Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type

2.1 Definition of the solution

We treat problem (P) by a system of weak formulations. To do so, we introduce the spaces H := L 2 ⁢ ( Ω ) , H Γ := L 2 ⁢ ( Γ ) , V := H 1 ⁢ ( Ω ) , V Γ := H 1 ⁢ ( Γ ) , W := H 2 ⁢ ( Ω ) , W Γ := H 2 ⁢ ( Γ ) , with usual norms and inner products; we denote them by | ⋅ | H , | ⋅ | H Γ and ( ⋅ , ⋅ ) H , ( ⋅ , ⋅ ) H Γ , and so on. Concerning these inner products and norms, we use the same notation for scalar and vectorial functions (e.g., typically gradients).

Moreover, we put 𝑯 := H × H Γ and 𝑽 := { ( z , z Γ ) ∈ V × V Γ : z Γ = z | Γ ⁢ a.e. on  ⁢ Γ } . Then 𝑯 and 𝑽 are Hilbert spaces with the inner products

and related norms. As a remark, if 𝒛 := ( z , z Γ ) ∈ 𝑽 , then z Γ is the trace z | Γ of z on Γ, while if 𝒛 := ( z , z Γ ) ∈ 𝑯 , then z ∈ H and z Γ ∈ H Γ are independent. Hereafter, we use the notation of a bold letter like 𝒛 to denote the pair which corresponds to the letter, that is, ( z , z Γ ) for 𝒛 .

It is easy to see that problem (P) has a structure of volume conservation on the boundary Γ. Indeed, integrating the last equation in (2.3) on Σ, and using (2.1) and (2.5), we obtain

hereafter, we put

where | Γ | := ∫ Γ 1 ⁢ 𝑑 Γ . The space 𝑽 * denotes the dual of 𝑽 , and 〈 ⋅ , ⋅ 〉 𝑽 * , 𝑽 denotes the duality pairing between 𝑽 * and 𝑽 . Moreover, it is understood that 𝑯 is embedded in 𝑽 * in the usual way, i.e., 〈 𝒖 , 𝒛 〉 𝑽 * , 𝑽 = ( 𝒖 , 𝒛 ) 𝑯 for all 𝒖 ∈ 𝑯 , 𝒛 ∈ 𝑽 . Then we obtain 𝑽 ⁢ ↪ ↪ 𝑯 ⁢ ↪ ↪ 𝑽 * , where “ ↪ ↪ ” stands for the dense and compact embedding, namely, ( 𝑽 , 𝑯 , 𝑽 * ) is a standard Hilbert triplet.

Under this setting, we define the solution of (P) as follows.

2.2 Main theorems

The first result states the existence of the solution. To this end, we assume the following:

1. f ∈ L 2 ⁢ ( 0 , T ; H ) and f Γ ∈ W 1 , 1 ⁢ ( 0 , T ; H Γ ) .

2. 𝒖 0 := ( u 0 , u 0 ⁢ Γ ) ∈ 𝑽 .

3. The maximal monotone graphs β and β Γ in ℝ × ℝ are the subdifferentials β = ∂ ⁡ β ^ and β Γ = ∂ ⁡ β ^ Γ of some proper lower semicontinuous and convex functions β ^ and β ^ Γ : ℝ → [ 0 , + ∞ ] satisfying β ^ ⁢ ( 0 ) = β ^ Γ ⁢ ( 0 ) = 0 , with some effective domains D ⁢ ( β ^ ) ⊃ D ⁢ ( β ) and D ⁢ ( β ^ Γ ) ⊃ D ⁢ ( β Γ ) , respectively. This implies that 0 ∈ β ⁢ ( 0 ) and 0 ∈ β Γ ⁢ ( 0 ) .

4. π , π Γ : ℝ → ℝ are Lipschitz continuous functions with Lipschitz constants L and L Γ , and they satisfy π ⁢ ( 0 ) = π Γ ⁢ ( 0 ) = 0 .

5. D ⁢ ( β Γ ) ⊆ D ⁢ ( β ) and there exist positive constants ϱ , c 0 > 0 such that

   (2.12) | β ∘ ⁢ ( r ) | ≤ ϱ ⁢ | β Γ ∘ ⁢ ( r ) | + c 0   for all  ⁢ r ∈ D ⁢ ( β Γ ) ,

   where β ∘ and β Γ ∘ denote the minimal sections of β and β Γ .

6. m Γ ∈ int ⁡ D ⁢ ( β Γ ) and the compatibility conditions β ^ ⁢ ( u 0 ) ∈ L 1 ⁢ ( Ω ) , β ^ Γ ⁢ ( u 0 ⁢ Γ ) ∈ L 1 ⁢ ( Γ ) hold.

The minimal section β ∘ of β is specified by β ∘ ( r ) := { r * ∈ β ( r ) : | r * | = min s ∈ β ⁢ ( r ) | s | } and the same definition applies to β Γ ∘ . These assumptions are the same as in [2, 5]. Concerning assumption (A1), let us note that the regularity conditions for f and f Γ are not symmetric. In fact, we need more regularity for the source term on the boundary, since the equation on the boundary is of Cahn–Hilliard type, while the equation on the bulk turns out to be of the simpler Allen–Cahn type. Of course, here the condition τ > 0 plays a role, and if the term τ ⁢ ∂ t ⁡ u is not present in (2.2), then we would certainly need higher regularity for f.

The second result states the continuous dependence on the data. The uniqueness of the component 𝒖 of the solution is obtained from this theorem. Here, we just use the following regularity properties on the data:

1. f ∈ L 2 ⁢ ( 0 , T ; V * ) and f Γ ∈ L 2 ⁢ ( 0 , T ; V Γ * ) ,

2. u 0 ∈ H and u 0 ⁢ Γ ∈ V Γ * .

Then we obtain the continuous dependence on the data as follows:

3.1 Moreau–Yosida regularization

For each λ ∈ ( 0 , 1 ] , we define β λ , β Γ , λ : ℝ → ℝ , along with the associated resolvent operators J λ , J Γ , λ : ℝ → ℝ , by

for all r ∈ ℝ , where ϱ > 0 is the same constant as in assumption (2.12). Note that the two definitions are not symmetric, since in the second one, it is λ ⁢ ϱ and not directly λ to be used as approximation parameter; let us note that this adaptation comes from the previous work [2]. Now, we easily have β λ ⁢ ( 0 ) = β Γ , λ ⁢ ( 0 ) = 0 . Moreover, the related Moreau–Yosida regularizations β ^ λ , β ^ Γ , λ of β ^ , β ^ Γ : ℝ → ℝ fulfill

It is well known that β λ is Lipschitz continuous with Lipschitz constant 1 / λ and β Γ , λ is also Lipschitz continuous with constant 1 / ( λ ⁢ ϱ ) . In addition, for each λ ∈ ( 0 , 1 ] , we have the standard properties

Let us point out that, using [2, Lemma 4.4], we have

for all λ ∈ ( 0 , 1 ] , with the same constants ϱ and c 0 as in (2.12).

Now for each ε ∈ ( 0 , 1 ] , let 𝒇 ε := ( f ε , f Γ , ε ) and 𝒖 0 , ε := ( u 0 , ε , u 0 ⁢ Γ , ε ) be smooth approximations for 𝒇 and 𝒖 0 , so that 𝒇 ε ∈ H 1 ⁢ ( 0 , T ; 𝑯 ) with 𝒇 ε ⁢ ( 0 ) ∈ 𝑽 and 𝒖 0 , ε ∈ 𝑾 ∩ 𝑽 with ( - Δ ⁢ u 0 , ε , ∂ 𝝂 ⁡ u 0 , ε - Δ Γ ⁢ u 0 ⁢ Γ , ε ) ∈ 𝑽 satisfying

where C 0 is a positive constant independent of ε , λ ∈ ( 0 , 1 ] . Indeed, 𝒇 ε and 𝒖 0 , ε satisfying (3.3)–(3.6) are given in Appendix A. Hereafter, we use C * := ( 1 + ε 1 / 2 / λ ) 1 / 2 , which satisfies C * ↘ 1 as ε ↘ 0 for λ ∈ ( 0 , 1 ] . Then we can solve the following auxiliary problem.

As a remark, the trace conditions is included in the regularities (3.7)–(3.8). The strategy of the proof of this proposition is based on the previous work [5, Theorems 2.2, 4.2]. It is given in Appendix A.

3.2 A priori estimates

In order to obtain the uniform estimates independent of the approximate parameter ε and λ, we use the following type of Poincaré–Wirtinger inequalities (see, e.g., [29, 22]): there exists a positive constant C P such that

Moreover, from the compactness inequality, recalled in [26, Chapter 1, Lemma 5.1] or [30, Section 8, Lemma 8], for each δ > 0 , there exists a positive constant C δ depending on δ such that

because we have the compact embeddings V ⁢ ↪ ↪ H 1 - α ⁢ ( Ω ) ⁢ ↪ ↪ H for α ∈ ( 0 , 1 ) , (see, e.g., [27, Chapter 1, Theorem 16.1]) and V Γ ⁢ ↪ ↪ H Γ ⁢ ↪ ↪ V Γ * , respectively. Next, from the standard theorem for the trace operators γ 0 ⁢ ( z ) := z | Γ of γ 0 : V → H 1 / 2 ⁢ ( Γ ) and γ 0 : H 1 - α ⁢ ( Ω ) → H ( 1 - α ) - 1 / 2 ⁢ ( Γ ) ⊂ H Γ for α ∈ ( 0 , 1 / 2 ) (see, e.g., [23, Chapter 2, Theorem 2.24], [29, Chapter 2, Theorem 5.5]), we see that there exists a positive constant c 1 such that

for α ∈ ( 0 , 1 / 2 ) . Moreover, the boundedness of some recovering operator ℛ : H 1 / 2 ⁢ ( Γ ) → V of the trace γ 0 gives us

where c 2 is a positive constant (see, e.g., [23, Chapter 2, Theorem 2.24], [29, Chapter 2, Theorem 5.7]).