Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain

Proposition A.1.

Let A ⊂ R 2 be a bounded L 2 -measurable set with H 1 ⁢ ( ∂ ⁡ A ) < + ∞ . Then A is a set of finite perimeter in R 2 .

Proof.

Since A ⁢ Δ ⁢ Int ⁡ ( A ¯ ) ⊂ A ¯ ∖ Int ⁡ ( A ) = ∂ ⁡ A , we have | A ⁢ Δ ⁢ Int ⁡ ( A ¯ ) | ≤ | ∂ ⁡ A | = 0 . Hence, it suffices to prove that the open set E := Int ⁡ ( A ¯ ) has finite perimeter in ℝ 2 . Note that, by construction,

We divide the proof of E ∈ BV ⁡ ( ℝ 2 , { 0 , 1 } ) into three steps.

Step 1. We claim that if E is simply connected, then E ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) . Indeed, in this case ∂ ⁡ E is a connected compact set with ℋ 1 ⁢ ( ∂ ⁡ E ) ≤ ℋ 1 ⁢ ( ∂ ⁡ A ) < + ∞ and, by [33, Lemma 3.12], it contains a closed curve Γ enclosing E ¯ . Since ℋ 1 ⁢ ( Γ ) < + ∞ , it is rectifiable in the sense of [33, Section 3.2]: its length ℋ 1 ⁢ ( Γ ) is well-approximated by the length of closed polygonal curves π k whose vertices lie on Γ, i.e., ℋ 1 ⁢ ( π k ) → ℋ 1 ⁢ ( Γ ) . Let E k be the set enclosed by π k and observe that

Since E k are Lipschitz sets, they are sets of finite perimeter and

for large k. Since E is open, for every x ∈ E there exists a ball B r ⁢ ( x ) ⊂ E and by the Kuratowski convergence of π k to Γ. It follows that B r ⁢ ( x ) ⊂ E k for large k, and hence χ E k ⁢ ( x ) = χ E ⁢ ( x ) = 1 . Similarly, χ E k ⁢ ( x ) = χ E ⁢ ( x ) = 0 for every x ∈ ℝ 2 ∖ E ¯ provided k is large enough. Therefore, χ E k → χ E a.e. in ℝ 2 , and hence E k → E in L 1 ⁢ ( ℝ 2 ) . Now, by the L 1 -lower semicontinuity of perimeter (see [52, Proposition 12.15]), E is a set of finite perimeter.

Step 2. We claim that if E is connected, then E ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) . Indeed, let E ′ be the smallest simply connected open set containing E (basically, E ′ is constructed by filling all “holes” in E), and let

be the union of all holes. Since

by step 1, we have E ′ ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) . Observing E = E ′ ∖ F ¯ , to conclude this step it is enough to prove that F has finite perimeter. Since every open set in ℝ 2 is a union of at most countably many connected components^[2], we have F = ⋃ j F j , where the { F j } are open, connected and F i ∩ F j = ∅ for i ≠ j . Since E is connected, each F j is simply connected. Hence, by step 1, F j ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) . Moreover, the set ∂ ⁡ F i ∩ ∂ ⁡ F j , i ≠ j , can have at most one point. Indeed, otherwise, by the definition of F and the connectedness of E, we could find a curve γ ⊂ ∂ ⁡ F i ∩ ∂ ⁡ F j ∩ ∂ ⁡ E with ℋ 1 ⁢ ( γ ) > 0 , which contradicts the equality E = Int ⁡ ( E ¯ ) . Therefore, observing ∂ ⁡ F = ⋃ ∂ ⁡ F j ⊂ ∂ ⁡ E , we obtain

Thus, F = ⋃ j F j has finite perimeter in ℝ 2 .

Step 3. Now, we prove that E ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) (without assuming any extra connectedness assumption). Let { E j } be the family of connected components of E. Since

by step 2, we have E j ∈ BV ⁡ ( ℝ 2 ; { 0 , 1 } ) . Therefore, since ∂ ⁡ E = ⋃ j ∂ ⁡ E j , we obtain that

Hence, by the finiteness of ℋ 1 ⁢ ( ∂ ⁡ E ) , the set E = ⋃ j E j has finite perimeter in ℝ 2 . ∎

Proposition A.2.

Let K ⊂ R 2 be such that H 1 ⁢ ( K ) < + ∞ and let { E t } t ∈ Υ be a family of sets parametrized by t ∈ Υ such that

and H 1 ⁢ ( K ∩ E t ) > 0 . Then Υ is at most countable.

Proof.

The proof runs along the lines of the proof of [52, Proposition 2.16]. For j ∈ ℕ , let Υ j ⊂ Υ be the set of all t ∈ Υ such that ℋ 1 ⁢ ( K ∩ E t ) > 1 j . Then, by (A.1), Υ j cannot contain more than j ⁢ ℋ 1 ⁢ ( K ) elements. Since Υ = ⋃ j Υ j , the set Υ is at most countable. ∎

Proposition A.3.

Let U ⊂ R d be a connected bounded open set and let u ∈ GSBD 2 ⁡ ( U ) be such that H d - 1 ⁢ ( J u ) = 0 . Then u ∈ H loc 1 ⁢ ( U ) .

Proof.

Indeed, for r > 0 let Q := x 0 + ( - r , r ) d ⊂ U be any cube centered at x ∈ U and let 0 < θ ′′ < θ ′ < 1 . For shortness, write

By [11, Proposition 3.1 (1)] (see also [10, Theorem 1.1]), there exists an ℒ 2 -measurable set ω ⊂ Q ′ and a rigid displacement a : ℝ d → ℝ d such that | ω | ≤ c * ⁢ r ⁢ ℋ d - 1 ⁢ ( J u ) = 0 and

where c * depends only on d. Hence, u ∈ L loc 2 ⁢ d / ( d - 1 ) ⁢ ( Q ) . Next, fix any mollifier ρ 1 ∈ C ∞ ⁢ ( B r ⁢ ( 0 ) ) with

where ρ ϵ ⁢ ( x ) := ρ 1 ⁢ ( x / ϵ ) , ϵ ∈ ( 0 , r ) . By [11, Proposition 3.1], there exists p ¯ > 0 depending on n and ϵ such that

where c depends on n, ρ 1 and ϵ. Hence,

Recall that u * ρ ϵ ∈ C ∞ ⁢ ( Q ′′ ) . Since e ⁢ ( u ) ∈ L 2 ⁢ ( Q ) , we have

in particular,

By the Poincaré–Korn inequality, u * ρ ϵ ∈ H 1 ⁢ ( Q ′′ ) . Since e ⁢ ( u ) * ρ ϵ → e ⁢ ( u ) in L 2 ⁢ ( Q ′′ ) as ϵ → 0 , in view of (A.2), there exists ϵ 0 > 0 such that

Moreover, by the Poincaré–Korn inequality, for any ϵ ∈ ( 0 , ϵ 0 ) there exists a rigid displacement a ϵ such that

where C is the Poincaré–Korn constant for a cube. Thus, the family { u * ρ ϵ } ϵ is uniformly bounded in H 1 ⁢ ( Q ′′ ) . Since u * ρ ϵ → u in L 2 ⁢ ( Q ′′ ) , there exists a rigid displacement a such that a ϵ → a in L 2 ⁢ ( Q ′′ ) . Then u * ρ ϵ - a ϵ weakly converges to u - a in H 1 ⁢ ( Q ′′ ) , i.e., u - a ∈ H 1 ⁢ ( Q ′′ ) . Since a is linear and θ ′′ is arbitrary, u ∈ H loc 1 ⁢ ( Q ) . Now, covering U with finitely many cubes of edgelength 2 ⁢ r , we get u ∈ H loc 1 ⁢ ( U ) . ∎