Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters

Proposition A.1.

Let { v ε } ε > 0 ⊂ W 1 , 2 ⁢ ( I ) be such that 0 ≤ v ε ≤ 1 , v ε → 1 in L 1 ⁢ ( I ) and a.e., and

Then, for any 0 < η < 1 , there exists an open set H η ⊂ I such that I ∖ H η is a collection of finitely many points in I, and for every set T ⋐ H η we have T ⊂ B ε η for all sufficiently small ε > 0 , where

Proof.

Using Theorem 2.7, we obtain that there exists C := C ⁢ ( ε 0 , k , Ω ) > 0 such that

Hence, by the arguments from [4, pp. 1020–1021], we conclude the proof. ∎

Lemma A.2.

The constant c k is positive and

Proof.

The proof employs the arguments used in [15, Lemma 2.5]. Moreover, by solving the associated Euler–Lagrange equation, we have also

∎

Proposition A.3 (Γ- lim inf ).

Given u ∈ L 1 ⁢ ( I ) , let ω ∈ P ⁢ ( I ) satisfy (3.1), and let

Then T ω - ⁢ ( u ) ≥ T ω ⁢ ( u ) .

Proof.

Assume that M := T ω - ⁢ ( u ) < ∞ , and choose u ε and v ε that are admissible for T ω - ⁢ ( u ) such that

Since inf x ∈ I ⁡ ω ⁢ ( x ) ≥ 1 , we have

By Theorem 2.7, we have

and by [4] we get also

The proof would be complete provided we show the following inequalities:

and

Up to a (not relabeled) subsequence, we have u ε → u and v ε → 1 a.e. in I, with

(A.7)

By Proposition A.1, we deduce that, for a fixed η ∈ ( 1 2 , 1 ) , there exists a set H η such that, for every T ⋐ H η , it holds

Here we used [13, Theorem 6.3.7] in the first inequality. By taking the limit T ↗ H η on the left-hand side of (A.8) first, and then the limit η ↗ 1 on the right-hand side, we get (A.5).

We next show (A.6). Let t ∈ S u be given, and for simplicity assume that t = 0 and t ∈ S ω . By the same arguments as in [4, p. 1021], we can prove that there exist { t n 1 } n = 1 ∞ , { t n 2 } n = 1 ∞ , and { s n } n = 1 ∞ such that

and, up to a subsequence, also

We conclude, using Lemma A.2, that

and, since ω is positive,

(A.9)

Moreover, if t ∈ S u ∖ S ω , we may use the above arguments to infer that (A.9) holds also with ω - ⁢ ( 0 ) replaced by ω ⁢ ( 0 ) , since t = 0 ∉ S ω implies ω - ⁢ ( 0 ) = ω ⁢ ( 0 ) .

Finally, since S u ⊂ I ∖ H η , by (A.4) we have that S u is a finite collection of points, and we may repeat the above arguments for all t ∈ S u by partitioning I into disjoint intervals, each of which containing at most one single point of S u , to deduce (A.6). ∎

Theorem A.4 ([4, Theorem 3.3]).

Let ν ∈ S N - 1 be given, and assume that u ∈ W 1 , 2 ⁢ ( Ω ) . Then, for H N - 1 -a.e. x ∈ Ω ν , u x , ν belongs to W 1 , 2 ⁢ ( Ω x , ν ) and u x , ν ′ ⁢ ( t ) = 〈 ∇ ⁡ u ⁢ ( x + t ⁢ ν ) , ν 〉 .

Proposition A.5 ([14, Proposition 3.6]).

Let ν ∈ S N - 1 be a fixed direction, let Γ ⊂ R N be such that H N - 1 ⁢ ( Γ ) < ∞ , and let P ν : R N → Π ν be a projection operator, where, by (A.10), Π ν ⊂ R N is a hyperplane in R N - 1 . Then

and, for H N - 1 -a.e. x ∈ Π ν ,

Lemma A.6 ([14, Lemma 3.9]).

Let τ > 0 and η > 0 be given. Let u ∈ SBV ⁡ ( Ω ) and assume that H N - 1 ⁢ ( S u ) < ∞ . The following statements hold:

1. There exist a set S ⊂ S u with ℋ N - 1 ⁢ ( S u ∖ S ) < η , and a countable collection 𝒬 of mutually disjoint, open cubes centered on elements of S u , such that ⋃ Q ∈ 𝒬 Q ⊂ Ω , and ℋ N - 1 ⁢ ( S ∖ ⋃ Q ∈ 𝒬 Q ) = 0 .

2. For every Q ∈ 𝒬 , there exists a direction vector ν Q ∈ 𝒮 N - 1 such that ℋ 0 ⁢ ( S ∩ Q x , ν Q ) = 1 for ℋ N - 1 -a.e. x ∈ Q ∩ S .

3. S ∩ Q is contained in a Lipschitz ( N - 1 ) - graph Γ Q , with Lipschitz constant not exceeding τ.

Proof of Proposition 3.1, with ω satisfying (3.1).

Assume that M := MS ω - ⁡ ( u ) < ∞ . Let

be such that u ε → u in L 1 ⁢ ( Ω ) , v ε → 1 in L 1 ⁢ ( Ω ) , and lim ε → 0 ⁡ AT ω , ε k ⁡ ( u ε , v ε ) = MS ω - ⁡ ( u ) . Since inf x ∈ Ω ⁡ ω ⁢ ( x ) ≥ 1 , we have

and by [4] we deduce that

We show separately that

and

Assume A to be an open subset of Ω. Fix ν ∈ 𝒮 N - 1 , and define A x , ν , A x , ν 1 , and A ν as in (A.10). For K ∈ ℝ + , set u K := K ∧ u ∨ - K , K ∈ ℕ . We observe, by Fubini’s Theorem, Fatou’s Lemma, Theorem A.4, equation (A.5), and [4, Theorem 2.3], that

(A.15)

Taking the limit K → ∞ , and using the dominated convergence theorem, we have

Let ϕ n ⁢ ( x ) := | 〈 ∇ ⁡ u ⁢ ( x ) , ν n 〉 | 2 ⁢ ω for ℒ N -a.e. x ∈ Ω , where { ν n } n = 1 ∞ is a dense subset of 𝒮 N - 1 , and let

Then μ is positive, super-additive on any pair of open sets A and B with disjoint closures, and, by [6, Lemma 15.2] and (A.16), we conclude (A.13). Now we prove (A.14). By Fubini’s Theorem, Fatou’s Lemma, (A.12), (A.6), and using similar arguments to the ones in (A.15), we have

Next, given arbitrary τ > 0 and η > 0 , we choose a set S ⊂ S u and a collection 𝒬 of mutually disjoint cubes according to Lemma A.6 with respect to S u . Fix one such cube Q ν S ⁢ ( x 0 , r 0 ) ∈ 𝒬 . By Lemma A.6, we have, up to rigid motions,

In (A.17), set A = Q ν S ⁢ ( x 0 , r 0 ) and ν = ν S ⁢ ( x 0 ) . Using the same notation as in (A.10), we obtain

(A.18)

Next, considering that ω x , ν - ⁢ ( t ) = ω - ⁢ ( x + t ⁢ ν ) (see [3, Remark 3.109]), we have that

(A.19)

which, together with (A.17) and (A.18), yields

(A.20)

Finally, (A.14) follows by the arbitrariness of η and τ. ∎

Proposition A.7.

Let ω ∈ W ⁢ ( Ω ) and τ ∈ ( 0 , 1 4 ) be given. Then there exist a set S ⊂ S ω and a countable family of disjoint cubes F = { Q ν S ω ⁢ ( x n , r n ) } n = 1 ∞ , with r n < τ , such that the following assertions hold:

1. ℋ N - 1 ⁢ ( S ω ∖ S ) < τ and

   S ⊂ ⋃ n = 1 ∞ Q ν S ω ⁢ ( x n , r n ) ⊂ Ω .

2. dist ⁡ ( Q ν S ω ⁢ ( x n , r n ) , Q ν S ω ⁢ ( x n ′ , r n ′ ) ) > 0 for n ≠ n ′ .

3. S ∩ Q ν S ω ⁢ ( x n , r n ) ⊂ R τ / 2 , ν S ω ⁢ ( x n , r n ) .

4. It holds

   ( 1 + τ 2 ) - 1 ⁢ r n N - 1 ≤ ℋ N - 1 ⁢ ( S ∩ Q ν S ω ⁢ ( x n , r n ) ) ≤ ( 1 + τ 2 ) ⁢ r n N - 1 .

5. ∑ n = 1 ∞ r n N - 1 ≤ 4 ⁢ ℋ N - 1 ⁢ ( S ω ) .

6. For each n ∈ ℕ , there exist t n ∈ ( 2.5 ⁢ τ ⁢ r n , 3.5 ⁢ τ ⁢ r n ) and 0 < t x n , r n < t n , depending on τ, r n , and x n , such that

   T x n , ν S ω ⁢ ( - t n ± t x n , r n ) ∩ Q ν S ω ⁢ ( x n , r n ) ⊂ Q ν S ω - ⁢ ( x n , r n ) ∖ R τ / 2 , ν S ω ⁢ ( x n , r n )

   and

   (A.21) sup 0 < t ≤ t x n , r n ⁡ 1 | I ⁢ ( t n , t ) | ⁢ ∫ I ⁢ ( t n , t ) ∫ Q ν S ω ⁢ ( x n , r n ) ∩ T x n , ν S ω ⁢ ( - l ) ω - ⁢ ( x ) ⁢ 𝑑 ℋ N - 1 ⁢ 𝑑 l ≤ ∫ S ∩ Q ν S ω ⁢ ( x n , r n ) ω - ⁢ 𝑑 ℋ N - 1 + O ⁢ ( τ ) ⁢ r N - 1 ,

   where we recall I ⁢ ( t n , t ) := ( - t n - t , - t n + t ) .

Proof.

The proof uses similar arguments to the ones in [14, Proposition 4.4]. ∎

Proposition A.8.

Let u ∈ SBV 2 ⁡ ( Ω ) ∩ L ∞ ⁢ ( Ω ) be given satisfying H N - 1 ⁢ ( S u ¯ ) < + ∞ , and let ω ∈ W ⁢ ( Ω ) . Then there exists a sequence { ( u ε , v ε ) } ε > 0 ⊂ W 1 , 2 ⁢ ( Ω ) × W 1 , 2 ⁢ ( Ω ) such that

Proof.

Without loss of generality, we assume that MS ω ⁡ ( u ) < + ∞ , which implies ℋ N - 1 ⁢ ( S u ) < + ∞ .

Step 1. Assume

Fix τ ∈ ( 0 , 1 4 ) . Applying Proposition A.7 to ω, we obtain a set S τ , a collection ℱ τ = { Q ν S ω ⁢ ( x n , r n ) } n = 1 ∞ , and corresponding t n ∈ ( 2.5 ⁢ τ ⁢ r n , 3.5 ⁢ τ ⁢ r n ) and t x n , r n , for which (A.21) holds. Extract a finite collection

from ℱ τ with M τ > 0 , large enough such that

and set

Note that

We observe that

where in the last inequality we used Proposition A.7 (v). We note the following facts:

1. u ¯ τ is a reflection of u ¯ within the set with measure less than O ⁢ ( τ ) .

2. It holds

   ℒ N ( { u ¯ ≠ u } ) ≤ ∑ m = 1 Y τ ℒ N ( Q m ) ≤ O ( τ ) .

3. u ∈ SBV 2 ⁡ ( Ω ) ∩ L ∞ ⁢ ( Ω ) .

Then

For brevity, in the rest of the proof we abbreviate Q ν S ω ⁢ ( x n , r n ) by Q n , T x n , ν S u by T x n , and T x n , ν S u ⁢ ( - t n ) by T x n ⁢ ( - t n ) . Note that the jump set of u ¯ τ is contained in

and S u ¯ τ ⊂ P τ and P τ are both unions of finitely many polyhedra. We also observe that, denoting by cl ⁡ ( ⋅ ) the closure of a set,

(A.26)

where we used Proposition A.7 (v), (A.23), and the assumption that ℋ N - 1 ⁢ ( S u ¯ ) < + ∞ .

Let ε > 0 be such that

Hence, by Proposition A.7 (vi), we have

We set u τ , ε := ( 1 - φ ε ) ⁢ u ¯ τ , where φ ε is such that φ ε ∈ C c ∞ ⁢ ( Ω ; [ 0 , 1 ] ) , φ ε ≡ 1 on ( S u ¯ τ ¯ ) ε 2 / 4 , and φ ε ≡ 0 in Ω ∖ ( S u ¯ τ ¯ ) ε 2 / 2 . By (A.26) we have ℋ N - 1 ⁢ ( S u ¯ τ ¯ ) < + ∞ , and hence { u τ , ε } ε > 0 ⊂ W 1 , 2 ⁢ ( Ω ) . Moreover, by the dominated convergence theorem and by (A.24), we conclude that u τ , ε → u in L 1 ⁢ ( Ω ) .

Consider the sequence { v τ , ε } ε > 0 ⊂ W 1 , 2 ⁢ ( Ω ) given by v τ , ε ⁢ ( x ) := v ~ ε ⁢ ( d τ ⁢ ( x ) ) , where d τ ⁢ ( x ) := dist ⁡ ( x , P τ ) and the v ~ ε are defined by

An explicit computation shows that

for ε 2 ≤ t ≤ ε + ε 2 and v ~ ε ∈ W loc 1 , 2 ⁢ ( ℝ ) , and we remark that

and

Moreover, since S u τ ⊂ P τ and by (A.24), we conclude that

Step 2. For the general case ℋ N - 1 ⁢ ( S u ∖ S ω ) > 0 , the proof follows by applying the same construction as in step 1 on S u , and by noticing that ω - ⁢ ( x ) = ω ⁢ ( x ) if x ∈ S u ∖ S ω . ∎