ⁡ E KWC ε ⁢ ( u ε , v ε ) .

The second term in the left-hand side equals σ ⁢ ∑ x ∈ Σ d i ⁢ | ξ i - | 2 since ξ i - = 0 on Σ 0 . Thus the left-hand side equals E KWC 0 ⁢ ( u , Ξ , M ) . The proof of the liminf inequality is now complete.

(ii) (limsup inequality) We take u ε = u . We notice that Theorem 2.1 extends to the case when E sMM 0 , b ⁢ ( Ξ , M ) , E sMM ε , b ⁢ ( v ) are replaced by

E sMM 0 , { b ℓ } ( Ξ , M ) : = E sMM 0 ( Ξ , M ) + ∑ ℓ = 1 ∞ b ℓ ( min Ξ ( a ℓ ) ) 2 ,

E sMM ε , { b ℓ } ( v ε ) : = E sMM ε ( v ε ) + ∑ ℓ = 1 ∞ b ℓ ( v ε ( a ℓ ) ) 2 ,

where we assume that ∑ ℓ = 1 ∞ b ℓ < ∞ with b ℓ ≥ 0 and a ℓ ∈ M ̊ for ℓ = 1 , 2 , … , m . Let { a ℓ } denote the jump discontinuity of u, that is, J u = { a ℓ } . Let b ℓ denote σ times the jump d ℓ = | u ⁢ ( a ℓ + 0 ) - u ⁢ ( a ℓ - 0 ) | , that is, b ℓ = σ ⁢ d ℓ . Note that ∑ b ℓ < ∞ . By Theorem 2.1 (ii) for E sMM ε , { b ℓ } , we see that there exist w ε → 𝑔 Ξ such that

(5.1) E sMM 0 , { b ℓ } ⁢ ( Ξ , M ) = lim ε → 0 ⁡ E sMM ε , { b ℓ } ⁢ ( w ε ) .

We notice that

E KWC ε ⁢ ( u , w ε ) = σ ⁢ ∫ M ̊ w ε 2 ⁢ | u x | + E sMM ε ⁢ ( w ε ) = σ ⁢ ∫ M ̊ ∖ Σ w ε 2 ⁢ | u x | + ∑ ℓ = 1 ∞ b ℓ ⁢ w ε ⁢ ( a ℓ ) 2 + E sMM ε ⁢ ( w ε ) = σ ⁢ ∫ M ̊ ∖ Σ w ε 2 ⁢ | u x | + E sMM ε , { b ℓ } ⁢ ( w ε ) .

By construction, w ε is bounded and w ε → 1 almost everywhere with respect to the measure | u x | outside Σ, i.e. | u x | ⌊ M ̊ ∖ Σ . Since w ε 2 - 1 tends to zero for all x outside Σ and it is bounded, the first term in the right-hand side converges to σ ⁢ ∫ M ∖ Σ | u x | by a bounded convergence theorem. Convergence (5.1) yields the desired result. ∎

Communicated by Irene Fonseca

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 26220702

Award Identifier / Grant number: 19H00639

Award Identifier / Grant number: 18H05323

Award Identifier / Grant number: 17H01091

Award Identifier / Grant number: 16H03948

Funding statement: The work of the first author was partly supported by Japan Society for the Promotion of Science (JSPS) through the grants KAKENHI No. 26220702, No. 19H00639, No. 18H05323, No. 17H01091, No. 16H03948 and by Arithmer, Inc. through collaborative grant. The work of the second author was partly supported by the Leading Graduate Program “Frontiers of Mathematical Sciences and Physics”, JSPS.

Acknowledgements

The authors are grateful to Professor Ken Shirakawa for letting us know his recent results before publication as well as development of researches on gradient flows of Kobayashi–Warren–Carter type energies.

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Received: 2020-11-25

Revised: 2021-02-16

Accepted: 2021-04-20

Published Online: 2021-06-23

Published in Print: 2023-01-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/acv-2020-0113

Keywords for this article

Gamma convergence; Modica–Mortola functional; Kobayashi–Warren–Carter energy

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