Pierluigi Colli ORCID logo and Takeshi Fukao ORCID logo

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

Open Access
De Gruyter | Published online: July 20, 2018

Abstract

The well-posedness of a system of partial differential equations with dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk Ω and on the boundary Γ. The Poisson equation for the chemical potential and the Allen–Cahn equation for the order parameter in the bulk Ω are considered as auxiliary conditions for solving the Cahn–Hilliard equation on the boundary Γ. Recently, the well-posedness of this equation with a dynamic boundary condition, both of Cahn–Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.

1 Introduction

In this paper, we treat the Cahn–Hilliard equation [1] on the boundary of some bounded smooth domain. Let 0 < T < + be some fixed time and let Ω d , d = 2 or 3, be a bounded domain occupied by a material, where the boundary Γ of Ω is supposed to be smooth enough. We start from the following equations of Cahn–Hilliard type on the boundary Γ:

(1.1) t u Γ - Δ Γ μ Γ = - 𝝂 μ on  Σ := Γ × ( 0 , T ) ,
(1.2) μ Γ = - Δ Γ u Γ + 𝒲 Γ ( u Γ ) - f Γ + 𝝂 u on  Σ ,

where t denotes the partial derivative with respect to time, and Δ Γ denotes the Laplace–Beltrami operator on Γ (see, e.g., [21, Chapter 3]). Here, the unknowns u Γ and μ Γ : Σ stand for the order parameter and the chemical potential, respectively. In the right-hand sides of (1.1) and (1.2), the outward normal derivative 𝝂 on Γ acts on functions μ , u : Q := ( 0 , T ) × Ω that satisfy the following trace conditions:

(1.3) μ | Σ = μ Γ , u | Σ = u Γ on  Σ ,

where μ | Σ and u | Σ are the traces of μ and u on Σ. Moreover, these functions μ and u solve the following equations in the bulk Ω:

(1.4) - Δ μ = 0 in  Q ,
(1.5) τ t u - Δ u + 𝒲 ( u ) = f in  Q ,

where τ > 0 is a positive constant and Δ denotes the Laplacian.

Note that the nonlinear terms 𝒲 Γ and 𝒲 are the derivatives of the functions 𝒲 Γ and 𝒲 , usually referred as double-well potentials, with two minima and a local unstable maximum in between. The prototype model is provided by 𝒲 Γ ( r ) = 𝒲 ( r ) = ( 1 / 4 ) ( r 2 - 1 ) 2 , so that 𝒲 Γ ( r ) = 𝒲 ( r ) = r 3 - r , r , is the sum of an increasing function with a power growth and another smooth function which breaks the monotonicity properties of the former and is related to the non-convex part of the potential 𝒲 Γ or 𝒲 .

Therefore, we can say that system (1.1)–(1.2) yields the Cahn–Hilliard equation on a smooth manifold Γ, and equations (1.4) and (1.5) reduce to the Poisson equation for μ and the Allen–Cahn equation for u in the bulk Ω, as auxiliary conditions for solving (1.1)–(1.2). In other word, (1.1)–(1.5) is a sort of transmission problem between the dynamics in the bulk Ω and the one on the boundary Γ. With the initial conditions

(1.6) u Γ ( 0 ) = u 0 Γ on  Γ , u ( 0 ) = u 0 in  Ω ,

problem (1.1)–(1.6) becomes an initial value problem of a Poisson–Allen–Cahn system with a dynamic boundary condition of Cahn–Hilliard type, named (P). Indeed, the interaction between μ and u appears only on (1.1)–(1.2), whereas (1.4) and (1.5) are independent equations. As a remark, if τ = 0 , then problem (P) turns out to be a quasi-static system. From (1.1), (1.4) and (1.6), we easily see that the mass conservation on the boundary holds as follows:

(1.7) Γ u Γ ( t ) 𝑑 Γ = Γ u 0 Γ 𝑑 Γ for all  t [ 0 , T ] .

Let us mention some related results: the papers [11, 14, 12, 13, 16] consider some quasi-static systems with dynamic boundary conditions (see also [6, 4]), the contributions [3, 5, 8, 9, 15, 17, 20, 28] set a Cahn–Hilliard equation on the boundary as the dynamic boundary condition, and a more complicated system of Cahn–Hilliard type on the boundary with a mass conservation condition is investigated in [18]. Especially, in this paper we will exploit the previous result in [5], on which equations (1.1)–(1.5) were replaced by

(1.8) ε t u - Δ μ = 0 in  Q ,
(1.9) ε μ = τ t u - Δ u + 𝒲 ( u ) - f in  Q ,
(1.10) u | Σ = u Γ , μ | Σ = μ Γ on  Σ ,
(1.11) t u Γ + 𝝂 μ - Δ Γ μ Γ = 0 on  Σ ,
(1.12) μ Γ = 𝝂 u - Δ Γ u Γ + 𝒲 Γ ( u Γ ) - f Γ on  Σ ,

where ε > 0 . Then from system (1.8)–(1.12) with (1.6), we obtain the following total mass conservation:

(1.13) ε Ω u ( t ) 𝑑 x + Γ u Γ ( t ) 𝑑 Γ = ε Ω u 0 𝑑 x + Γ u 0 Γ 𝑑 Γ for all  t [ 0 , T ] .

Our essential idea of the existence proof is to be able to pass to the limit as ε 0 in system (1.8)–(1.12), with (1.6). To be more precise about our arguments, let us introduce a brief outline of the paper along with a short description of the various items.

In Section 2, we present the main results of the well-posedness of system (1.1)–(1.6). A solution to problem (P) is suitably defined. The main theorems are concerned with the existence of the solution (Theorem 2.3) and the continuous dependence on the given data (Theorem 2.4), the second theorem entailing the uniqueness property.

In Section 3, we consider the approximate problem for (P), with two approximation parameters ε and λ, by substituting the maximal monotone graphs with their Yosida regularizations in terms of the parameter λ. Moreover, we obtain uniform estimates with suitable growth order. Here, we can apply the results that have been shown in [5].

In Section 4, we prove the existence result. The proof is split in several steps. In the first step, we obtain uniform estimates with respect to ε. Then, combining them with the previous estimates of Section 3, we can pass to the limit as ε 0 . In the second step, we improve suitable estimates in order to make them independent of λ. Then we can pass to the limit as λ 0 and conclude the existence proof. The last part of this section is devoted to the proof of the continuous dependence.

Finally, in Appendix A, the approximate problem for (P) and some auxiliary results are discussed.

2 Main results

In this section, our main result is stated. At first, we give our target system (P) some equations and conditions as follows: for any fixed constant τ > 0 , we have

(2.1) - Δ μ = 0 a.e. in  Q ,
(2.2) τ t u - Δ u + ξ + π ( u ) = f , ξ β ( u ) a.e. in  Q ,
(2.3) u Γ = u | Σ , μ Γ = μ | Σ , t u Γ + 𝝂 μ - Δ Γ μ Γ = 0 a.e. on  Σ ,
(2.4) μ Γ = 𝝂 u - Δ Γ u Γ + ξ Γ + π Γ ( u Γ ) - f Γ , ξ Γ β Γ ( u Γ ) a.e. on  Σ ,
(2.5) u ( 0 ) = u 0 a.e. in  Ω , u Γ ( 0 ) = u 0 Γ a.e. on  Γ ,

where f : Q , f Γ : Σ are given sources, u 0 : Ω , u 0 Γ : Γ are known initial data, β stands for the subdifferential of the convex part β ^ and π stands for the derivative of the concave perturbation π ^ of a double well potential 𝒲 ( r ) = β ^ ( r ) + π ^ ( r ) , defined for all r in the domain of β ^ . The same setting holds for β Γ and π Γ .

Typical examples of the nonlinearities β, π are given by

  • β ( r ) = r 3 , π ( r ) = - r for all r , with D ( β ) = for the prototype double well potential

    𝒲 ( r ) = 1 4 ( r 2 - 1 ) 2 .

  • β ( r ) = ln ( ( 1 + r ) / ( 1 - r ) ) , π ( r ) = - 2 c r for all r D ( β ) , with D ( β ) = ( - 1 , 1 ) for the logarithmic double well potential

    𝒲 ( r ) = ( ( 1 + r ) ln ( 1 + r ) + ( 1 - r ) ln ( 1 - r ) ) - c r 2 ,

    where c > 0 is a large constant which breaks convexity.

  • β ( r ) = I [ - 1 , 1 ] ( r ) , π ( r ) = - r for all r D ( β ) , with D ( β ) = [ - 1 , 1 ] for the singular potential

    𝒲 ( r ) = I [ - 1 , 1 ] ( r ) - 1 2 r 2 ,

    where I [ - 1 , 1 ] is the indicator function on [ - 1 , 1 ] .

Similar choices can be considered for β Γ , π Γ and the related potential 𝒲 Γ . What is important in our approach is that the potential on the boundary should dominate the potential in the bulk, that is, we prescribe a compatibility condition between β and β Γ (see the later assumption A5) that forces the growth of β to be controlled by the growth of β Γ . A similar approach was taken in previous analyses, see [2, 6, 4, 5, 9, 28].

As a remark, τ > 0 plays the role of a viscous parameter. Indeed, if τ = 0 , then equations (2.1) and (2.2) become the stationary problem in Q, namely, the quasi-static system. A natural question arises whether one can investigate also the case τ = 0 or, in our framework, also study the singular limit as τ 0 . In our opinion, this is not a trivial question and deserves some attention and efforts. For the moment, we can just highlight it as open problem.

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

( 𝒖 , 𝒛 ) 𝑯 := ( u , z ) H + ( u Γ , z Γ ) H Γ for all  𝒖 := ( u , u Γ ) , 𝒛 := ( z , z Γ ) 𝑯 ,
( 𝒖 , 𝒛 ) 𝑽 := ( u , z ) V + ( u Γ , z Γ ) V Γ for all  𝒖 := ( u , u Γ ) , 𝒛 := ( z , z Γ ) 𝑽

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

(2.6) Γ u Γ ( t ) 𝑑 Γ = Γ u 0 Γ 𝑑 Γ for all  t [ 0 , T ] ;

hereafter, we put

(2.7) m Γ := 1 | Γ | Γ u 0 Γ 𝑑 Γ ,

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.

Definition 2.1.

The triplet ( 𝒖 , 𝝁 , 𝝃 ) is called the solution of (P) if 𝒖 = ( u , u Γ ) , 𝝁 = ( μ , μ Γ ) , 𝝃 = ( ξ , ξ Γ ) satisfy

u H 1 ( 0 , T ; H ) C ( [ 0 , T ] ; V ) L 2 ( 0 , T ; W ) ,
u Γ H 1 ( 0 , T ; V Γ * ) L ( 0 , T ; V Γ ) L 2 ( 0 , T ; W Γ ) ,
μ L 2 ( 0 , T ; V ) , μ Γ L 2 ( 0 , T ; V Γ ) ,
ξ L 2 ( 0 , T ; H ) , ξ Γ L 2 ( 0 , T ; H Γ ) ,
u | Σ = u Γ , μ | Σ = μ Γ a.e. on  Σ ,
ξ β ( u ) a.e. in  Q , ξ Γ β Γ ( u Γ ) a.e. on  Σ ,

solve

(2.8) u Γ ( t ) , z Γ V Γ * , V Γ + Ω μ ( t ) z d x + Γ Γ μ Γ ( t ) Γ z Γ d Γ = 0

for all 𝒛 := ( z , z Γ ) 𝑽 and a.a. t ( 0 , T ) , and

(2.9) τ t u - Δ u + ξ + π ( u ) = f a.e. in  Q ,
(2.10) μ Γ = 𝝂 u - Δ Γ u Γ + ξ Γ + π Γ ( u Γ ) - f Γ a.e. on  Σ ,
(2.11) u ( 0 ) = u 0 a.e. in  Ω , u Γ ( 0 ) = u 0 Γ a.e. on  Γ .

Remark 2.2.

Taking 𝒛 := ( 1 , 1 ) in the weak formulation (2.8), we see that (2.8) and (2.11) imply the mass conservation (2.6) on the boundary. Moreover, for any z 𝒟 ( Ω ) , taking 𝒛 := ( z , 0 ) in (2.8) and using the trace condition on μ, we deduce that

- Δ μ ( t ) = 0 a.e. in  Ω , μ | Γ ( t ) = μ Γ ( t ) a.e. on  Γ ,

for a.a. t ( 0 , T ) , whence the regularities μ L 2 ( 0 , T ; V ) , μ Γ L 2 ( 0 , T ; V Γ ) allow us to infer the higher regularity μ L 2 ( 0 , T ; H 3 / 2 ( Ω ) ) . Moreover, the boundedness Δ μ ( = 0 ) in L 2 ( 0 , T ; H ) gives us the property 𝝂 μ L 2 ( 0 , T ; H Γ ) , as well as

u Γ ( t ) + 𝝂 μ ( t ) , z Γ V Γ * , V Γ + Γ Γ μ Γ ( t ) Γ z Γ d Γ = 0 for all  z Γ V Γ ,

this is the weak formulation of (2.3).

2.2 Main theorems

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

  1. (A1)

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

  2. (A2)

    𝒖 0 := ( u 0 , u 0 Γ ) 𝑽 .

  3. (A3)

    The maximal monotone graphs β and β Γ in × are the subdifferentials β = β ^ and β Γ = β ^ Γ of some proper lower semicontinuous and convex functions β ^ and β ^ Γ : [ 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. (A4)

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

  5. (A5)

    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. (A6)

    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.

Theorem 2.3.

Under assumptions (A1)(A6), there exists a solution of problem (P).

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. (A1)’

    f L 2 ( 0 , T ; V * ) and f Γ L 2 ( 0 , T ; V Γ * ) ,

  2. (A2)’

    u 0 H and u 0 Γ V Γ * .

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

Theorem 2.4.

Under assumptions (A3)(A4), let, for i = 1 , 2 , 𝐟 ( i ) , 𝐮 0 ( i ) satisfy (A1)’, (A2)’ and assume that the corresponding solutions ( 𝐮 ( i ) , 𝛍 ( i ) , 𝛏 ( i ) ) exist. Then there exists a positive constant C > 0 , depending on L, L Γ and T, such that

| u ( 1 ) ( t ) - u ( 2 ) ( t ) | H 2 + | u Γ ( 1 ) ( t ) - u Γ ( 2 ) ( t ) | V Γ * 2 + 0 t | u ( 1 ) ( s ) - u ( 2 ) ( s ) | V 2 d s + 0 t | u Γ ( 1 ) ( s ) - u Γ ( 2 ) ( s ) | V Γ 2 d s
(2.13) C { | u 0 ( 1 ) - u 0 ( 2 ) | H 2 + | u 0 Γ ( 1 ) - u 0 Γ ( 2 ) | V Γ * 2 + 0 t | f ( 1 ) ( s ) - f ( 2 ) ( s ) | V * 2 d s + 0 t | f Γ ( 1 ) ( s ) - f Γ ( 2 ) ( s ) | V Γ * 2 d s }

for all t [ 0 , T ] .

3 Approximate problem and uniform estimates

In this section, we first consider an approximate problem for (P), and then we obtain uniform estimates. For each ε , λ ( 0 , 1 ] , we introduce an approximate problem ( P ; λ , ε ) where the proof of the well-posedness of ( P ; λ , ε ) is given in Appendix A.

3.1 Moreau–Yosida regularization

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

β λ ( r ) := 1 λ ( r - J λ ( r ) ) := 1 λ ( r - ( I + λ β ) - 1 ( r ) ) ,
β Γ , λ ( r ) := 1 λ ϱ ( r - J Γ , λ ( r ) ) := 1 λ ϱ ( r - ( I + λ ϱ β Γ ) - 1 ( r ) )

for all 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 : fulfill

β ^ λ ( r ) := inf s { 1 2 λ | r - s | 2 + β ^ ( s ) } = 1 2 λ | r - J λ ( r ) | 2 + β ^ ( J λ ( r ) ) = 0 r β λ ( s ) 𝑑 s ,
β ^ Γ , λ ( r ) := inf s { 1 2 λ ϱ | r - s | 2 + β ^ Γ ( s ) } = 0 r β Γ , λ ( s ) 𝑑 s for all  r .

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

| β λ ( r ) | | β ( r ) | , | β Γ , λ ( r ) | | β Γ ( r ) | ,
(3.1) 0 β ^ λ ( r ) β ^ ( r ) , 0 β ^ Γ , λ ( r ) β ^ Γ ( r ) for all  r .

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

(3.2) | β λ ( r ) | ϱ | β Γ , λ ( r ) | + c 0 for all  r ,

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

(3.3) 𝒇 ε 𝒇 strongly in  L 2 ( 0 , T ; 𝑯 )  as  ε 0 ,
(3.4) | f Γ , ε - f Γ | L 2 ( 0 , T ; H Γ ) ε 1 / 2 C 0 for all  ε ( 0 , 1 ] ,
(3.5) 𝒖 0 , ε 𝒖 0 strongly in  𝑽  as  ε 0 ,
(3.6) Ω β ^ λ ( u 0 , ε ) 𝑑 x C 0 , Γ β ^ Γ , λ ( u 0 Γ , ε ) 𝑑 Γ ( 1 + ε 1 / 2 λ ) C 0 for all  ε ( 0 , 1 ] ,

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.

Proposition 3.1.

Under assumptions (A1)(A6), for each ε , λ ( 0 , 1 ] , there exists a unique pair

(3.7) 𝒖 λ , ε := ( u λ , ε , u Γ , λ , ε ) W 1 , ( 0 , T ; 𝑯 ) H 1 ( 0 , T ; 𝑽 ) L ( 0 , T ; 𝑾 ) ,
(3.8) 𝝁 λ , ε := ( μ λ , ε , μ Γ , λ , ε ) L ( 0 , T ; 𝑽 ) L 2 ( 0 , T ; 𝑾 ) ,

satisfying

(3.9) ε t u λ , ε - Δ μ λ , ε = 0 a.e. in  Q ,
(3.10) ε μ λ , ε = τ t u λ , ε - Δ u λ , ε + β λ ( u λ , ε ) + π ( u λ , ε ) - f ε a.e. in  Q ,
(3.11) t u Γ , λ , ε + 𝝂 μ λ , ε - Δ Γ μ Γ , λ , ε = 0 a.e. on  Σ ,
(3.12) μ Γ , λ , ε = ε t u Γ , λ , ε + 𝝂 u λ , ε - Δ Γ u Γ , λ , ε + β Γ , λ ( u Γ , λ , ε ) + π Γ ( u Γ , λ , ε ) - f Γ , ε a.e. on  Σ ,
(3.13) u λ , ε ( 0 ) = u 0 , ε a.e. in  Ω , u Γ , λ , ε ( 0 ) = u 0 Γ , ε a.e. on  Γ .

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

(3.14) | z | H 2 C P { Ω | z | 2 d x + Γ | z | Γ | 2 d Γ } for all  z V ,
(3.15) | z Γ | H Γ 2 C P Γ | Γ z Γ | 2 𝑑 Γ for all  z Γ V Γ , with  Γ z Γ 𝑑 Γ = 0 ,
(3.16) | 𝒛 | 𝑽 2 C P { Ω | z | 2 d x + Γ | Γ z Γ | 2 d Γ } for all  𝒛 𝑽 , with  Γ z Γ 𝑑 Γ = 0 .

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

(3.17) | z | H 1 - α ( Ω ) 2 δ | z | V 2 + C δ | z | H 2 for all  z V  and all  α ( 0 , 1 ) ,
(3.18) | z Γ | H Γ 2 δ | z Γ | V Γ 2 + C δ | z Γ | V Γ * 2 for all  z Γ V Γ ,

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

(3.19) | z Γ | H Γ 2 | z Γ | H 1 / 2 ( Γ ) 2 c 1 | z | V 2 , | z Γ | H Γ 2 c 1 | z | H 1 - α ( Ω ) 2 for all  𝒛 𝑽 ,

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

(3.20) | z Γ | V 2 c 2 | z Γ | H 1 / 2 ( Γ ) 2 c 2 | z Γ | V Γ 2 for all  z Γ V Γ ,

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

Lemma 3.2.

There exist two positive constants M 1 and M 2 , depending on τ but independent of ε and λ ( 0 , 1 ] , such that

(3.21) | u λ , ε | L ( 0 , T ; V ) C * M 1 , | u Γ , λ , ε | L ( 0 , T ; H Γ ) c 1 1 / 2 C * M 1 ,
| t u λ , ε | L 2 ( 0 , T ; H ) + ε 1 / 2 | t u Γ , λ , ε | L 2 ( 0 , T ; H Γ ) + | Γ u Γ , λ , ε | L ( 0 , T ; H Γ ) + | β ^ λ ( u λ , ε ) | L ( 0 , T ; L 1 ( Ω ) )
(3.22) + | β ^ Γ , λ ( u Γ , λ , ε ) | L ( 0 , T ; L 1 ( Γ ) ) + | μ λ , ε | L 2 ( 0 , T ; H ) + | Γ μ Γ , λ , ε | L 2 ( 0 , T ; H Γ ) C * M 2 .

Proof.

We test (3.10) at time s by s u λ , ε , the time derivative of u λ , ε . Integrating the resultant over Ω × ( 0 , t ) leads to

τ 0 t | s u λ , ε ( s ) | H 2 d s + 1 2 | u λ , ε ( t ) | H 2 + Ω β ^ λ ( u λ , ε ( t ) ) d x + Ω π ^ ( u λ , ε ( t ) ) d x
- 0 t ( 𝝂 u λ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ 𝑑 s - 0 t ( f ε ( s ) , s u λ , ε ( s ) ) H 𝑑 s
(3.23) = ε 0 t ( μ λ , ε ( s ) , s u λ , ε ( s ) ) H 𝑑 s + 1 2 | u 0 , ε | H 2 + Ω β ^ λ ( u 0 , ε ) 𝑑 x + Ω π ^ ( u 0 , ε ) 𝑑 x

for all t [ 0 , T ] . Next, testing (3.12) by s u Γ , λ , ε and integrating the resultant over Γ × ( 0 , t ) , we obtain

- 0 t ( 𝝂 u λ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ d s = ε 0 t | s u Γ , λ , ε ( s ) | H Γ 2 d s - 0 t ( μ Γ , λ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ d s
+ 1 2 | u Γ , λ , ε ( t ) | H Γ 2 - 1 2 | u 0 Γ , ε | H Γ 2 + Γ β ^ Γ , λ ( u Γ , λ , ε ( t ) ) 𝑑 Γ
+ Γ π ^ Γ ( u Γ , λ , ε ( t ) ) 𝑑 Γ - Γ β ^ Γ , λ ( u 0 Γ , ε ) 𝑑 Γ
(3.24) - Γ π ^ Γ ( u 0 Γ , ε ) 𝑑 Γ - 0 t ( f Γ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ 𝑑 s

for all t [ 0 , T ] . On the other hand, testing (3.9) by μ λ , ε , testing (3.11) by μ Γ , λ , ε , and adding them, we infer that

(3.25) ε 0 t ( μ λ , ε ( s ) , s u λ , ε ( s ) ) H d s + 0 t ( μ Γ , λ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ d s = - 0 t | μ λ , ε ( s ) | H 2 d s - 0 t | Γ μ Γ , λ , ε ( s ) | H Γ 2 d s

for all t [ 0 , T ] . Combining (3.23)–(3.25) and using (3.1), we have

τ 0 t | s u λ , ε ( s ) | H 2 d s + ε 0 t | s u Γ , λ , ε ( s ) | H Γ 2 d s + 1 2 | u λ , ε ( t ) | H 2 + Ω β ^ λ ( u λ , ε ( t ) ) d x
+ 1 2 | u Γ , λ , ε ( t ) | H Γ 2 + Γ β ^ Γ , λ ( u Γ , λ , ε ( t ) ) d Γ + 0 t | μ λ , ε ( s ) | H 2 d s + 0 t | Γ μ Γ , λ , ε ( s ) | H Γ 2 d s
1 2 | u 0 , ε | H 2 + Ω β ^ λ ( u 0 , ε ) d x + Ω | π ^ ( u λ , ε ( t ) ) | d x + Ω | π ^ ( u 0 , ε ) | d x
+ 1 2 | u 0 Γ , ε | H Γ 2 + Γ β ^ Γ , λ ( u 0 Γ , ε ) d Γ + Γ | π ^ Γ ( u Γ , λ , ε ( t ) ) | d Γ + Γ | π ^ Γ ( u 0 Γ , ε ) | d Γ
(3.26) + 0 t ( f ε ( s ) , s u λ , ε ( s ) ) H 𝑑 s + 0 t ( f Γ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ 𝑑 s

for all t [ 0 , T ] . Therefore, in order to estimate the right-hand side of (3.26), we prepare the estimate of | u λ , ε ( s ) | H . Indeed, from the Young inequality, we see that

1 2 | u λ , ε ( s ) | H 2 = 0 t ( s u λ , ε ( s ) , u λ , ε ( s ) ) H 𝑑 s + 1 2 | u 0 , ε | H 2
(3.27) δ ~ 2 0 t | s u λ , ε ( s ) | H 2 d s + 1 2 δ ~ 0 t | u λ , ε ( s ) | H 2 d s + 1 2 | u 0 , ε | H 2

for all t [ 0 , T ] and some δ ~ > 0 . Now, from (A4), we can use the fact that | π ( r ) | = | π ( r ) - π ( 0 ) | L | r | , and then we deduce

| π ^ ( r ) | 0 r | π ( l ) | d l L 2 r 2 for all  r .

Therefore, by taking δ ~ := τ / ( 5 L ) in (3.27), we have

(3.28) Ω | π ^ ( u λ , ε ( t ) ) | d x L 2 Ω | u λ , ε ( t ) | 2 d x τ 10 0 t | s u λ , ε ( s ) | H 2 d s + 5 L 2 2 τ 0 t | u λ , ε ( s ) | H 2 d s + L 2 | u 0 , ε | H 2

and, analogously,

(3.29) Ω | π ^ ( u 0 , ε ) | d x L 2 Ω | u 0 , ε | 2 d x , Γ | π ^ Γ ( u 0 Γ , ε ) | d Γ L Γ 2 Γ | u 0 Γ , ε | 2 d Γ .

Additionally, by using (3.17), (3.19) and (3.27) with δ ~ := τ / ( 10 C δ ) , we get

Γ | π ^ Γ ( u Γ , λ , ε ( t ) ) | d Γ L Γ 2 Γ | u Γ , λ , ε ( t ) | 2 d Γ
c 1 L Γ 2 | u λ , ε ( t ) | H 1 - α ( Ω ) 2
δ | u λ , ε ( t ) | V 2 + C δ | u λ , ε ( t ) | H 2
(3.30) δ | u λ , ε ( t ) | V 2 + τ 10 0 t | s u λ , ε ( s ) | H 2 d s + 10 C δ 2 τ 0 t | u λ , ε ( s ) | H 2 d s + C δ | u 0 , ε | H 2

for some α ( 0 , 1 / 2 ) and δ > 0 . Moreover, from the Young inequality, it turns out that

(3.31) 0 t ( f ε ( s ) , s u λ , ε ( s ) ) H d s τ 10 0 t | s u λ , ε ( s ) | H 2 d s + 5 2 τ 0 t | f ε ( s ) | H 2 d s ,
(3.32) 0 t ( f Γ , ε ( s ) , s u Γ , λ , ε ( s ) ) H Γ d s ε 2 0 t | s u Γ , λ , ε ( s ) | H Γ 2 d s + 1 2 ε 0 t | f Γ , ε ( s ) | H Γ 2 d s

for all t [ 0 , T ] . Therefore, collecting (3.28)–(3.32), adding ( 1 / 2 ) | u λ , ε ( s ) | H 2 to both sides of (3.26) and using (3.27) with δ ~ := τ / 5 , we obtain

τ 2 0 t | s u λ , ε ( s ) | H 2 d s + ε 2 0 t | s u Γ , λ , ε ( s ) | H Γ 2 d s + 1 2 | u λ , ε ( t ) | V 2 + Ω β ^ λ ( u λ , ε ( t ) ) d x
+ 1 2 | u Γ , λ , ε ( t ) | H Γ 2 + Γ β ^ Γ , λ ( u Γ , λ , ε ( t ) ) d Γ + 0 t | μ λ , ε ( s ) | H 2 d s + 0 t | Γ μ Γ , λ , ε ( s ) | H Γ 2 d s
5 2 τ 0 t | u λ , ε ( s ) | H 2 d s + 1 2 | u 0 , ε | V 2 + Ω β ^ λ ( u 0 , ε ) d x + 5 L 2 2 τ 0 t | u λ , ε ( s ) | H 2 d s + L | u 0 , ε | H 2
+ 5 2 τ 0 t | f ε ( s ) | H 2 d s + 1 2 | u 0 Γ , ε | V Γ 2 + Γ β ^ Γ , λ ( u 0 Γ , ε ) d Γ + δ | u λ , ε ( t ) | V 2
(3.33) + 10 C δ 2 τ 0 t | u λ , ε ( s ) | H 2 d s + C δ | u 0 , ε | H 2 + L Γ 2 | u 0 Γ , ε | H Γ 2 + 1 2 ε 0 t | f Γ , ε ( s ) | H Γ 2 d s

for all t [ 0 , T ] . Thus, taking δ := 1 / 4 , and using (3.3)–(3.6) and the Gronwall inequality, we see that there exists a positive constant M 1 , depending on | u 0 | V , C 0 , L, | f | L 2 ( 0 , T ; H ) , | u 0 Γ | V Γ , L Γ , | f Γ | L 2 ( 0 , T ; H Γ ) and τ, in which M 1 + as τ 0 , independent of ε and λ, such that

| u λ , ε | L ( 0 , T ; V ) C * M 1 .

Moreover, using this, (3.19) implies

| u Γ , λ , ε | L ( 0 , T ; H Γ ) c 1 1 / 2 C * M 1 ,

and from (3.33) we obtain estimate (3.22). ∎

By comparison, the following estimates for Δ μ λ , ε and 𝒖 λ , ε := ( t u λ , ε , t u Γ , λ , ε ) are obtained.

Lemma 3.3.

Let M 2 be the same constant as in Lemma 3.2. Then, for each ε and λ ( 0 , 1 ] , the following estimates hold:

(3.34) | Δ μ λ , ε | L 2 ( 0 , T ; H ) ε C * M 2 ,
(3.35) | 𝒖 λ , ε | L 2 ( 0 , T ; 𝑽 * ) 2 C * M 2 .

Proof.

From (3.9), we easily see that

| Δ μ λ , ε | L 2 ( 0 , T ; H ) = | ε t u λ , ε | L 2 ( 0 , T ; H ) ε C * M 2 .

Next, we separate 𝒖 λ , ε as follows:

𝒖 λ , ε = ( ( 1 - ε ) t u λ , ε , 0 ) + ( ε t u λ , ε , t u Γ , λ , ε ) .

Then the first term of the right-hand side is estimated as follows:

(3.36) | ( ( 1 - ε ) t u λ , ε , 0 ) | L 2 ( 0 , T ; 𝑽 * ) = | ( 1 - ε ) t u λ , ε | L 2 ( 0 , T ; H ) ( 1 - ε ) C * M 2 C * M 2 .

Moreover, for the second term, we see that

(3.37) | ( ε t u λ , ε , t u Γ , λ , ε ) | L 2 ( 0 , T ; 𝑽 * ) C * M 2 .

Indeed, from (3.9) and (3.11), we have

0 T ( ε t u λ , ε ( t ) , η ( t ) ) H 𝑑 t + 0 T ( t u Γ , λ , ε ( t ) , η Γ ( t ) ) H Γ 𝑑 t = 0 T ( μ λ , ε ( t ) , η ( t ) ) H 𝑑 t + 0 T ( Γ μ Γ , λ , ε ( t ) , Γ η Γ ( t ) ) H Γ 𝑑 t

for all 𝜼 L 2 ( 0 , T ; 𝑽 ) . Therefore, estimate (3.22) for μ λ , ε and μ Γ , λ , ε imply (3.37). Thus, using (3.36)–(3.37), we show (3.35). ∎

We have obtained the uniform estimates (3.21), (3.22), (3.34), (3.35) provided in Lemmas 3.2 and 3.3 independent of ε and λ ( 0 , 1 ] , actually C * 1 as ε 0 . They can be used throughout this paper by considering the limiting procedure ε 0 for each fixed λ ( 0 , 1 ] , and next the limiting procedure λ 0 .

4 Proof of the main theorem

In this section, we prove Theorem 2.3.

4.1 Additional uniform estimate independent of ε ( 0 , 1 ]

In this subsection, we obtain additional uniform estimates independent of ε ( 0 , 1 ] , which may depend on λ ( 0 , 1 ] . Therefore, we use them only in the nest subsection to consider the limiting procedure as ε 0 .

Lemma 4.1.

There exist two positive constants M 3 ( λ ) and M 4 ( λ ) , depending on λ ( 0 , 1 ] but independent of ε ( 0 , 1 ] , such that

(4.1) | β λ ( u λ , ε ) | L ( 0 , T ; H ) M 3 ( λ ) , | β Γ , λ ( u Γ , λ , ε ) | L ( 0 , T ; H Γ ) M 3 ( λ ) ,
(4.2) | μ Γ , λ , ε | L 2 ( 0 , T ; V Γ ) M 4 ( λ ) , | μ λ , ε | L 2 ( 0 , T ; V ) M 4 ( λ ) .

Proof.

From the Lipschitz continuity of β λ and β Γ , λ , we see from (3.21) that there exists a positive constant M 3 ( λ ) , which is proportional to the Lipschitz constants 1 / λ of β λ and 1 / ( λ ϱ ) of β Γ , λ , such that (4.1) holds. Next, let us point out the variational equality, deduced from (3.10) and (3.12):

ε ( μ λ , ε ( s ) , z ) H + ( μ Γ , λ , ε ( s ) , z Γ ) H Γ = τ ( s u λ , ε ( s ) , z ) H + ( u λ , ε ( s ) , z ) H
+ ( β λ ( u λ , ε ( s ) ) + π ( u λ , ε ( s ) ) - f ε ( s ) , z ) H
+ ε ( s u Γ , λ , ε ( s ) , z Γ ) H Γ + ( Γ u Γ , λ , ε ( s ) , Γ z Γ ) H Γ
(4.3) + ( β Γ , λ ( u Γ , λ , ε ( s ) ) + π Γ ( u Γ , λ , ε ( s ) ) - f Γ , ε ( s ) , z Γ ) H Γ

for all 𝒛 𝑽 and a.a. s ( 0 , T ) . Taking now 𝒛 := 𝝁 λ , ε ( s ) in (4.3), integrating with respect to time, we infer, with the help of Poincaré–Wirtinger inequality (3.14), that

ε 0 t | μ λ , ε ( s ) | H 2 d s + 0 t | μ Γ , λ , ε ( s ) | H Γ 2 d s
C P τ | s u λ , ε | L 2 ( 0 , T ; H ) { | μ λ , ε | L 2 ( 0 , T ; H ) 2 + | μ Γ , λ , ε | L 2 ( 0 , T ; H Γ ) 2 } 1 / 2
    + | u λ , ε | L 2 ( 0 , T ; H ) | μ λ , ε | L 2 ( 0 , T ; H )
    + C P | β λ ( u λ , ε ) + π ( u λ , ε ) - f ε | L 2 ( 0 , T ; H ) { | μ λ , ε | L 2 ( 0 , T ; H ) 2 + | μ Γ , λ , ε | L 2 ( 0 , T ; H Γ ) 2 } 1 / 2
    + ε | s u Γ , λ , ε | L 2 ( 0 , T ; H Γ ) | μ Γ , λ , ε | L 2 ( 0 , T ; H Γ ) + | Γ u Γ , λ , ε | L 2 ( 0 , T ; H Γ ) | Γ μ Γ , λ , ε | L 2 ( 0 , T ; H Γ )
    + | β Γ , λ ( u Γ , λ , ε ) + π Γ ( u Γ , λ , ε ) - f Γ , ε | L 2 ( 0 , T ; H Γ ) | μ Γ , λ , ε | L 2 ( 0 , T ; H Γ )

for all t [ 0 , T ] . Then, using the Young inequality along with (3.3), (3.21), (3.22) and (4.1), we deduce the uniform estimate of { μ Γ , λ , ε } ε ( 0 , 1 ] in L 2 ( 0 , T ; V Γ ) . Next, with the help of the Poincaré–Wirtinger inequality (3.14) again, we deduce the uniform estimate of { μ λ , ε } ε ( 0 , 1 ] in L 2 ( 0 , T ; H ) . Thus, combining the resultant with (3.22), we see that there exists a positive constant M 4 ( λ ) , depending on M 3 ( λ ) but independent of ε ( 0 , 1 ] , such that (4.2) holds. ∎

Lemma 4.2.

There exists a positive constant M 5 ( λ ) , depending on λ ( 0 , 1 ] but independent of ε ( 0 , 1 ] , such that

(4.4) | u λ , ε | L 2 ( 0 , T ; W ) + | u Γ , λ , ε | L 2 ( 0 , T ; W Γ ) M 5 ( λ ) .

Proof.

We can compare the terms in (3.10) and conclude that { | Δ u λ , ε | L 2 ( 0 , T ; H ) } ε ( 0 , 1 ] is uniformly bounded. Hence, applying the theory of elliptic regularity (see, e.g., [23, Theorem 3.2, p. 1.79]), we have that

| u λ , ε | L 2 ( 0 , T ; H 3 / 2 ( Ω ) ) M ~ 5 ( λ )

for some positive constant M ~ 5 ( λ ) , and owing to both the uniform bounds, we see that

| 𝝂 u λ , ε | L 2 ( 0 , T ; H Γ ) M ~ 5 ( λ ) .

Next, by comparison in (3.12), { | Δ Γ u Γ , λ , ε | L 2 ( 0 , T ; H Γ ) } ε ( 0 , 1 ] is uniformly bounded and, consequently (see, e.g., [21, Section 4.2]),

| u Γ , λ , ε | L 2 ( 0 , T ; W Γ ) ( | u Γ , λ , ε | L 2 ( 0 , T ; V Γ ) 2 + | Δ Γ u Γ , λ , ε | L 2 ( 0 , T ; H Γ ) 2 ) 1 / 2 M 5 ( λ )

for some constant M 5 ( λ ) . Then, using the theory of elliptic regularity (see, e.g., [23, Theorem 3.2, p. 1.79]), we get (4.4). ∎

4.2 Passage to the limit as ε 0

In this subsection, we pass to the limit in the approximating problem as ε 0 . Indeed, owing to the estimates stated in Lemmas 3.2, 3.3, 4.1 and 4.2, there exist a subsequence of ε (not relabeled) and some limit functions u λ , u Γ , λ , μ λ and μ Γ , λ such that

(4.5) u λ , ε u λ weakly star in  H 1 ( 0 , T ; H ) L ( 0 , T ; V ) L 2 ( 0 , T ; W ) ,
(4.6) μ λ , ε μ λ weakly in  L 2 ( 0 , T ; V ) ,
(4.7) u Γ , λ , ε u Γ , λ weakly star in  H 1 ( 0 , T ; V Γ * ) L ( 0 , T ; V Γ ) L 2 ( 0 , T ; W Γ ) ,
(4.8) ε t u Γ , λ , ε 0 strongly in  L 2 ( 0 , T ; H Γ ) ,
(4.9) μ Γ , λ , ε μ Γ , λ weakly in  L 2 ( 0 , T ; V Γ )

as ε 0 . From (4.5) and (4.7), using well-known compactness results (see, e.g., [30, Section 8, Corollary 4]), we obtain

(4.10) u λ , ε u λ strongly in  C ( [ 0 , T ] ; H ) L 2 ( 0 , T ; V ) ,
(4.11) u Γ , λ , ε u Γ , λ strongly in  C ( [ 0 , T ] ; H Γ ) L 2 ( 0 , T ; V Γ )

as ε 0 . Then (4.5), (4.10) and (4.11) imply that

(4.12) ε t u λ , ε 0 strongly in  L 2 ( 0 , T ; H ) ,
(4.13) β λ ( u λ , ε ) β λ ( u λ ) strongly in  C ( [ 0 , T ] ; H ) ,
(4.14) β Γ , λ ( u Γ , λ , ε ) β Γ , λ ( u Γ , λ ) strongly in  C ( [ 0 , T ] ; H Γ )
(4.15) π ( u λ , ε ) π ( u λ ) strongly in  C ( [ 0 , T ] ; H ) ,
(4.16) π Γ ( u Γ , λ , ε ) π Γ ( u Γ , λ ) strongly in  C ( [ 0 , T ] ; H Γ )

as ε 0 . We point out that (3.5), (4.10) and (4.11) entail that

(4.17) u λ ( 0 ) = u 0 a.e. in  Ω , u Γ , λ ( 0 ) = u 0 Γ a.e. on  Γ .

From (3.9) and (3.11) it follows that

ε ( s u λ , ε ( s ) , z ) H + ( s u Γ , λ , ε ( s ) , z Γ ) H Γ + ( μ λ , ε ( s ) , z ) H + ( Γ μ Γ , λ , ε ( s ) , Γ z Γ ) H Γ = 0

for all 𝒛 𝑽 and a.a. s ( 0 , T ) . Then using (4.5)–(4.16) we can pass to the limit in this variational equality and in (4.3) obtaining

(4.18) ( s u Γ , λ ( s ) , z Γ ) H Γ + ( μ λ ( s ) , z ) H + ( Γ μ Γ , λ ( s ) , Γ z Γ ) H Γ = 0

and

( μ Γ , λ ( s ) , z Γ ) H Γ = τ ( t u λ ( s ) , z ) H + ( u λ ( s ) , z ) H + ( β λ ( u λ ( s ) ) + π ( u λ ( s ) ) - f ( s ) , z ) H
(4.19) + ( Γ u Γ , λ ( s ) , Γ z Γ ) H Γ + ( β Γ , λ ( u Γ , λ ( s ) ) + π Γ ( u Γ , λ ( s ) ) - f Γ ( s ) , z Γ ) H Γ

for all 𝒛 𝑽 . Moreover, from the a priori estimates shown in Lemmas 3.2 and 3.3, we have

(4.20) | u λ | L ( 0 , T ; V ) M 1 , | u Γ , λ | L ( 0 , T ; H Γ ) c 1 1 / 2 M 1 ,
| t u λ | L 2 ( 0 , T ; H ) + | Γ u Γ , λ | L ( 0 , T ; H Γ ) + | β ^ λ ( u λ ) | L ( 0 , T ; L 1 ( Ω ) )
(4.21) + | β ^ Γ , λ ( u Γ , λ ) | L ( 0 , T ; L 1 ( Γ ) ) + | μ λ | L 2 ( 0 , T ; H ) + | Γ μ Γ , λ | L 2 ( 0 , T ; H Γ ) M 2 ,
(4.22) | t u Γ , λ | L 2 ( 0 , T ; V Γ * ) 2 M 2 ,

because C * 1 as ε 0 . As a remark, taking z = 1 , z Γ = 1 in (4.18) and using (4.11), (4.17), we obtain that

(4.23) Γ u λ ( t ) 𝑑 Γ = Γ u 0 Γ 𝑑 Γ for all  t [ 0 , T ] .

4.3 Proof of Theorem 2.3

In this subsection, we prove the main theorem. To do so, we are going to produce estimates independent of λ, and then pass to the limit as λ 0 . The point of emphasis is the effective usage of the mean value zero function.

Lemma 4.3.

There exists a positive constant M 6 , independent of λ ( 0 , 1 ] , such that

(4.24) | β λ ( u λ ) | L 2 ( 0 , T ; L 1 ( Ω ) ) + | β Γ , λ ( u Γ , λ ) | L 2 ( 0 , T ; L 1 ( Γ ) ) M 6 .

Proof.

Recall (2.7) and take

z := u λ ( s ) - m Γ , z Γ := u Γ , λ ( s ) - m Γ

in (4.19). Then we have

τ ( t u λ ( s ) , u λ ( s ) - m Γ ) H + | u λ ( s ) | H 2 + ( β λ ( u λ ( s ) ) , u λ ( s ) - m Γ ) H
+ ( π ( u λ ( s ) ) - f ( s ) , u λ ( s ) - m Γ ) H + | Γ u Γ , λ ( s ) | H Γ 2 + ( β Γ , λ ( u Γ , λ ( s ) ) , u Γ , λ ( s ) - m Γ ) H Γ
(4.25) + ( π Γ ( u Γ , λ ( s ) ) - f Γ ( s ) , u Γ , λ ( s ) - m Γ ) H Γ = ( μ Γ , λ ( s ) , u Γ , λ ( s ) - m Γ ) H Γ .

Let now ( y , y Γ ) be the solution of the following problem:

(4.26) Ω y z d x + Γ Γ y Γ Γ z Γ d Γ = Γ ( u Γ , λ - m Γ ) z Γ 𝑑 Γ for all  𝒛 𝑽 ,
y Γ = y | Γ a.e. on  Γ , Γ y Γ 𝑑 Γ = 0 a.e. in  ( 0 , T ) .

This problem has one and only one solution, see [5, Appendix, Lemma A] and repeat the proof with the different condition. Moreover, due to (3.14) and (3.15), there exists a positive constant M ~ 6 , independent of λ ( 0 , 1 ] , such that

(4.27) | y ( s ) | V 2 + | y Γ ( s ) | V Γ 2 M ~ 6 | u Γ , λ ( s ) - m Γ | H Γ 2

for a.a. s ( 0 , T ) . Taking z := μ λ ( s ) , z Γ := μ Γ , λ in (4.26), we see that the right-hand side of (4.26) is equal to

( y ( s ) , μ λ ( s ) ) H + ( Γ y Γ ( s ) , Γ μ Γ , λ ( s ) ) H Γ

and, consequently, we use (4.18) with z := y ( s ) , z Γ := y Γ ( s ) to conclude that

( μ Γ , λ ( s ) , u Γ , λ ( s ) - m Γ ) H Γ = - ( s u Γ , λ ( s ) , y Γ ( s ) ) H Γ .

Then, from (4.25) and the properties

β λ ( r ) ( r - m Γ ) δ 0 | β λ ( r ) |