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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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

Pierluigi ColliORCID iD: https://orcid.org/0000-0002-7921-5041 / Takeshi FukaoORCID iD: https://orcid.org/0000-0003-4899-6890
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/anona-2018-0055

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.

Keywords: Cahn–Hilliard equation; bulk condition; dynamic boundary condition; well-posedness

MSC 2010: 35K61; 35K25; 35D30; 58J35; 80A22

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 Ωd, 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:

tuΓ-ΔΓμΓ=-𝝂μon Σ:=Γ×(0,T),(1.1)μΓ=-ΔΓuΓ+𝒲Γ(uΓ)-fΓ+𝝂uon Σ,(1.2)

where t denotes the partial derivative with respect to time, and ΔΓ denotes the Laplace–Beltrami operator on Γ (see, e.g., [21, Chapter 3]). Here, the unknowns uΓ and μΓ: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:

μ|Σ=μΓ,u|Σ=uΓon Σ,(1.3)

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:

-Δμ=0in Q,(1.4)τtu-Δu+𝒲(u)=fin Q,(1.5)

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

Note that the nonlinear terms 𝒲Γ and 𝒲 are the derivatives of the functions 𝒲Γ and 𝒲, usually referred as double-well potentials, with two minima and a local unstable maximum in between. The prototype model is provided by 𝒲Γ(r)=𝒲(r)=(1/4)(r2-1)2, so that 𝒲Γ(r)=𝒲(r)=r3-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 𝒲Γ 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

uΓ(0)=u0Γon Γ,u(0)=u0in Ω,(1.6)

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:

ΓuΓ(t)𝑑Γ=Γu0Γ𝑑Γfor all t[0,T].(1.7)

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

εtu-Δμ=0in Q,(1.8)εμ=τtu-Δu+𝒲(u)-fin Q,(1.9)u|Σ=uΓ,μ|Σ=μΓon Σ,(1.10)tuΓ+𝝂μ-ΔΓμΓ=0on Σ,(1.11)μΓ=𝝂u-ΔΓuΓ+𝒲Γ(uΓ)-fΓon Σ,(1.12)

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

εΩu(t)𝑑x+ΓuΓ(t)𝑑Γ=εΩu0𝑑x+Γu0Γ𝑑Γfor all t[0,T].(1.13)

Our essential idea of the existence proof is to be able to pass to the limit as ε0 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 ε0. In the second step, we improve suitable estimates in order to make them independent of λ. Then we can pass to the limit as λ0 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

-Δμ=0a.e. in Q,(2.1)τtu-Δu+ξ+π(u)=f,ξβ(u)a.e. in Q,(2.2)uΓ=u|Σ,μΓ=μ|Σ,tuΓ+𝝂μ-ΔΓμΓ=0a.e. on Σ,(2.3)μΓ=𝝂u-ΔΓuΓ+ξΓ+πΓ(uΓ)-fΓ,ξΓβΓ(uΓ)a.e. on Σ,(2.4)u(0)=u0a.e. in Ω,uΓ(0)=u0Γa.e. on Γ,(2.5)

where f:Q, fΓ:Σ are given sources, u0:Ω, u0Γ: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 βΓ and πΓ.

Typical examples of the nonlinearities β, π are given by

  • β(r)=r3, π(r)=-r for all r, with D(β)= for the prototype double well potential

    𝒲(r)=14(r2-1)2.

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

    𝒲(r)=((1+r)ln(1+r)+(1-r)ln(1-r))-cr2,

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

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

    𝒲(r)=I[-1,1](r)-12r2,

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

Similar choices can be considered for βΓ,πΓ and the related potential 𝒲Γ. 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 βΓ (see the later assumption A5) that forces the growth of β to be controlled by the growth of βΓ. 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 τ0. 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:=L2(Ω), HΓ:=L2(Γ), V:=H1(Ω), VΓ:=H1(Γ), W:=H2(Ω), WΓ:=H2(Γ), with usual norms and inner products; we denote them by ||H, ||HΓ and (,)H, (,)HΓ, 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×HΓ and 𝑽:={(z,zΓ)V×VΓ:zΓ=z|Γa.e. on Γ}. Then 𝑯 and 𝑽 are Hilbert spaces with the inner products

(𝒖,𝒛)𝑯:=(u,z)H+(uΓ,zΓ)HΓfor all 𝒖:=(u,uΓ),𝒛:=(z,zΓ)𝑯,(𝒖,𝒛)𝑽:=(u,z)V+(uΓ,zΓ)VΓfor all 𝒖:=(u,uΓ),𝒛:=(z,zΓ)𝑽

and related norms. As a remark, if 𝒛:=(z,zΓ)𝑽, then zΓ is the trace z|Γ of z on Γ, while if 𝒛:=(z,zΓ)𝑯, then zH and zΓHΓ 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,zΓ) 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

ΓuΓ(t)𝑑Γ=Γu0Γ𝑑Γfor all t[0,T];(2.6)

hereafter, we put

mΓ:=1|Γ|Γu0Γ𝑑Γ,(2.7)

where |Γ|:=Γ1𝑑Γ. 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,𝒛𝑽*,𝑽=(𝒖,𝒛)𝑯 for all 𝒖𝑯, 𝒛𝑽. Then we obtain 𝑽𝑯𝑽*, 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,uΓ), 𝝁=(μ,μΓ), 𝝃=(ξ,ξΓ) satisfy

uH1(0,T;H)C([0,T];V)L2(0,T;W),uΓH1(0,T;VΓ*)L(0,T;VΓ)L2(0,T;WΓ),μL2(0,T;V),μΓL2(0,T;VΓ),ξL2(0,T;H),ξΓL2(0,T;HΓ),u|Σ=uΓ,μ|Σ=μΓa.e. on Σ,ξβ(u)a.e. in Q,ξΓβΓ(uΓ)a.e. on Σ,

solve

uΓ(t),zΓVΓ*,VΓ+Ωμ(t)zdx+ΓΓμΓ(t)ΓzΓdΓ=0(2.8)

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

τtu-Δu+ξ+π(u)=fa.e. in Q,(2.9)μΓ=𝝂u-ΔΓuΓ+ξΓ+πΓ(uΓ)-fΓa.e. on Σ,(2.10)u(0)=u0a.e. in Ω,uΓ(0)=u0Γa.e. on Γ.(2.11)

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 z𝒟(Ω), taking 𝒛:=(z,0) in (2.8) and using the trace condition on μ, we deduce that

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

for a.a. t(0,T), whence the regularities μL2(0,T;V), μΓL2(0,T;VΓ) allow us to infer the higher regularity μL2(0,T;H3/2(Ω)). Moreover, the boundedness Δμ(=0) in L2(0,T;H) gives us the property 𝝂μL2(0,T;HΓ), as well as

uΓ(t)+𝝂μ(t),zΓVΓ*,VΓ+ΓΓμΓ(t)ΓzΓdΓ=0for all zΓVΓ,

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:

  • (A1)

    fL2(0,T;H) and fΓW1,1(0,T;HΓ).

  • (A2)

    𝒖0:=(u0,u0Γ)𝑽.

  • (A3)

    The maximal monotone graphs β and βΓ in × are the subdifferentials β=β^ and βΓ=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(βΓ), respectively. This implies that 0β(0) and 0βΓ(0).

  • (A4)

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

  • (A5)

    D(βΓ)D(β) and there exist positive constants ϱ,c0>0 such that

    |β(r)|ϱ|βΓ(r)|+c0for all rD(βΓ),(2.12)

    where β and βΓ denote the minimal sections of β and βΓ.

  • (A6)

    mΓintD(βΓ) and the compatibility conditions β^(u0)L1(Ω), β^Γ(u0Γ)L1(Γ) hold.

The minimal section β of β is specified by β(r):={r*β(r):|r*|=minsβ(r)|s|} and the same definition applies to βΓ. These assumptions are the same as in [2, 5]. Concerning assumption (A1), let us note that the regularity conditions for f and fΓ 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 τtu 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:

  • (A1)’

    fL2(0,T;V*) and fΓL2(0,T;VΓ*),

  • (A2)’

    u0H and u0ΓVΓ*.

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), 𝐮0(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, LΓ and T, such that

|u(1)(t)-u(2)(t)|H2+|uΓ(1)(t)-uΓ(2)(t)|VΓ*2+0t|u(1)(s)-u(2)(s)|V2ds+0t|uΓ(1)(s)-uΓ(2)(s)|VΓ2dsC{|u0(1)-u0(2)|H2+|u0Γ(1)-u0Γ(2)|VΓ*2+0t|f(1)(s)-f(2)(s)|V*2ds+0t|fΓ(1)(s)-fΓ(2)(s)|VΓ*2ds}(2.13)

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 βλ,βΓ,λ:, along with the associated resolvent operators Jλ,JΓ,λ:, by

βλ(r):=1λ(r-Jλ(r)):=1λ(r-(I+λβ)-1(r)),βΓ,λ(r):=1λϱ(r-JΓ,λ(r)):=1λϱ(r-(I+λϱβΓ)-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 λϱ 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 βλ(0)=βΓ,λ(0)=0. Moreover, the related Moreau–Yosida regularizations β^λ,β^Γ,λ of β^,β^Γ: fulfill

β^λ(r):=infs{12λ|r-s|2+β^(s)}=12λ|r-Jλ(r)|2+β^(Jλ(r))=0rβλ(s)𝑑s,β^Γ,λ(r):=infs{12λϱ|r-s|2+β^Γ(s)}=0rβΓ,λ(s)𝑑sfor all r.

It is well known that βλ is Lipschitz continuous with Lipschitz constant 1/λ and βΓ,λ is also Lipschitz continuous with constant 1/(λϱ). In addition, for each λ(0,1], we have the standard properties

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

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

|βλ(r)|ϱ|βΓ,λ(r)|+c0for all r,(3.2)

for all λ(0,1], with the same constants ϱ and c0 as in (2.12).

Now for each ε(0,1], let 𝒇ε:=(fε,fΓ,ε) and 𝒖0,ε:=(u0,ε,u0Γ,ε) be smooth approximations for 𝒇 and 𝒖0, so that 𝒇εH1(0,T;𝑯) with 𝒇ε(0)𝑽 and 𝒖0,ε𝑾𝑽 with (-Δu0,ε,𝝂u0,ε-ΔΓu0Γ,ε)𝑽 satisfying

𝒇ε𝒇strongly in L2(0,T;𝑯) as ε0,(3.3)|fΓ,ε-fΓ|L2(0,T;HΓ)ε1/2C0for all ε(0,1],(3.4)𝒖0,ε𝒖0strongly in 𝑽 as ε0,(3.5)Ωβ^λ(u0,ε)𝑑xC0,Γβ^Γ,λ(u0Γ,ε)𝑑Γ(1+ε1/2λ)C0for all ε(0,1],(3.6)

where C0 is a positive constant independent of ε,λ(0,1]. Indeed, 𝒇ε and 𝒖0,ε satisfying (3.3)–(3.6) are given in Appendix A. Hereafter, we use C*:=(1+ε1/2/λ)1/2, which satisfies C*1 as ε0 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

𝒖λ,ε:=(uλ,ε,uΓ,λ,ε)W1,(0,T;𝑯)H1(0,T;𝑽)L(0,T;𝑾),(3.7)𝝁λ,ε:=(μλ,ε,μΓ,λ,ε)L(0,T;𝑽)L2(0,T;𝑾),(3.8)

satisfying

εtuλ,ε-Δμλ,ε=0a.e. in Q,(3.9)εμλ,ε=τtuλ,ε-Δuλ,ε+βλ(uλ,ε)+π(uλ,ε)-fεa.e. in Q,(3.10)tuΓ,λ,ε+𝝂μλ,ε-ΔΓμΓ,λ,ε=0a.e. on Σ,(3.11)μΓ,λ,ε=εtuΓ,λ,ε+𝝂uλ,ε-ΔΓuΓ,λ,ε+βΓ,λ(uΓ,λ,ε)+πΓ(uΓ,λ,ε)-fΓ,εa.e. on Σ,(3.12)uλ,ε(0)=u0,εa.e. in Ω,uΓ,λ,ε(0)=u0Γ,εa.e. on Γ.(3.13)

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 CP such that

|z|H2CP{Ω|z|2dx+Γ|z|Γ|2dΓ}for all zV,(3.14)|zΓ|HΓ2CPΓ|ΓzΓ|2𝑑Γfor all zΓVΓ, with ΓzΓ𝑑Γ=0,(3.15)|𝒛|𝑽2CP{Ω|z|2dx+Γ|ΓzΓ|2dΓ}for all 𝒛𝑽, with ΓzΓ𝑑Γ=0.(3.16)

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 Cδ depending on δ such that

|z|H1-α(Ω)2δ|z|V2+Cδ|z|H2for all zV and all α(0,1),(3.17)|zΓ|HΓ2δ|zΓ|VΓ2+Cδ|zΓ|VΓ*2for all zΓVΓ,(3.18)

because we have the compact embeddings VH1-α(Ω)H for α(0,1), (see, e.g., [27, Chapter 1, Theorem 16.1]) and VΓHΓVΓ*, respectively. Next, from the standard theorem for the trace operators γ0(z):=z|Γ of γ0:VH1/2(Γ) and γ0:H1-α(Ω)H(1-α)-1/2(Γ)HΓ 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 c1 such that

|zΓ|HΓ2|zΓ|H1/2(Γ)2c1|z|V2,|zΓ|HΓ2c1|z|H1-α(Ω)2for all 𝒛𝑽,(3.19)

for α(0,1/2). Moreover, the boundedness of some recovering operator :H1/2(Γ)V of the trace γ0 gives us

|zΓ|V2c2|zΓ|H1/2(Γ)2c2|zΓ|VΓ2for all zΓVΓ,(3.20)

where c2 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 M1 and M2, depending on τ but independent of ε and λ(0,1], such that

|uλ,ε|L(0,T;V)C*M1,|uΓ,λ,ε|L(0,T;HΓ)c11/2C*M1,(3.21)|tuλ,ε|L2(0,T;H)+ε1/2|tuΓ,λ,ε|L2(0,T;HΓ)+|ΓuΓ,λ,ε|L(0,T;HΓ)+|β^λ(uλ,ε)|L(0,T;L1(Ω))+|β^Γ,λ(uΓ,λ,ε)|L(0,T;L1(Γ))+|μλ,ε|L2(0,T;H)+|ΓμΓ,λ,ε|L2(0,T;HΓ)C*M2.(3.22)

Proof.

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

τ0t|suλ,ε(s)|H2ds+12|uλ,ε(t)|H2+Ωβ^λ(uλ,ε(t))dx+Ωπ^(uλ,ε(t))dx-0t(𝝂uλ,ε(s),suΓ,λ,ε(s))HΓ𝑑s-0t(fε(s),suλ,ε(s))H𝑑s=ε0t(μλ,ε(s),suλ,ε(s))H𝑑s+12|u0,ε|H2+Ωβ^λ(u0,ε)𝑑x+Ωπ^(u0,ε)𝑑x(3.23)

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

-0t(𝝂uλ,ε(s),suΓ,λ,ε(s))HΓds=ε0t|suΓ,λ,ε(s)|HΓ2ds-0t(μΓ,λ,ε(s),suΓ,λ,ε(s))HΓds+12|uΓ,λ,ε(t)|HΓ2-12|u0Γ,ε|HΓ2+Γβ^Γ,λ(uΓ,λ,ε(t))𝑑Γ+Γπ^Γ(uΓ,λ,ε(t))𝑑Γ-Γβ^Γ,λ(u0Γ,ε)𝑑Γ-Γπ^Γ(u0Γ,ε)𝑑Γ-0t(fΓ,ε(s),suΓ,λ,ε(s))HΓ𝑑s(3.24)

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

ε0t(μλ,ε(s),suλ,ε(s))Hds+0t(μΓ,λ,ε(s),suΓ,λ,ε(s))HΓds=-0t|μλ,ε(s)|H2ds-0t|ΓμΓ,λ,ε(s)|HΓ2ds(3.25)

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

τ0t|suλ,ε(s)|H2ds+ε0t|suΓ,λ,ε(s)|HΓ2ds+12|uλ,ε(t)|H2+Ωβ^λ(uλ,ε(t))dx+12|uΓ,λ,ε(t)|HΓ2+Γβ^Γ,λ(uΓ,λ,ε(t))dΓ+0t|μλ,ε(s)|H2ds+0t|ΓμΓ,λ,ε(s)|HΓ2ds12|u0,ε|H2+Ωβ^λ(u0,ε)dx+Ω|π^(uλ,ε(t))|dx+Ω|π^(u0,ε)|dx+12|u0Γ,ε|HΓ2+Γβ^Γ,λ(u0Γ,ε)dΓ+Γ|π^Γ(uΓ,λ,ε(t))|dΓ+Γ|π^Γ(u0Γ,ε)|dΓ+0t(fε(s),suλ,ε(s))H𝑑s+0t(fΓ,ε(s),suΓ,λ,ε(s))HΓ𝑑s(3.26)

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

12|uλ,ε(s)|H2=0t(suλ,ε(s),uλ,ε(s))H𝑑s+12|u0,ε|H2δ~20t|suλ,ε(s)|H2ds+12δ~0t|uλ,ε(s)|H2ds+12|u0,ε|H2(3.27)

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)|0r|π(l)|dlL2r2for all r.

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

Ω|π^(uλ,ε(t))|dxL2Ω|uλ,ε(t)|2dxτ100t|suλ,ε(s)|H2ds+5L22τ0t|uλ,ε(s)|H2ds+L2|u0,ε|H2(3.28)

and, analogously,

Ω|π^(u0,ε)|dxL2Ω|u0,ε|2dx,Γ|π^Γ(u0Γ,ε)|dΓLΓ2Γ|u0Γ,ε|2dΓ.(3.29)

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

Γ|π^Γ(uΓ,λ,ε(t))|dΓLΓ2Γ|uΓ,λ,ε(t)|2dΓc1LΓ2|uλ,ε(t)|H1-α(Ω)2δ|uλ,ε(t)|V2+Cδ|uλ,ε(t)|H2δ|uλ,ε(t)|V2+τ100t|suλ,ε(s)|H2ds+10Cδ2τ0t|uλ,ε(s)|H2ds+Cδ|u0,ε|H2(3.30)

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

0t(fε(s),suλ,ε(s))Hdsτ100t|suλ,ε(s)|H2ds+52τ0t|fε(s)|H2ds,(3.31)0t(fΓ,ε(s),suΓ,λ,ε(s))HΓdsε20t|suΓ,λ,ε(s)|HΓ2ds+12ε0t|fΓ,ε(s)|HΓ2ds(3.32)

for all t[0,T]. Therefore, collecting (3.28)–(3.32), adding (1/2)|uλ,ε(s)|H2 to both sides of (3.26) and using (3.27) with δ~:=τ/5, we obtain

τ20t|suλ,ε(s)|H2ds+ε20t|suΓ,λ,ε(s)|HΓ2ds+12|uλ,ε(t)|V2+Ωβ^λ(uλ,ε(t))dx+12|uΓ,λ,ε(t)|HΓ2+Γβ^Γ,λ(uΓ,λ,ε(t))dΓ+0t|μλ,ε(s)|H2ds+0t|ΓμΓ,λ,ε(s)|HΓ2ds52τ0t|uλ,ε(s)|H2ds+12|u0,ε|V2+Ωβ^λ(u0,ε)dx+5L22τ0t|uλ,ε(s)|H2ds+L|u0,ε|H2+52τ0t|fε(s)|H2ds+12|u0Γ,ε|VΓ2+Γβ^Γ,λ(u0Γ,ε)dΓ+δ|uλ,ε(t)|V2+10Cδ2τ0t|uλ,ε(s)|H2ds+Cδ|u0,ε|H2+LΓ2|u0Γ,ε|HΓ2+12ε0t|fΓ,ε(s)|HΓ2ds(3.33)

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 M1, depending on |u0|V, C0, L, |f|L2(0,T;H), |u0Γ|VΓ, LΓ, |fΓ|L2(0,T;HΓ) and τ, in which M1+ as τ0, independent of ε and λ, such that

|uλ,ε|L(0,T;V)C*M1.

Moreover, using this, (3.19) implies

|uΓ,λ,ε|L(0,T;HΓ)c11/2C*M1,

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

By comparison, the following estimates for Δμλ,ε and 𝒖λ,ε:=(tuλ,ε,tuΓ,λ,ε) are obtained.

Lemma 3.3.

Let M2 be the same constant as in Lemma 3.2. Then, for each ε and λ(0,1], the following estimates hold:

|Δμλ,ε|L2(0,T;H)εC*M2,(3.34)|𝒖λ,ε|L2(0,T;𝑽*)2C*M2.(3.35)

Proof.

From (3.9), we easily see that

|Δμλ,ε|L2(0,T;H)=|εtuλ,ε|L2(0,T;H)εC*M2.

Next, we separate 𝒖λ,ε as follows:

𝒖λ,ε=((1-ε)tuλ,ε,0)+(εtuλ,ε,tuΓ,λ,ε).

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

|((1-ε)tuλ,ε,0)|L2(0,T;𝑽*)=|(1-ε)tuλ,ε|L2(0,T;H)(1-ε)C*M2C*M2.(3.36)

Moreover, for the second term, we see that

|(εtuλ,ε,tuΓ,λ,ε)|L2(0,T;𝑽*)C*M2.(3.37)

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

0T(εtuλ,ε(t),η(t))H𝑑t+0T(tuΓ,λ,ε(t),ηΓ(t))HΓ𝑑t=0T(μλ,ε(t),η(t))H𝑑t+0T(ΓμΓ,λ,ε(t),ΓηΓ(t))HΓ𝑑t

for all 𝜼L2(0,T;𝑽). Therefore, estimate (3.22) for μλ,ε and μΓ,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 ε0. They can be used throughout this paper by considering the limiting procedure ε0 for each fixed λ(0,1], and next the limiting procedure λ0.

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 ε0.

Lemma 4.1.

There exist two positive constants M3(λ) and M4(λ), depending on λ(0,1] but independent of ε(0,1], such that

|βλ(uλ,ε)|L(0,T;H)M3(λ),|βΓ,λ(uΓ,λ,ε)|L(0,T;HΓ)M3(λ),(4.1)|μΓ,λ,ε|L2(0,T;VΓ)M4(λ),|μλ,ε|L2(0,T;V)M4(λ).(4.2)

Proof.

From the Lipschitz continuity of βλ and βΓ,λ, we see from (3.21) that there exists a positive constant M3(λ), which is proportional to the Lipschitz constants 1/λ of βλ and 1/(λϱ) of βΓ,λ, such that (4.1) holds. Next, let us point out the variational equality, deduced from (3.10) and (3.12):

ε(μλ,ε(s),z)H+(μΓ,λ,ε(s),zΓ)HΓ=τ(suλ,ε(s),z)H+(uλ,ε(s),z)H+(βλ(uλ,ε(s))+π(uλ,ε(s))-fε(s),z)H+ε(suΓ,λ,ε(s),zΓ)HΓ+(ΓuΓ,λ,ε(s),ΓzΓ)HΓ+(βΓ,λ(uΓ,λ,ε(s))+πΓ(uΓ,λ,ε(s))-fΓ,ε(s),zΓ)HΓ(4.3)

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

ε0t|μλ,ε(s)|H2ds+0t|μΓ,λ,ε(s)|HΓ2dsCPτ|suλ,ε|L2(0,T;H){|μλ,ε|L2(0,T;H)2+|μΓ,λ,ε|L2(0,T;HΓ)2}1/2   +|uλ,ε|L2(0,T;H)|μλ,ε|L2(0,T;H)   +CP|βλ(uλ,ε)+π(uλ,ε)-fε|L2(0,T;H){|μλ,ε|L2(0,T;H)2+|μΓ,λ,ε|L2(0,T;HΓ)2}1/2   +ε|suΓ,λ,ε|L2(0,T;HΓ)|μΓ,λ,ε|L2(0,T;HΓ)+|ΓuΓ,λ,ε|L2(0,T;HΓ)|ΓμΓ,λ,ε|L2(0,T;HΓ)   +|βΓ,λ(uΓ,λ,ε)+πΓ(uΓ,λ,ε)-fΓ,ε|L2(0,T;HΓ)|μΓ,λ,ε|L2(0,T;HΓ)

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 {μΓ,λ,ε}ε(0,1] in L2(0,T;VΓ). Next, with the help of the Poincaré–Wirtinger inequality (3.14) again, we deduce the uniform estimate of {μλ,ε}ε(0,1] in L2(0,T;H). Thus, combining the resultant with (3.22), we see that there exists a positive constant M4(λ), depending on M3(λ) but independent of ε(0,1], such that (4.2) holds. ∎

Lemma 4.2.

There exists a positive constant M5(λ), depending on λ(0,1] but independent of ε(0,1], such that

|uλ,ε|L2(0,T;W)+|uΓ,λ,ε|L2(0,T;WΓ)M5(λ).(4.4)

Proof.

We can compare the terms in (3.10) and conclude that {|Δuλ,ε|L2(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

|uλ,ε|L2(0,T;H3/2(Ω))M~5(λ)

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

|𝝂uλ,ε|L2(0,T;HΓ)M~5(λ).

Next, by comparison in (3.12), {|ΔΓuΓ,λ,ε|L2(0,T;HΓ)}ε(0,1] is uniformly bounded and, consequently (see, e.g., [21, Section 4.2]),

|uΓ,λ,ε|L2(0,T;WΓ)(|uΓ,λ,ε|L2(0,T;VΓ)2+|ΔΓuΓ,λ,ε|L2(0,T;HΓ)2)1/2M5(λ)

for some constant M5(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 ε0

In this subsection, we pass to the limit in the approximating problem as ε0. 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 uλ,uΓ,λ, μλ and μΓ,λ such that

uλ,εuλweakly star in H1(0,T;H)L(0,T;V)L2(0,T;W),(4.5)μλ,εμλweakly in L2(0,T;V),(4.6)uΓ,λ,εuΓ,λweakly star in H1(0,T;VΓ*)L(0,T;VΓ)L2(0,T;WΓ),(4.7)εtuΓ,λ,ε0strongly in L2(0,T;HΓ),(4.8)μΓ,λ,εμΓ,λweakly in L2(0,T;VΓ)(4.9)

as ε0. From (4.5) and (4.7), using well-known compactness results (see, e.g., [30, Section 8, Corollary 4]), we obtain

uλ,εuλstrongly in C([0,T];H)L2(0,T;V),(4.10)uΓ,λ,εuΓ,λstrongly in C([0,T];HΓ)L2(0,T;VΓ)(4.11)

as ε0. Then (4.5), (4.10) and (4.11) imply that

εtuλ,ε0strongly in L2(0,T;H),(4.12)βλ(uλ,ε)βλ(uλ)strongly in C([0,T];H),(4.13)βΓ,λ(uΓ,λ,ε)βΓ,λ(uΓ,λ)strongly in C([0,T];HΓ)(4.14)π(uλ,ε)π(uλ)strongly in C([0,T];H),(4.15)πΓ(uΓ,λ,ε)πΓ(uΓ,λ)strongly in C([0,T];HΓ)(4.16)

as ε0. We point out that (3.5), (4.10) and (4.11) entail that

uλ(0)=u0a.e. in Ω,uΓ,λ(0)=u0Γa.e. on Γ.(4.17)

From (3.9) and (3.11) it follows that

ε(suλ,ε(s),z)H+(suΓ,λ,ε(s),zΓ)HΓ+(μλ,ε(s),z)H+(ΓμΓ,λ,ε(s),ΓzΓ)HΓ=0

for all 𝒛𝑽 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

(suΓ,λ(s),zΓ)HΓ+(μλ(s),z)H+(ΓμΓ,λ(s),ΓzΓ)HΓ=0(4.18)

and

(μΓ,λ(s),zΓ)HΓ=τ(tuλ(s),z)H+(uλ(s),z)H+(βλ(uλ(s))+π(uλ(s))-f(s),z)H+(ΓuΓ,λ(s),ΓzΓ)HΓ+(βΓ,λ(uΓ,λ(s))+πΓ(uΓ,λ(s))-fΓ(s),zΓ)HΓ(4.19)

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

|uλ|L(0,T;V)M1,|uΓ,λ|L(0,T;HΓ)c11/2M1,(4.20)|tuλ|L2(0,T;H)+|ΓuΓ,λ|L(0,T;HΓ)+|β^λ(uλ)|L(0,T;L1(Ω))+|β^Γ,λ(uΓ,λ)|L(0,T;L1(Γ))+|μλ|L2(0,T;H)+|ΓμΓ,λ|L2(0,T;HΓ)M2,(4.21)|tuΓ,λ|L2(0,T;VΓ*)2M2,(4.22)

because C*1 as ε0. As a remark, taking z=1, zΓ=1 in (4.18) and using (4.11), (4.17), we obtain that

Γuλ(t)𝑑Γ=Γu0Γ𝑑Γfor all t[0,T].(4.23)

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 λ0. The point of emphasis is the effective usage of the mean value zero function.

Lemma 4.3.

There exists a positive constant M6, independent of λ(0,1], such that

|βλ(uλ)|L2(0,T;L1(Ω))+|βΓ,λ(uΓ,λ)|L2(0,T;L1(Γ))M6.(4.24)

Proof.

Recall (2.7) and take

z:=uλ(s)-mΓ,zΓ:=uΓ,λ(s)-mΓ

in (4.19). Then we have

τ(tuλ(s),uλ(s)-mΓ)H+|uλ(s)|H2+(βλ(uλ(s)),uλ(s)-mΓ)H+(π(uλ(s))-f(s),uλ(s)-mΓ)H+|ΓuΓ,λ(s)|HΓ2+(βΓ,λ(uΓ,λ(s)),uΓ,λ(s)-mΓ)HΓ+(πΓ(uΓ,λ(s))-fΓ(s),uΓ,λ(s)-mΓ)HΓ=(μΓ,λ(s),uΓ,λ(s)-mΓ)HΓ.(4.25)

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

Ωyzdx+ΓΓyΓΓzΓdΓ=Γ(uΓ,λ-mΓ)zΓ𝑑Γfor all 𝒛𝑽,(4.26)yΓ=y|Γa.e. on Γ,ΓyΓ𝑑Γ=0a.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

|y(s)|V2+|yΓ(s)|VΓ2M~6|uΓ,λ(s)-mΓ|HΓ2(4.27)

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

(y(s),μλ(s))H+(ΓyΓ(s),ΓμΓ,λ(s))HΓ

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

(μΓ,λ(s),uΓ,λ(s)-mΓ)HΓ=-(suΓ,λ(s),yΓ(s))HΓ.

Then, from (4.25) and the properties

βλ(r)(r-mΓ)δ0|βλ(r)|-c3,βΓ,λ(r)(r-mΓ)δ0|βΓ,λ(r)|-c3

for all r, λ(0,1] and some positive constants δ0 and c3 which are provided from [19, Section 5] with the assumptions D(βΓ)D(β) of (A5) and mΓintD(βΓ) of (A6), we deduce

|uλ(s)|H2+δ0|βλ(uλ(s))|L1(Ω)+|ΓuΓ,λ(s)|HΓ2+δ0|βΓ,λ(uΓ,λ(s))|L1(Γ)c3T(|Ω|+|Γ|)+|τtuλ(s)+π(uλ(s))-f(s)|H|uλ(s)-mΓ|H   +|πΓ(uΓ,λ(s))-fΓ(s)|HΓ|uΓ,λ(s)-mΓ|HΓ+|suΓ,λ(s)|VΓ*|yΓ(s)|VΓ.

Now, squaring both sides, we obtain

(δ0|βλ(uλ(s))|L1(Ω)+δ0|βΓ,λ(uΓ,λ(s))|L1(Γ))24c32T2(|Ω|+|Γ|)2+12(τ2|tuλ(s)|H2+|π(uλ(s))|H2+|f(s)|H2)|uλ(s)-mΓ|H2   +8(|πΓ(uΓ,λ(s))|HΓ2+|fΓ(s)|HΓ2)|uΓ,λ(s)-mΓ|HΓ2+|suΓ,λ(s)|VΓ*2|yΓ(s)|VΓ2(4.28)

for a.a. s(0,T) Here, by virtue of (4.20), (3.15), (4.21) and (4.27), we have that

|uλ-mΓ|L(0,T;H)22M12+2mΓ|Ω|,|uΓ,λ-mΓ|L(0,T;HΓ)2CPM22,|yΓ|L(0,T;VΓ)2CPM22M~6,

because Γ(uΓ,λ(s)-mΓ)𝑑Γ=0 for a.a. s(0,T). Therefore, we integrate (4.28) over [0,T] with respect to time. Then the right-hand side can be bounded due to (4.20), (4.21) and (4.22). Thus, there exists a positive constant M6, independent of λ(0,1], such that (4.24) holds. ∎

Put

ωλ(t):=1|Γ|ΓμΓ,λ(t)𝑑t

for a.a. t(0,T). Then we obtain the following estimate.

Lemma 4.4.

There exist two positive constants M7 and M8, independent of λ(0,1], such that

|ωλ|L2(0,T)M7,(4.29)|μΓ,λ|L2(0,T;VΓ)M8,|μλ|L2(0,T;V)M8.(4.30)

Proof.

Taking z:=1/|Γ| and zΓ:=1/|Γ| in (4.19), we obtain

|ωλ(t)|2=7|Γ|2{τ2|tuλ(t)|L1(Ω)2+|βλ(uλ(t))|L1(Ω)2+|π(uλ(t))|L1(Ω)2+|fλ(t)|L1(Ω)2+|βΓ,λ(uΓ,λ(t))|L1(Γ)2+|πΓ(uΓ,λ(t))|L1(Γ)2+|fΓ,λ(t)|L1(Γ)2}

for a.a. t(0,T), that is, there exists a positive constant M7, independent of λ(0,1], such that (4.29) holds. Next, using (3.15) we obtain

|μΓ,λ(t)|HΓ22|μΓ,λ(t)-ωλ(t)|HΓ2+2|ωλ(t)|HΓ22CP|ΓμΓ,λ(t)|HΓ2+2|Γ||ωλ(t)|2

for a.a. t(0,T). Thus, (4.21) and (4.29) imply the first estimate of (4.30) for some positive constant M8 independent of λ(0,1]. Moreover, by using (3.14) with the above, (4.21) ensures the validity of the second estimate in (4.30). ∎

Lemma 4.5.

There exist a positive constant M9, independent of λ(0,1], such that

|βλ(uλ)|L2(0,T;H)+|βλ(uΓ,λ)|L2(0,T;HΓ)M9,(4.31)|Δuλ|L2(0,T;H)+|uλ|L2(0,T;H3/2(Ω))+|𝝂uλ|L2(0,T;HΓ)M9,(4.32)|βΓ,λ(uΓ,λ)|L2(0,T;HΓ)M9,(4.33)|uλ|L2(0,T;W)+|uΓ,λ|L2(0,T;WΓ)M9.(4.34)

Estimates (4.31)–(4.32) take advantage from writing (4.19) as the combination of the equations

τtuλ-Δuλ+βλ(uλ)+π(uλ)=fa.e. in Q,(4.35)μΓ,λ=𝝂uλ-ΔΓuΓ,λ+βΓ,λ(uΓ,λ)+πΓ(uΓ,λ)-fΓa.e. on Σ,(4.36)

which are rigorous due to the regularity of uλ and uΓ,λ stated in (4.4). Recalling [2, Lemmas 4.4, 4.5], we observe that the proof is essentially the same as in these lemmas. Therefore, we omit the details for (4.31)–(4.34).

We have collected all information which enables us to pass to the limit as λ0.

Proof of Theorem 2.3.

Thanks to (4.20)–(4.22), (4.30), (4.31), (4.33) and (4.34), we see that there exist a subsequence of λ (not relabeled) and some limit functions u,uΓ, μ, μΓ, ξ and ξΓ such that

uλuweakly star in H1(0,T;H)L(0,T;V)L2(0,T;W),(4.37)μλμweakly in L2(0,T;V),(4.38)uΓ,λuΓweakly star in H1(0,T;VΓ*)L(0,T;VΓ)L2(0,T;WΓ),(4.39)

μΓ,λμΓweakly in L2(0,T;VΓ),(4.40)βλ(uλ)ξweakly in L2(0,T;H),(4.41)βΓ,λ(uΓ,λ)ξΓweakly in L2(0,T;HΓ)(4.42)

as λ0. From (4.37) and (4.39), using well-known compactness results (see, e.g., [30, Section 8, Corollary 4]), we obtain

uλustrongly in C([0,T];H)L2(0,T;V),(4.43)uΓ,λuΓstrongly in C([0,T];HΓ)L2(0,T;VΓ),(4.44)

which imply that

π(uλ)π(u)strongly in C([0,T];H),(4.45)πΓ(uΓ,λ)πΓ(uΓ)strongly in C([0,T];HΓ)(4.46)

as λ0 and

ξβ(u)a.e. in Q,ξΓβΓ(uΓ)a.e. on Σ.(4.47)

due to the maximal monotonicity of β and βΓ. Finally, we can pass to the limit as λ0 in the variational equality (4.18) as well as in (4.35)–(4.36) obtaining (2.8)–(2.10), and in the initial conditions (4.17) obtaining (2.11). Thus, we arrive at the conclusion. ∎

Remark 4.6.

Thanks to the strong convergence (4.44) and equality (4.23), we deduce that

Γu(t)𝑑Γ=Γu0Γ𝑑Γfor all t[0,T].(4.48)

4.4 Proof of Theorem 2.4

Proof of Theorem 2.4.

Let us note that the solutions 𝒖(i):=(u(i),uΓ(i)), 𝝁(i):=(μ(i),μΓ(i)), 𝝃(i):=(ξ(i),ξΓ(i)) of (P) satisfy

suΓ(i)(s),zΓVΓ*,VΓ+(μ(i)(s),z)H+(ΓμΓ(i)(s),ΓzΓ)HΓ=0,(4.49)(μΓ(i)(s),zΓ)HΓ=τ(tu(i)(s),z)H+(u(i)(s),z)H+(ξ(i)(s)+π(u(i)(s))-f(i)(s),z)H      +(ΓuΓ(i)(s),ΓzΓ)HΓ+(ξΓ(i)(s)+πΓ(uΓ(i)(s))-fΓ(i)(s),zΓ)HΓ(4.50)

for all 𝒛:=(z,zΓ)𝑽 and i=1,2. Let us use the notation

u¯:=u(1)-u(2),u¯Γ:=uΓ(1)-uΓ(2),μ¯:=μ(1)-μ(2),μ¯Γ:=μΓ(1)-μΓ(2),ξ¯:=ξ(1)-ξ(2),ξ¯Γ:=ξΓ(1)-ξΓ(2),f¯:=f(1)-f(2),f¯Γ:=fΓ(1)-fΓ(2),u¯0:=u0(1)-u0(2),u¯0Γ:=u0Γ(1)-u0Γ(2).

We take the difference of equations (4.50), test it by (u¯,u¯Γ) and integrate over [0,t] with respect to s, obtaining

τ2|u¯(t)|H2+0t|u¯(s)|H2ds+0t|Γu¯Γ(s)|HΓ2ds+0t(ξ¯(s),u¯(s))Hds+0t(ξ¯Γ(s),u¯Γ(s))HΓds-0t(μ¯Γ(s),u¯Γ(s))HΓds=τ2|u¯0|H2-0t(π(u(1)(s))-π(u(2)(s)),u¯(s))Hds-0t(πΓ(uΓ(1)(s))-πΓ(uΓ(2)(s)),u¯Γ(s))HΓds   +0t(f¯(s),u¯(s))H𝑑s+0t(f¯Γ(s),u¯Γ(s))HΓ𝑑s(4.51)

for all t[0,T]. We start by discussing the last term in the left-hand side of (4.51). Let (y¯,y¯Γ)H1(0,T;𝑽) be a solution of the problem

Ωy¯zdx+ΓΓy¯ΓΓzΓdΓ=u¯Γ,zΓVΓ*,VΓ(4.52)

for all 𝒛:=(z,zΓ)𝑽, with

Γy¯Γ𝑑Γ=0(4.53)

a.e. in (0,T). Now, testing (4.52) by 𝒛=(μ¯(s),μ¯Γ(s)), we have

-0t(μ¯Γ(s),u¯Γ(s))HΓ𝑑s=-0t(y¯(s),μ¯(s))H𝑑s-0t(Γy¯Γ(s),Γμ¯Γ(s))HΓ𝑑s

for all t[0,T]. On the other hand, considering the difference of (4.49) and testing then by 𝒛=(y¯(s),y¯Γ(s)), we infer that

-0t(μ¯(s),y¯(s))H𝑑s-0t(Γμ¯Γ(s),Γy¯Γ(s))HΓ𝑑s=0tsu¯Γ(s),y¯Γ(s)VΓ*,VΓ𝑑s

for all t[0,T]. Therefore, we can differentiate (4.52) with respect to time, then test by 𝒛=(y¯(s),y¯Γ(s)) and integrate the resultant over [0,t] with respect to s, obtaining

12|y¯(t)|H2+12|y¯Γ(t)|HΓ2-12|y¯(0)|H2-12|y¯Γ(0)|HΓ2=0tsu¯Γ(s),y¯Γ(s)VΓ*,VΓds

for all t[0,T]. Thus, we have the contribution

12|y¯(t)|H2+12|y¯Γ(t)|HΓ2c4|u¯Γ(t)|VΓ*2(4.54)

on the left-hand side and

12|y¯(0)|H2+12|y¯Γ(t)|HΓ2c5|u¯0Γ|VΓ*2(4.55)

on the right-hand side for all t[0,T], for some positive constants c4 and c5. Notice that (4.54) follows from (4.52), since we have

|u¯Γ(t)|VΓ*=supzΓVΓ|zΓ|VΓ1|u¯Γ(t),zΓVΓ*,VΓ|supzΓVΓ|zΓ|VΓ1|Ωy¯(t)zΓdx+ΓΓy¯Γ(t)ΓzΓdΓ|supzΓVΓ|zΓ|VΓ1{c21/2|y¯(t)|H|zΓ|VΓ+|Γy¯Γ(t)|HΓ|zΓ|VΓ}

for all t[0,T], where we have to use (3.20) and the fact that (zΓ,zΓ)𝑽 for all zΓVΓ. Moreover, (4.55) follows from (3.16) and (4.52) at time 0, taking 𝒛=(y¯(0),y¯Γ(0)); of course, c5 depends on CP. Next, for all t[0,T], we note that

0t(ξ¯(s),u¯(s))H𝑑s0,0t(ξ¯Γ(s),u¯Γ(s))HΓ𝑑s0,

by the monotonicity,

0t(π(u(1)(s))-π(u(2)(s)),u¯(s))HdsL0t|u¯(s)|H2ds

and this will be treated by the Gronwall inequality,

0t(πΓ(uΓ(1)(s))-πΓ(uΓ(2)(s)),u¯Γ(s))HΓdsLΓ0t|u¯Γ(s)|HΓ2dsδ0t|u¯Γ(s)|VΓ2ds+Cδ0t|u¯Γ(s)|VΓ*2ds,

due to the compactness inequality (3.18), also for this term we will use the Gronwall inequality (the last two terms can be simply treated by the Young inequality), and finally

0t(f¯(s),u¯(s))Hdsδ~0t|u¯(s)|V2ds+14δ~0t|f¯(s)|V*2ds,0t(f¯Γ(s),u¯Γ(s))HΓdsδ0t|u¯Γ(s)|VΓ2ds+14δ0t|f¯Γ(s)|VΓ*2ds

for all δ,δ~>0. Now we take advantage of (3.16) Indeed, u¯Γ satisfies the zero mean value condition from (4.48). Moreover, by (4.55), we can recover from (4.51) the following inequality:

τ2|u¯(t)|H2+c4|u¯Γ(t)|VΓ*2+12CP0t|u¯(s)|V2ds+12CP0t|u¯Γ(s)|VΓ2dsτ2|u¯0|H2+c5|u¯0Γ|VΓ*2+L0t|u¯(s)|H2ds+Cδ0t|u¯Γ(s)|VΓ*2ds+CP20t|f¯(s)|V*2ds+CP0t|f¯Γ(s)|VΓ*2ds

for all t[0,T], that is, δ:=1/(4CP) and δ~:=1/(2CP). Then the continuous dependence (2.13) follows from the application of the Gronwall inequality. ∎

A Appendix

We use the same notation as in the previous sections. We also use the following notation of function spaces. For each fixed ε(0,1], define a subspace 𝑯0ε of 𝑯 by 𝑯0ε:={𝒛𝑯:mε(𝒛)=0}, where mε:𝑯,

mε(𝒛):=1ε|Ω|+|Γ|{εΩz𝑑x+ΓzΓ𝑑Γ}for all 𝒛𝑯.

Moreover, define an inner product of 𝑯 by

((𝒛,𝒛~))𝑯:=ε(z,z~)H+(zΓ,z~Γ)HΓfor all 𝒛𝑯.

Then we see that the induced norm 𝑯 and the standard norm ||𝑯 are equivalent, because

𝒛𝑯2|𝒛|𝑯21ε𝒛𝑯2for all 𝒛𝑯.

Next, we define 𝑽0ε:=𝑽𝑯0ε, with |𝒛|𝑽0ε:=a(𝒛,𝒛) for all 𝒛𝑽0ε, where we use a(,):𝑽×𝑽, a bilinear form defined by

a(𝒛,𝒛~):=Ωzz~dx+ΓΓzΓΓz~ΓdΓfor all 𝒛,𝒛~𝑽.

Let us define a linear bounded operator 𝑭:𝑽0ε(𝑽0ε)* by

𝑭𝒛,𝒛~(𝑽0ε)*,𝑽0ε:=a(𝒛,𝒛~)for all 𝒛,𝒛~𝑽0ε.

Then there exists a positive constant cP(ε) such that

𝒛𝑯2|𝒛|𝑯2cP(ε)|𝒛|𝑽0ε2for all 𝒛𝑽0ε,

see, e.g., [5, Appendix]. Thus, we see that ||𝑽0ε and the standard ||𝑽 are equivalent norms of 𝑽0ε, and 𝑭 is the duality mapping from 𝑽0ε to (𝑽0ε)*. We also define the inner product in (𝑽0ε)* by

(𝒛1*,𝒛2*)(𝑽0ε)*:=𝒛1*,𝑭-1𝒛2*(𝑽0ε)*,𝑽0εfor all 𝒛1*,𝒛2*(𝑽0ε)*.

Thanks to [5, Appendix] again, we obtain 𝑽0ε𝑯0ε(𝑽0ε)*.

Lemma A.1.

The operator 𝐏ε:𝐇𝐇0ε defined by

𝑷ε𝒛:=𝒛-mε(𝒛)𝟏for all 𝒛𝑯

is the projection from 𝐇 to 𝐇0ε with respect to 𝐇-norm, namely,

𝒛-𝑷ε𝒛𝑯𝒛-𝒚𝑯for all 𝒚𝑯0ε.(A.1)

Proof.

Firstly, we see that

mε(𝑷ε𝒛)=1ε|Ω|+|Γ|{εΩ(z-mε(𝒛))𝑑x+Γ(zΓ-mε(𝒛))𝑑Γ}=mε(𝒛)-mε(𝒛)ε|Ω|+|Γ|{ε|Ω|+|Γ|}=0for all 𝒛𝑯.

Next, we infer that

((𝒛-𝑷ε𝒛,𝒚-𝑷ε𝒛))𝑯=((mε(𝒛)𝟏,𝒚-𝒛+mε(𝒛)𝟏))𝑯=mε(𝒛){εΩ(y-z)𝑑x+Γ(yΓ-zΓ)𝑑Γ}+mε(𝒛)2{ε|Ω|+|Γ|}=mε(𝒛)mε(𝒚){ε|Ω|+|Γ|}=0for all 𝒚𝑯0ε.

Therefore, we deduce that

𝒛-𝑷ε𝒛𝑯2=((𝒛-𝑷ε𝒛,𝒛-𝒚+𝒚-𝑷ε𝒛))𝑯=((𝒛-𝑷ε𝒛,𝒛-𝒚))𝑯12𝒛-𝑷ε𝒛𝑯2+12𝒛-𝒚𝑯2,

i.e., (A.1) holds. ∎

We also easily obtain the following conditions:

((𝒛*,𝑷ε𝒛))𝑯=((𝒛*,𝒛))𝑯for all 𝒛*𝑯0ε and 𝒛𝑯,|𝑷ε𝒛|𝑽0ε|𝒛|𝑽for all 𝒛𝑽.

Incidentally, we note that another possibility of projection is given by

𝑷~ε𝒛:=(z-1|Ω|Ωz𝑑x,zΓ-1|Γ|ΓzΓ𝑑Γ)for all 𝒛𝑯,

and 𝑷~ε is actually the projection from 𝑯 to 𝑯0ε with respect to the standard norm (cf. (A.1)). However, this choice is not suitable from the viewpoint of the trace condition. Indeed, 𝑷~ε𝒛𝑽 even if 𝒛𝑽.

Next, we prepare suitable approximations for 𝒇 and 𝒖0, which satisfy assumptions (A1) and (A2), respectively.

Lemma A.2.

For each ε(0,1], the solution of the following Cauchy problem

{ε𝒇ε(s)+𝒇ε(s)=𝒇(s)in 𝑯, for a.a. s(0,T),𝒇ε(0)=𝟎in 𝑯(A.2)

satisfies (3.3) and (3.4).

Proof.

Note that 𝒇ε(0)=𝟎𝑽. Multiplying the above equation by the solution 𝒇ε:=(fε,fΓ,ε)H1(0,T;𝑯), integrating it over [0,t] and using the Young inequality, we have

ε2|𝒇ε(t)|𝑯2+120t|𝒇ε(s)|𝑯2ds120t|𝒇(s)|𝑯2ds(A.3)

for all t[0,T]. Thus, there exist a subsequence of ε (not relabeled) and some limit function 𝒇~L2(0,T;𝑯) such that

𝒇ε𝒇~weakly in L2(0,T;𝑯)

as ε0. Moreover, from (A.2) and (A.3), there exists a positive constant M10 such that

|ε𝒇ε|L2(0,T;𝑯)=|𝒇-𝒇ε|L2(0,T;𝑯)M10,ε1/2|𝒇ε|L(0,T;𝑯)M10,

that is, there exist a subsequence of ε (not relabeled) and some limit function 𝒇˙L2(0,T;𝑯) such that

ε𝒇ε𝒇˙weakly in L2(0,T;𝑯),ε𝒇ε𝟎strongly in C([0,T];𝑯)

as ε0, that is, 𝒇˙=𝟎. From (A.2), this implies that 𝒇~=𝒇. Therefore, by using (A.3) again, we have

lim supε0|𝒇ε|L2(0,T;𝑯)|𝒇|L2(0,T;𝑯),

which gives us the convergence of norms and consequently the strong convergence (3.3). Next we show the required order of the convergence (3.4). Recall the equation for fΓ,εH1(0,T;HΓ) as follows:

{εfΓ,ε(s)+fΓ,ε(s)=fΓ(s)in HΓ, for a.a. s(0,T),fΓ,ε(0)=0in HΓ.(A.4)

From (A.4) we see that

|fΓ-fΓ,ε|L2(0,T;HΓ)=ε|fΓ,ε|L2(0,T;HΓ),

therefore it is enough to prove that there exists a positive constant C0 such that

ε1/2|fΓ,ε|L2(0,T;HΓ)C0.

Testing (A.4) by fΓ,ε(s) and integrating the resultant with respect to time, we obtain

ε0t|fΓ,ε(s)|HΓ2ds+12|fΓ,ε(t)|HΓ2=(fΓ(t),fΓ,ε(t))HΓ-0t(fΓ(s),fΓ,ε(s))HΓds|fΓ(t)|HΓ2+14|fΓ,ε(t)|HΓ2+0t|fΓ(s)|HΓ|fΓ,ε(s)|HΓds

for all t[0,T]. Therefore, using the Gronwall inequality, we see that there exists a positive constant C0, depending on |fΓ|C([0,T];HΓ) and |fΓ|L1(0,T;HΓ) but independent of ε(0,1], such that

ε1/2|fΓ,ε|L2(0,T;HΓ)+|fΓ,ε|L(0,T;HΓ)C0

for all ε(0,1]. ∎

Lemma A.3.

For each ε(0,1], the solution 𝐮0,ε:=(u0,ε,u0Γ,ε)𝐖𝐕 of the following elliptic system

u0,ε-εΔu0,ε=u0a.e. in Ω,(A.5)(u0,ε)|Γ=u0Γ,ε,u0Γ,ε+ε𝝂u0,ε-εΔΓu0Γ,ε=u0Γa.e. on Γ(A.6)

satisfies (3.5) and (3.6) with the required regularity (-Δu0,ε,𝛎u0,ε-ΔΓu0Γ,ε)𝐕.

Proof.

The strategy of the proof is similar to the one of [7, Lemma A.1]. By virtue of [5, Lemma C], there exists 𝒖0,ε:=(u0,ε,u0Γ,ε)𝑾𝑽 such that 𝒖0,ε satisfies system (A.5)–(A.6). Moreover, assumption (A2) gives us the required regularity (-Δu0,ε,𝝂u0,ε-ΔΓu0Γ,ε)𝑽. Next we show (3.5). Indeed, testing equation (A.5) by u0,ε and the second equation in (A.6) by u0Γ,ε, adding them, and using the Young inequality, we obtain

12Ω|u0,ε|2dx+12Γ|u0Γ,ε|2dΓ+εΩ|u0,ε|2dx+εΓ|Γu0Γ,ε|2dΓ12Ω|u0|2dx+12Γ|u0Γ|2dΓ.

Therefore, {u0,ε}ε(0,1] is bounded in H, {u0Γ,ε}ε(0,1] is bounded in HΓ, {ε1/2u0,ε}ε(0,1] is bounded in H, and {ε1/2Γu0,ε}ε(0,1] is bounded in HΓ, respectively. These give us (see (A.5)–(A.6))

u0,εu0weakly in H,εu0,ε0strongly in V,u0Γ,εu0Γweakly in HΓ,εu0Γ,ε0strongly in VΓ

as ε0. Moreover, we have

lim supε0(Ω|u0,ε|2dx+Γ|u0Γ,ε|2dΓ)Ω|u0|2dx+Γ|u0Γ|2dΓ,

which entails that

𝒖0,ε𝒖0strongly in 𝑯

as ε0. Next, from the definition of the subdifferential together with (A.5) and (A.6), we see that

Ωβ^λ(u0,ε)𝑑x-Ωβ^λ(u0)𝑑xΩ(u0,ε-u0)βλ(u0,ε)𝑑x=-Ωεβλ(u0,ε)|u0,ε|2𝑑x+εΓ𝝂u0,εβλ(u0Γ,ε)𝑑ΓΓ(u0Γ-u0Γ,ε)βλ(u0Γ,ε)𝑑Γ-εΓβλ(u0Γ,ε)|u0Γ,ε|2𝑑ΓΓβ^λ(u0Γ)𝑑Γ-Γβ^λ(u0Γ,ε)𝑑Γ,

that is,

Ωβ^λ(u0,ε)𝑑x+Γβ^λ(u0Γ,ε)𝑑ΓΩβ^(u0)𝑑x+Γβ^(u0Γ)𝑑Γ

for all ε(0,1], where (3.1) has been used. Now, due to (3.2), we obtain that

β^λ(r)=0rβλ(s)𝑑sϱ0rβΓ,λ(s)𝑑s+c0|r|=ϱβ^Γ,λ(r)+c0|r|

for all r. Indeed, in the case r0, because of the fact that βλ(0)=βΓ,λ(0)=0, we see from (3.2) that

0rβλ(s)ds=0r|βλ(s)|dsϱ0r|βΓ,λ(s)|ds+c0r=ϱ0rβΓ,λ(s)ds+c0|r|.

In the case r<0, we have

0rβλ(s)ds=r0|βλ(s)|dsϱr0|βΓ,λ(s)|ds-c0r=ϱ0rβΓ,λ(s)ds+c0|r|.

Then it turns out that

Γβ^λ(u0Γ)𝑑ΓϱΓβ^Γ,λ(u0Γ)𝑑Γ+c0|u0Γ|L1(Γ)ϱΓβ^Γ(u0Γ)𝑑Γ+c0|u0Γ|L1(Γ).

Therefore, β^(u0Γ)L1(Γ), due to the Fatou lemma and the almost everywhere convergence of β^λ(u0Γ) to β^(u0Γ). Concerning our approximation, testing (A.5) by -Δu0,ε and using (A.6), we get

Ω(u0,ε-u0)u0εdx+1εΓ|u0Γ,ε-u0Γ|2dΓ+εΩ|Δu0ε|2dx+ΓΓu0Γ,εΓ(u0Γ,ε-u0Γ)dΓ=0.

Therefore, using the Young inequality, we deduce

12Ω|u0,ε|2dx+12Γ|Γu0Γ,ε|2dΓ+1εΓ|u0Γ,ε-u0Γ|2dΓ+εΩ|Δu0ε|2dx12|u0|2dx+12Γ|Γu0Γ|2dΓ.

Thus, we obtain that there exists a positive constant C~0 such that

|u0Γ,ε-u0Γ|HΓε1/2C~0,ε1/2|Δu0ε|HC~0for all ε(0,1],𝒖0,ε𝒖0strongly in 𝑽

as ε0, because, 𝒖0,ε𝒖0 weakly in 𝑽 and |𝒖0,ε|𝑽|𝒖0|𝑽 as ε0. Then, from (A.6), we can also infer that

ε1/2|𝝂u0,ε-ΔΓu0Γ,ε|HΓC~0.

Now, if we test (A.6) by βΓ,λ(u0Γ,ε), then we obtain

Γβ^Γ,λ(u0Γ,ε)𝑑Γ-Γβ^Γ,λ(u0Γ)𝑑ΓΓ(-ε𝝂u0,ε+εΔΓu0Γ,ε)βΓ,λ(u0Γ,ε)𝑑ΓC~0ε1/2|βΓ,λ(u0Γ,ε)|HΓC~0ε1/2λ|u0Γ,ε|HΓ,

hence, from (3.1), there exists a positive constant C¯0, independent of ε,λ(0,1], such that

Γβ^Γ,λ(u0Γ,ε)𝑑ΓΓβ^Γ,λ(u0Γ)𝑑Γ+C¯0ε1/2λΓβ^Γ(u0Γ)𝑑Γ+C¯0ε1/2λ.

Thus, we get the conclusion. ∎

Proof of Proposition 3.1.

For each ε,λ(0,1], let us consider the Cauchy problem for the following equivalent evolution equation:

𝒗(t)+𝑭(𝑷ε𝝁(t))=𝟎in (𝑽0ε)*, for a.a. t(0,T),(A.7)𝑴𝝁(t)=𝑻𝒗(t)+𝑴φ(𝒗(t))+𝜷λ(𝒖(t))+𝝅(𝒖(t))-𝒇ε(t)in 𝑯, for a.a. t(0,T),(A.8)𝒖(t)=𝒗(t)+mε(𝒖0,ε)𝟏,𝒗(0)=𝒗0:=𝒖0,ε-mε(𝒖0,ε)in 𝑯0ε(A.9)

(cf. [5, Section 2.3], see also [25, 24]), where

𝑴:=(ε001),𝑻:=(τ00ε),

and we define φ:𝑯0ε[0,+] by

φ(𝒛):={12Ω|z|2dx+12Γ|ΓzΓ|2dΓif 𝒛𝑽0ε,+if 𝒛𝑯0ε𝑽0ε.

Then we see that φ is proper, lower semicontinuous and convex on 𝑯0ε, and the subdifferential φ on 𝑯0ε is characterized by φ(𝒛)=(-(1/ε)Δz,𝝂z-ΔΓzΓ) with 𝒛D(φ)=𝑾𝑽0ε. The Cauchy problem (A.7)–(A.9) can be solved by applying [5, Sections 4.2–4.4], based on the abstract theory of doubly nonlinear evolution equation [10], because all assumptions for 𝒇ε and 𝒖0,ε in order to obtain the strong solution are satisfied. ∎

Acknowledgements

The authors are grateful to the anonymous referees for reviewing the original manuscript and for many valuable comments that helped to clarify and refine this paper.

References

  • [1]

    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I: Interfacial free energy, J. Chem. Phys. 2 (1958), 258–267.  Google Scholar

  • [2]

    L. Calatroni and P. Colli, Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal. 79 (2013), 12–27.  CrossrefWeb of ScienceGoogle Scholar

  • [3]

    L. Cherfils, S. Gatti and A. Miranville, A variational approach to a Cahn–Hilliard model in a domain with nonpermeable walls, J. Math. Sci. (N.Y.) 189 (2013), no. 4, 604–636.  CrossrefGoogle Scholar

  • [4]

    P. Colli and T. Fukao, Cahn–Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl. 429 (2015), no. 2, 1190–1213.  CrossrefWeb of ScienceGoogle Scholar

  • [5]

    P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn–Hilliard type with singular potentials, Nonlinear Anal. 127 (2015), 413–433.  Web of ScienceCrossrefGoogle Scholar

  • [6]

    P. Colli and T. Fukao, The Allen–Cahn equation with dynamic boundary conditions and mass constraints, Math. Methods Appl. Sci. 38 (2015), no. 17, 3950–3967.  CrossrefWeb of ScienceGoogle Scholar

  • [7]

    P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems, J. Differential Equations 260 (2016), no. 9, 6930–6959.  CrossrefWeb of ScienceGoogle Scholar

  • [8]

    P. Colli, G. Gilardi and J. Sprekels, On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl. 419 (2014), no. 2, 972–994.  CrossrefWeb of ScienceGoogle Scholar

  • [9]

    P. Colli, G. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal. 4 (2015), no. 4, 311–325.  Web of ScienceGoogle Scholar

  • [10]

    P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), no. 5, 737–756.  CrossrefGoogle Scholar

  • [11]

    J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions, Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse 1991), Lecture Notes Pure Appl. Math. 155, Dekker, New York (1994), 173–183.  Google Scholar

  • [12]

    M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal. 11 (2012), no. 3, 1285–1301.  Web of ScienceGoogle Scholar

  • [13]

    M. Fila, K. Ishige and T. Kawakami, Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 2059–2078.  Web of ScienceCrossrefGoogle Scholar

  • [14]

    M. Fila and P. Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition, Math. Methods Appl. Sci. 20 (1997), no. 15, 1325–1333.  CrossrefGoogle Scholar

  • [15]

    C. G. Gal, A Cahn–Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci. 29 (2006), no. 17, 2009–2036.  CrossrefGoogle Scholar

  • [16]

    C. G. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1351–1380. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    C. G. Gal and H. Wu, Asymptotic behavior of a Cahn–Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst. 22 (2008), no. 4, 1041–1063.  CrossrefWeb of ScienceGoogle Scholar

  • [18]

    H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 (2016), no. 6, 1149–1189.  Web of ScienceCrossrefGoogle Scholar

  • [19]

    G. Gilardi, A. Miranville and G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal. 8 (2009), no. 3, 881–912.  Web of ScienceCrossrefGoogle Scholar

  • [20]

    G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn–Hilliard model in a domain with non-permeable walls, Phys. D 240 (2011), no. 8, 754–766.  CrossrefGoogle Scholar

  • [21]

    A. Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. 47, American Mathematical Society, Providence, 2009.  Google Scholar

  • [22]

    E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1635, Springer, Berlin, 1996.  Google Scholar

  • [23]

    H. Kardestuncer and D. H. Norrie, Finite Element Handbook, McGraw-Hill, New York, 1987.  Google Scholar

  • [24]

    N. Kenmochi, M. Niezgódka and I. Pawłow, Subdifferential operator approach to the Cahn–Hilliard equation with constraint, J. Differential Equations 117 (1995), no. 2, 320–356.  CrossrefGoogle Scholar

  • [25]

    M. Kubo, The Cahn–Hilliard equation with time-dependent constraint, Nonlinear Anal. 75 (2012), no. 14, 5672–5685.  CrossrefWeb of ScienceGoogle Scholar

  • [26]

    J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.  Google Scholar

  • [27]

    J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 182, Springer, New York, 1972.  Google Scholar

  • [28]

    C. Liu and H. Wu, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary conditions: Derivation and analysis, preprint (2017), https://arxiv.org/abs/1710.08318.  

  • [29]

    J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967.  Google Scholar

  • [30]

    J. Simon, Compact sets in the spaces Lp(0,T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.  Google Scholar

About the article

Received: 2018-03-12

Revised: 2018-05-15

Accepted: 2018-06-11

Published Online: 2018-07-20

Published in Print: 2019-03-01


The present paper benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia for PC; the support of the JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Number 17K05321 for TF.


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 16–38, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0055.

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