A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions

Nonlinear Analysis 158 (2017) 32–59

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Nonlinear Analysis www.elsevier.com/locate/na

A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions Pierluigi Colli a , Gianni Gilardi a , Ryota Nakayashiki b,∗ , Ken Shirakawa c a

Dipartimento di Matematica, Universit` a di Pavia, via Ferrata 5, 27100, Pavia, Italy Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan c Department of Mathematics, Faculty of Education, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan b

article

info

Article history: Received 31 October 2016 Accepted 31 March 2017 Communicated by Enzo Mitidieri MSC: 35K55 35K59 82C26 Keywords: Quasi-linear Allen–Cahn equation Dynamic boundary conditions Non-smooth energy functional Initial–boundary value problem Well-posedness Continuous dependence

abstract In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE)ε for ε ≥ 0. For each ε ≥ 0, the system (ACE)ε consists of an Allen–Cahn type equation in a bounded spacial domain Ω , and another Allen–Cahn type equation on the smooth boundary Γ := ∂Ω , and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the non-smooth energy proposed by Visintin in his monography “Models of phase transitions”: hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L2 -based solutions to our systems, and to see some robustness of (ACE)ε with respect to ε ≥ 0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE)ε for each ε ≥ 0, and the continuous dependence of solutions to (ACE)ε for the variations of ε ≥ 0, respectively. © 2017 Elsevier Ltd. All rights reserved.

0. Introduction Let 0 < T < ∞, κ > 0 and N ∈ N be fixed constants. Let Q := (0, T ) × Ω be a product set of a timeinterval (0, T ) and a bounded spatial domain Ω ⊂ RN . Let Γ := ∂Ω be the boundary of Ω with sufficient smoothness (when N > 1), and let nΓ be the unit outer normal to Γ . Besides, we put Σ := (0, T ) × Γ . In this paper, we fix a constant ε ≥ 0 to consider the following system of PDEs, denoted by (ACE)ε . (ACE)ε :  ∂t u − div

 ∇u 2 + κ ∇u + β(u) + g(u) ∋ θ |∇u|

in Q,

∗ Corresponding author. E-mail addresses: [email protected] (P. Colli), [email protected] (G. Gilardi), [email protected] (R. Nakayashiki), [email protected] (K. Shirakawa).

http://dx.doi.org/10.1016/j.na.2017.03.020 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

(0.1)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

∂t uΓ − ε2 ∆Γ uΓ + u(0, ·) = u0

33

 ∇u  + κ2 ∇u · nΓ + βΓ (uΓ ) + gΓ (uΓ ) ∋ θΓ and u|Γ = uΓ |∇u| |Γ

in Ω ,

and uΓ (0, ·) = uΓ ,0

on Γ .

on Σ ,

(0.2) (0.3)

The system (ACE)ε is a modified version of an Allen–Cahn type equation, proposed in [37, Chapter VI], and the principal modifications are in the points that: – the quasi-linear (singular) diffusion in (0.1) includes the regularization term κ2 ∇u with a small constant κ > 0; – the boundary data uΓ is governed by the dynamic boundary condition (0.2). In general, “Allen–Cahn type equation” is a collective term to call gradient flows (systems) of governing energies, which include some double-well type potentials to reproduce the bi-stability of different phases, such as solid–liquid phases. The governing energy is called free-energy, and in the case of (ACE)ε , the corresponding free-energy is provided as follows.    1 κ2 2 1 2 |∇u| + |∇u| + B(u) + G(u) dx [u, uΓ ] ∈ H (Ω ) × H (Γ ) → Fε (u, uΓ ) := 2 Ω    2 ε |∇Γ uΓ |2 + BΓ (uΓ ) + GΓ (uΓ ) dΓ ∈ (−∞, ∞], (0.4) + 2 Γ with the effective domain:  D(Fε ) :=

[z, zΓ ]

1

z ∈ H 1 (Ω ), zΓ ∈ H 2 (Γ ), εzΓ ∈ H 1 (Γ ), 1 and z|Γ = zΓ in H 2 (Γ )

 .

In the context, “|Γ ” denotes the trace (boundary-value) on Γ for a Sobolev function, dΓ denotes the areaelement on Γ , ∇Γ denotes the surface gradient on Γ , and ∆Γ denotes the Laplacian on the surface, i.e., the so-called Laplace–Beltrami operator. B : R → [0, ∞] and BΓ : R → [0, ∞] are given proper l.s.c. and convex functions, and β = ∂B and βΓ = ∂BΓ are the subdifferentials of B and BΓ , respectively. G : R → R and GΓ : R → R are C 1 -functions, that have locally Lipschitz differentials g and gΓ , respectively. θ : Q → R and θΓ : Σ → R are given heat sources of (relative) temperature, and u0 : Ω → R and uΓ ,0 : Γ → R are initial data for the components u and uΓ , respectively. In (0.4), the functions: σ ∈ R → B(σ) + G(σ) ∈ (−∞, ∞] and σ ∈ R → BΓ (σ) + GΓ (σ) ∈ (−∞, ∞], correspond to the double-well potentials, and for instance, the setting: B(σ) = BΓ (σ) = I[−1,1] (σ)

1 and G(σ) = GΓ (σ) = − σ 2 , 2

for σ ∈ R,

with use of the indicator function:  0, if σ ∈ [−1, 1], σ ∈ R → I[−1,1] (σ) := ∞, otherwise, is known as one of representative choices of the components (cf. [37]). This paper is concerned with the existence and uniqueness of the solution to (0.1)–(0.3), as well as with some continuous dependence results, also with respect to ε. Actually, our mathematical treatment of (ACE)ε is unified for the cases ε > 0 and ε = 0. Though the two cases could exhibit different features from various

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points of view (as for the long-time behavior, see, e.g., [15] for the case of Laplace operators), we note that the presence or absence of the term  ε2 |∇Γ uΓ |2 dΓ 2 Γ brings here the gap of effective domains D(Fε ) between the cases when ε > 0 and ε = 0. More precisely, the domains D(Fε ) when ε > 0 will uniformly coincide with a convex subset in H 1 (Ω ) × H 1 (Γ ), and this convex set will be a proper subset of the domain D(F0 ) when ε = 0 which will be located in the wider space 1 H 1 (Ω ) × H 2 (Γ ). In the case when the diffusion in (0.1) is just given by the usual Laplacian, the corresponding Allen–Cahn equation has been studied by a number of researches (cf., e.g., [5,7,16,17,24]), and some qualitative results for L2 -based solutions were obtained by means of the theories of parabolic PDEs, in [28,30]. To investigate dynamic boundary conditions, our approach exploits techniques similar to those employed in [5] and resumed in other solvability studies and optimal control theories, the reader may see [7–13]. Still about dynamic boundary conditions, let us point out that there has been a recent growing interest about the justification and the study of phase field models, as well as systems of Allen–Cahn and Cahn–Hilliard type, including dynamic boundary conditions. Without trying to be exhaustive, let us mention at least the papers [6,16,18,20–23,29,31]. Nevertheless, the mathematical analysis for our system (ACE)ε will not be just an analogy work with ∇u the previous ones. In fact, due to the singularity of the diffusion −div( |∇u| + κ2 ∇u) in (0.1), it will not 2 be so easy to apply the theories of [28,30], and to see the L -based expression of the first variation of the free-energy. In view of this, we set the goal in this paper to show the following two Main Theorems, which are concerned with qualitative properties of the systems (ACE)ε for ε ≥ 0. Main Theorem 1: the well-posedness for (ACE)ε , for all ε ≥ 0. Main Theorem 2: the continuous dependence of solutions to (ACE)ε with respect to the value of ε ≥ 0, and especially the (right-hand) continuity at ε = 0. The content of this paper is as follows. Main Theorems 1 and 2 are stated in Section 2, and these results are discussed on the basis of the preliminaries prepared in Section 1, and Key-Lemmas 1–3. Based on this, we give the proofs of the Key-Lemmas and Main Theorems in the remaining Sections 4 and 5, respectively. 1. Preliminaries In this section, we outline some basic notations and known facts, as preliminaries of our study. Notation 1 (Notations in Real Analysis). For arbitrary a, b ∈ [−∞, ∞], we define: a ∨ b := max{a, b}

and a ∧ b := min{a, b}.

Let d ∈ N be any fixed dimension. Then, we simply denote by |x| and x · y the Euclidean norm of x ∈ Rd and the standard scalar product of x, y ∈ Rd , respectively. Besides, we denote by Bd and Sd−1 the d-dimensional unit open ball centered at the origin, and its boundary, respectively. For any d ∈ N, the d-dimensional Lebesgue measure is denoted by Ld , and d-dimensional Hausdorff measure is denoted by Hd . Unless otherwise specified, the measure theoretical phrases, such as “a.e.”, “dt”, “dx”, and so on, are with respect to the Lebesgue measure in each corresponding dimension. Also, in the observations on a smooth surface S, the phrase “a.e.” is with respect to the Hausdorff measure in each corresponding Hausdorff dimension, and the area element on S is denoted by dS.

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Additionally, we mention about the following elementary fact, which is used, frequently, in the proofs of Key-Lemmas and Main Theorems. (Fact 0) Let m ∈ N be a fixed finite number. If {α1 , . . . , αm } ⊂ R and {akn }∞ n=1 , k = 1, . . . , m, fulfill that lim akn ≥ αk ,

k = 1, . . . , m,

and

n→∞

lim

n→∞

m  k=1

akn ≤

m 

αk

k=1

then, it holds that lim akn = αk ,

n→∞

k = 1, . . . , m.

Notation 2 (Notations of Functional Analysis). For an abstract Banach space X, we denote by |·|X the norm of X, and denote by X ∗ ⟨·, ·⟩X the duality pairing between X and the dual space X ∗ of X. Let IX : X → X be the identity map from X onto X. In particular, when X is a Hilbert space, we denote by (·, ·)X the inner product in X. Here and in the sequel, Ω denotes an open subset of RN , which we assume to be bounded and smooth. Moreover, Γ and nΓ denote its boundary ∂Ω and the outward unit normal vector field on Γ , respectively. Let ∆N be the Laplace operator, subject to the Neumann-zero boundary condition, which is defined as:   1 ∆N : v ∈ D(∆N ) := z ∈ H 2 (Ω ) (∇z)|Γ · nΓ = 0 in H 2 (Γ ) ⊂ L2 (Ω ) → ∆N v := ∆v ∈ L2 (Ω ).

(1.1)

In this paper, we identify the unbounded closed operator −∆N with its linear and continuous extension from H 1 (Ω ) into H 1 (Ω )∗ , by setting:  ∇z · ∇w dx, for all [z, w] ∈ H 1 (Ω ) × H 1 (Ω ). H 1 (Ω)∗ ⟨∆N z, w⟩H 1 (Ω) = Ω

Remark 1.1. Note that the boundary Γ = ∂Ω has no boundary as (N − 1)-dimensional surface. Therefore, for any s > 0, the dual space H −s (Γ ) := H s (Γ )∗ of the Sobolev space H s (Γ ) coincides with the closure of the class of the smooth functions on Γ in the topology of H s (Γ )∗ . Notation 3 (Notations of Surface-Differentials). Since Ω is bounded and smooth, there exists a function dΓ : Ω → R such that dΓ is smooth on Ω

and dΓ (x) = inf |x − y| for x in a neighborhood of Γ . y∈Γ

(1.2)

We notice that nΓ (x) = −∇dΓ (x) for every x ∈ Γ . On this basis, let ∇Γ be the operator of surface-gradient on Γ , which is defined as: ∇Γ : ϕ ∈ C 1 (Γ ) → ∇Γ ϕ := ∇ϕex − (∇dΓ ⊗ ∇dΓ )∇ϕex ∈ C 0 (Γ )N ,

(1.3)

by using the extension ϕex ∈ C 1 (Ω ) of each ϕ ∈ C 1 (Γ ). Let divΓ be the operator of surface-divergence, which is defined as: divΓ : ω ∈ C 1 (Γ )N → divΓ ω := divω ex − ∇(ω ex · ∇dΓ ) · ∇dΓ ∈ C 0 (Γ ),

(1.4)

by using the extension ω ex ∈ C 1 (Ω )N of each ω ∈ C 1 (Γ )N . It is known that the definition formulas (1.3)–(1.4) are well-defined, and the values ∇Γ ϕ and divΓ ω are settled independently of the choices of the extensions ϕex ∈ C 1 (Ω ) and ω ex ∈ C 1 (Ω )N of ϕ ∈ C 1 (Γ ) and ω ∈ C 1 (Γ )N , respectively, and of the function dΓ satisfying (1.2).

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On the basis of (1.3)–(1.4), the Laplace–Beltrami operator ∆Γ , i.e., the surface-Laplacian on Γ is defined as follows: ∆Γ : ϕ ∈ C 2 (Γ ) → ∆Γ ϕ := divΓ (∇Γ ϕ) ∈ C 0 (Γ ), by using the extension ϕex ∈ C 2 (Ω ) of each ϕ ∈ C 2 (Γ ). 2 2 Remark 1.2. Let us define a closed subspace Ldiv (Ω ) in L2 (Ω )N and a closed subspace Ltan (Γ ) in L2 (Γ )N , by putting:   2 Ldiv (Ω ) := ω ∈ L2 (Ω )N div ω ∈ L2 (Ω ) , and   2 Ltan (Γ ) := ω ∈ L2 (Γ )N ω · nΓ = 0 a.e. on Γ , respectively.

Then, on account of the general theories as in [26,34], we can see the following facts (cf. [26]). 1

(Fact 1) The mapping ν ∈ H 1 (Ω )N → ν|Γ · nΓ ∈ H 2 (Γ ) can be extended as a linear and continuous 1 2 operator [ (·) · nΓ ]Γ from Ldiv (Ω ) into H − 2 (Γ ), such that:   2 div ν z dx + ν · ∇z dx, for all ν ∈ Ldiv (Ω ) and z ∈ H 1 (Ω ). H −1/2 (Γ ) ⟨[ν · nΓ ]Γ , z|Γ ⟩H 1/2 (Γ ) = Ω

Ω

(Fact 2) The surface gradient ∇Γ can be extended as a linear and continuous operator from H 1 (Γ ) into 2 Ltan (Γ ). The extension is derived in the definition process of the space H 1 (Γ ) as the completion of C 1 (Γ ). Then, the topology of the completion is taken with respect to the norm, induced by the following bi-linear form:    [ϕ, ψ] ∈ C 1 (Γ )2 → ϕψ + ∇Γ ϕ · ∇Γ ψ dΓ . Γ

The inner product in (·, ·)H 1 (Γ ) is given as the extension of the above bi-linear form. Hence, in this 2 paper, we identify the operator ∇Γ with the extension from H 1 (Γ ) into Ltan (Γ ). 2 (Γ ) (Fact 3) The surface divergence divΓ can be extended as a linear and continuous operator from Ltan −1 1 ∗ into H (Γ ) (= H (Γ ) ), via the following Green-type formula:  2 1 −1 ⟨ div w, z ⟩ = − w · ∇Γ z dΓ , for all z ∈ H 1 (Γ ) and w ∈ Ltan (Γ ). Γ H (Γ ) H (Γ ) Γ

Hence, in this paper, we regard the Laplace–Beltrami operator ∆Γ = divΓ ◦ ∇Γ as a linear and continuous operator from H 1 (Γ ) into H −1 (Γ ). In particular, the operator −∆Γ forms a duality map between H 1 (Γ ) and H −1 (Γ ).

Notation 4 (Notations in Convex Analysis). For any proper lower semi-continuous (l.s.c. from now on) and convex function Ψ defined on a Hilbert space X, we denote by D(Ψ ) its effective domain, and denote by ∂Ψ its subdifferential. The subdifferential ∂Ψ is a set-valued map corresponding to a weak differential of Ψ , and it turns out to be a maximal monotone graph in the product space X × X. More precisely, for each z0 ∈ X, the value ∂Ψ (z0 ) is defined as a set of all elements z0∗ ∈ X which satisfy the following variational inequality: (z0∗ , z − z0 )X ≤ Ψ (z) − Ψ (z0 ),

for any z ∈ D(Ψ ).

The set D(∂Ψ ) := {z ∈ X | ∂Ψ (z) ̸= ∅} is called the domain of ∂Ψ . We often use the notation “[z0 , z0∗ ] ∈ ∂Ψ in X × X”, to mean that “z0∗ ∈ ∂Ψ (z0 ) in X with z0 ∈ D(∂Ψ )”, by identifying the operator ∂Ψ with its graph in X ×X. Let us refer to [2–4,27] for definitions, properties, results about subdifferentials and maximal monotone operators.

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Remark 1.3. As one of representatives of the subdifferentials, we exemplify the following set-valued function N Sgn : RN → 2R , given as: ω  , if ω ̸= 0, ω ∈ RN → Sgn(ω) := |ω|  N B , otherwise. It is known that the set-valued function Sgn coincides with the subdifferential of the Euclidean norm √ | · | : ω ∈ RN → |ω| = ω · ω ∈ [0, ∞). Also, it is known that (cf. [3,4]) the operator −∆N defined in (1.1) coincides with the subdifferential of the proper l.s.c. and convex function ΨN on L2 (Ω ), defined as:   1 |∇z|2 dx, if z ∈ H 1 (Ω ), z ∈ L2 (Ω ) → ΨN (z) := 2 Ω  ∞, otherwise. More precisely, we have: ∂ΨN (z) = {−∆N z} in L2 (Ω ),

for any z ∈ D(∂ΨN ) = D(∆N ).

Remark 1.4 (Time-dependent Subdifferentials). It is often useful to consider the subdifferentials under timedependent settings of convex functions. With regard to this topic, certain general theories were established ˆ by some researchers (e.g., Kenmochi [27] and Otani [33]). So, referring to, e.g., [27, Chapter 2] or [35, Remark 1.1 (Fact 1)], we can see the following fact. (Fact 4) Let E0 be a convex subset in a Hilbert space X, let I ⊂ [0, ∞) be a time-interval, and for any t ∈ I, let Ψ t : X → (−∞, ∞] be a proper l.s.c. and convex function such that D(Ψ t ) = E0 for all ˆ I : L2 (I; X) → (−∞, ∞], by putting: t ∈ I. Based on this, let us define a convex function Ψ   Ψ t (ζ(t)) dt, if Ψ (·) (ζ) ∈ L1 (I), ˆ I (ζ) := ζ ∈ L2 (I; X) → Ψ  I ∞, otherwise. ˆ I ), i.e., if the function t ∈ I → Ψ t (z) is integrable for any z ∈ E0 , then it holds Here, if E0 ⊂ D(Ψ that: ˆ I in L2 (I; X) × L2 (I; X), iff [ζ, ζ ∗ ] ∈ ∂ Ψ I ˆ ) and [ζ(t), ζ ∗ (t)] ∈ ∂Ψ t in X × X, a.e. t ∈ I. ζ ∈ D(Ψ

Finally, we mention about notions of convergence for functionals. Definition 1.1 (Mosco Convergence: Cf. [32]). Let X be an abstract Hilbert space. Let Ψ : X → (−∞, ∞] be a proper l.s.c. and convex function, and let {Ψn }∞ n=1 be a sequence of proper l.s.c. and convex functions Ψn : X → (−∞, ∞], n ∈ N. Then, it is said that Ψn → Ψ on X, in the sense of Mosco [32], as n → ∞, iff the following two conditions are fulfilled. † † (M1) The condition of lower-bound: limn→∞ Ψn (zn† ) ≥ Ψ (z † ), if z † ∈ X, {zn† }∞ n=1 ⊂ X, and zn → z weakly in X as n → ∞. (M2) The condition of optimality: for any z ‡ ∈ D(Ψ ), there exists a sequence {zn‡ }∞ n=1 ⊂ X such that ‡ ‡ ‡ ‡ zn → z in X and Ψn (zn ) → Ψ (z ), as n → ∞.

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Remark 1.5 (Cf. [2, Proposition 2.68 and Theorem 3.26]). For a proper l.s.c. and convex function Ψ : H → (−∞, ∞] on a Hilbert space H, it is known that the sequence {Ψ λ }λ>0 of Moreau–Yosida regularizations:  1  z ∈ H → Ψ λ (z) := inf |˜ z − z|2H + Ψ (˜ z ) z˜ ∈ H , for λ > 0, 2λ converges to Ψ on H, in the sense of Mosco, as λ ↓ 0. Definition 1.2 (Γ -convergence: Cf. [14]). Let X be an abstract Hilbert space, Ψ : X → (−∞, ∞] be a proper functional, and {Ψn }∞ n=1 be a sequence of proper functionals Ψn : X → (−∞, ∞], n ∈ N. We say that Ψn → Ψ on X, in the sense of Γ -convergence [14], as n → ∞ iff the following two conditions are fulfilled. (Γ 1) (Γ 2)

† † The condition of lower-bound: limn→∞ Ψn (zn† ) ≥ Ψ (z † ) if z † ∈ X, {zn† }∞ n=1 ⊂ X, and zn → z (strongly) in X as n → ∞. The condition of optimality: for any z ‡ ∈ D(Ψ ), there exists a sequence {zn‡ }∞ n=1 ⊂ X such that ‡ ‡ ‡ ‡ zn → z in X and Ψn (zn ) → Ψ (z ) as n → ∞.

Remark 1.6. Of course, the Γ -convergence recalled in Definition 1.2 is the one associated with the strong topology of X. Note that if the functionals are convex, then the Mosco convergence just introduced implies Γ -convergence, i.e., the Γ -convergence of convex functions can be regarded as a weak version of Mosco convergence. Additionally, as a basic matter of the Mosco convergence, we can see the following fact (see [27, Chapter 2], or [35, Remark 1.5 (Fact 7)], for example). (Fact 5) Let X, Ψ and {Ψn }∞ n=1 be as in Definition 1.1. Besides, let us assume that: Ψn → Ψ on X, in the sense of Γ -convergence,

as n → ∞,

and  [z, z ∗ ] ∈ X × X, [zn , zn∗ ] ∈ ∂Ψn in X × X, n ∈ N, zn → z in X and zn∗ → z ∗ weakly in X, as n → ∞. Then, it holds that: [z, z ∗ ] ∈ ∂Ψ

in X × X,

and Ψn (zn ) → Ψ (z),

as n → ∞.

2. Statements of main theorems First, we configure the base-space of solutions to the systems (ACE)ε , for ε ≥ 0. In any case of ε ≥ 0, the base-space is settled by a product Hilbert space: H := L2 (Ω ) × L2 (Γ ), endowed with the inner product: ([z1 , zΓ ,1 ], [z2 , zΓ ,2 ])H := (z1 , z2 )L2 (Ω) + (zΓ ,1 , zΓ ,2 )L2 (Γ ) ,

for any [zk , zΓ ,k ], k = 1, 2.

Next, we prescribe the assumptions in our study. (A0) N ∈ N and 0 < T < ∞ are fixed constants, and Ω is a bounded domain in RN with a smooth boundary Γ . In particular, it fulfills the condition (1.2) in Notation 3. (A1) B : D(B) ⊂ R → [0, ∞] and BΓ : D(BΓ ) ⊂ R → [0, ∞] are proper l.s.c. and convex functions and β = ∂B ⊂ R × R and βΓ = ∂BΓ ⊂ R × R are the subdifferentials of B and BΓ , respectively. Furthermore, the convex functions B and BΓ , and the subdifferentials β and βΓ fulfill the following conditions:

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(a1) B(0) = 0 and BΓ (0) = 0, and hence [0, 0] ∈ β and [0, 0] ∈ βΓ on R2 ; (a2) there exists an interval IB ⊂ R, such that: intIB ̸= ∅, D(β) = D(βΓ ) = IB , and B, BΓ ∈ C(IB ) ∩ L∞ (IB ); (a3) there exist positive constants ak , bk , k = 0, 1, such that: a0 |[βΓ ]◦ (τ )| − b0 ≤ |[β]◦ (τ )| ≤ a1 |[βΓ ]◦ (τ )| + b1 , for any τ ∈ IB , ◦ where [β] and [βΓ ]◦ are the minimal sections for β and βΓ , respectively. 2,∞ (A2) G : R → R and GΓ : R → R are Wloc -functions such that the differentials g = G′ and gΓ = GΓ ′ are Lipschitz continuous on IB . (A3) The forcing pair [θ, θΓ ] belongs to L2 (0, T ; H ) and the initial pair [u0 , uΓ ,0 ] belongs to a class D∗ , defined as: D∗ :=



[z, zΓ ] ∈ H

z ∈ IB a.e. in Ω and zΓ ∈ IB a.e. on Γ



.

(2.1)

In addition, let us set:  Vε :=

[z, zΓ ] ∈ H

1

z ∈ H 1 (Ω ), zΓ ∈ H 2 (Γ ), εzΓ ∈ H 1 (Γ ), and z|Γ = zΓ a.e. on Γ

 ,

for any ε ≥ 0,

and let us define the projection function TB : R → IB , by putting: r ∈ R → TB r := (r ∨ (inf IB )) ∧ (sup IB ) ∈ IB .

(2.2)

Then, we easily check the following facts. (Fact 6) If ε > 0, then the space Vε is a closed linear subspace in H 1 (Ω ) × H 1 (Γ ). Otherwise (i.e., if ε = 0), 1 the corresponding space V0 is a closed linear subspace in H 1 (Ω ) × H 2 (Γ ). Hence, when ε > 0 (resp. ε = 0), the space Vε (resp. V0 ) forms a Hilbert space endowed with the inner product in 1 H 1 (Ω ) × H 1 (Γ ) (resp. H 1 (Ω ) × H 2 (Γ )). (Fact 7) For any ε ≥ 0, let us put:   Dε := [z, zΓ ] ∈ Vε B(z) ∈ L1 (Ω ) and BΓ (zΓ ) ∈ L1 (Γ ) . (2.3) Then, the closures of Dε , for ε ≥ 0, in the topology of H coincide with the class D∗ given in (2.1), i.e. D∗ = Dε in H ,

for any ε ≥ 0.

Based on the above (A1)–(A3) and (Fact 6)–(Fact 7), the solutions to (ACE)ε , for ε ≥ 0, are defined as follows. Definition 2.1 (Definition of Solutions). A pair [u, uΓ ] of functions u : [0, T ] → L2 (Ω ) and uΓ : [0, T ] → L2 (Γ ) is called a solution to (ACE)ε , iff [u, uΓ ] fulfills the following conditions. 1,2 (S1) [u, uΓ ] ∈ C([0, T ]; H ) ∩ Wloc ((0, T ]; H ) ∩ L2 (0, T ; Vε ) ∩ L∞ loc ((0, T ]; Vε ), [u(0), uΓ (0)] = [u0 , uΓ ,0 ] in H . (S2) There exist functions νu : Q → RN , ξ : (0, T ) → L2 (Ω ) and ξΓ : (0, T ) → L2 (Γ ) such that:

νu ∈ L∞ (Q) and νu ∈ Sgn(∇u) a.e. in Q, ξ ∈ L2loc ((0, T ]; L2 (Ω )) and ξ ∈ β(u) a.e. in Q, ξΓ ∈ L2loc ((0, T ]; L2 (Γ )) and ξΓ ∈ βΓ (uΓ ) a.e. in Σ ,

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and 





2

  ∂t u(t)z dx + (νu (t) + κ ∇u(t)) · ∇z dx + ξ(t) + g(u(t)) z dx Ω Ω Ω      + ∂t uΓ (t)zΓ dΓ + ∇Γ (εuΓ (t)) · ∇Γ (εzΓ ) dΓ + ξΓ (t) + gΓ (uΓ (t)) zΓ dΓ Γ Γ  Γ  = θ(t)z dx + θΓ (t)zΓ dΓ , for any [z, zΓ ] ∈ Vε . Ω

Γ

Now our Main Theorems are stated as follows. Main Theorem 1 (Well-Posedness). Let us assume (A1)–(A3) and let us fix an arbitrary ε ≥ 0. Then, the following items hold. (I-1) (Existence and uniqueness) The system (ACE)ε admits a unique solution [u, uΓ ] and there exists a constant C1 > 0, independent of the initial value [u0 , uΓ ,0 ] and the forcing term [θ, θΓ ], such that: 2

2

|[u, uΓ ]|C([0,T ];H ) + |[∇u, ∇Γ (εuΓ )]|L2 (0,T ;H N ) √ 2 2 √     + sup  t [∇u(t), ∇Γ (εuΓ (t))] +  t [∂t u, ∂t uΓ ] 2 L (0,T ;H )



2

HN

t∈(0,T ) 2



≤ C1 1 + |[u0 , uΓ ,0 ]|H + |[θ, θΓ ]|L2 (0,T ;H ) .

(2.4)

Moreover, if [u0 , uΓ ,0 ] ∈ Dε , then there exists a constant C2 > 0, independent of the initial value [u0 , uΓ ,0 ] and the forcing term [θ, θΓ ], such that: 2

2

|[∂t u, ∂t uΓ ]|L2 (0,T ;H ) + sup |[∇u(t), ∇Γ (εuΓ (t))]|H N t∈(0,T )

≤ C2

 1

2

2

+ |[u0 , uΓ ,0 ]|H + |[∇u0 , ∇Γ (εuΓ ,0 )]|H N 2 + |B(u0 )|L1 (Ω) + |BΓ (uΓ ,0 )|L1 (Γ ) + |[θ, θΓ ]|L2 (0,T ;H )

 .

(2.5)

(I-2) (Continuous-dependence) For k = 1, 2, let [uk , ukΓ ] denote two solutions to the problem (ACE)ε corresponding to the forcing pairs [θk , θΓk ] ∈ L2 (0, T ; H ) and initial pairs [uk0 , ukΓ ,0 ] ∈ D∗ , respectively. Then, there exists a positive constant C3 , independent of the choices of [θk , θΓk ] and [uk0 , ukΓ ,0 ], k = 1, 2, such that:  1    [u − u2 , u1Γ − u2Γ ]2 [∇(u1 − u2 ), ∇Γ (ε(u1Γ − u2Γ ))]2 2 + C([0,T ];H ) L (0,T ;H N )      2 2 ≤ C3 [u10 − u20 , u1Γ ,0 − u2Γ ,0 ]H + [θ1 − θ2 , θΓ1 − θΓ2 ]L2 (0,T ;H ) . (2.6)

Main Theorem 2 (ε-Dependence of Solutions). Let ε0 ≥ 0 be a fixed constant. Let {[θε , θΓε ]}ε≥0 ⊂ L2 (0, T ; H ) be a sequence of forcing pairs, let {[uε0 , uεΓ ,0 ] ∈ Dε }ε≥0 ⊂ H be a sequence of initial pairs and for any ε ≥ 0, let [uε , uεΓ ] be a solution to (ACE)ε corresponding to the forcing pair [θε , θΓε ] ∈ L2 (0, T ; H ) and the initial pair [uε0 , uεΓ ,0 ] ∈ Dε . If:  [θε , θΓε ] → [θε0 , θΓε0 ] weakly in L2 (0, T ; H ), as ε → ε0 , (2.7) [uε0 , uεΓ ,0 ] → [uε00 , uεΓ0,0 ] in H ,

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then: [uε , uεΓ ] → [uε0 , uεΓ0 ] in C([0, T ]; H ) and in L2 (0, T ; V0 ), as ε → ε0 .

(2.8)

In particular, if ε0 > 0, then: uεΓ → uεΓ0

in L2 (0, T ; H 1 (Γ )) as ε → ε0 .

(2.9)

3. Key-Lemmas In this section, we specify the essential points in the proofs of Main Theorems, in forms of Key-Lemmas. In any case of ε ≥ 0, the keypoint will be to reformulate the system (ACE)ε as the following Cauchy problem (CP)ε for an evolution equation: (CP)ε :  U ′ (t) + ∂Φε (U (t)) + G(U (t)) ∋ Θ(t) U (0) = U0 in H .

in H , a.e. t ∈ (0, T ),

In the context: – the unknown U ∈ C([0, T ]; H ) corresponds to the solution pair [u, uΓ ] of the system (ACE)ε , i.e., U (t) = [u(t), uΓ (t)] in H , for any t ∈ [0, T ] with the initial pair U0 = [u0 , uΓ ,0 ] in H ; – ∂Φε denotes the subdifferential of a proper l.s.c. and convex function Φε : H → [0, ∞], defined as U = [u, uΓ ] ∈ H → Φε (U ) = Φε (u, uΓ )     κ2  2  |∇u| dx + B(u) dx |∇u| +    Ω  2 Ω 1 := + |∇Γ (εuΓ )|2 dΓ + BΓ (uΓ ) dΓ , if U = [u, uΓ ] ∈ Dε ,   2  Γ Γ   ∞, otherwise, for ε ≥ 0;

(3.1)

– G : H → H is a Lipschitz continuous operator, defined as U = [u, uΓ ] ∈ H → G(U ) = G(u, uΓ ) := [g(u), gΓ (uΓ )] ∈ H , where g and gΓ can be meant as Lipschitz continuous extensions outside IB of the functions g and gΓ defined in (A2); – the forcing term Θ corresponds to the forcing pair [θ, θΓ ] of (ACE)ε , i.e., Θ = [θ, θΓ ] in L2 (0, T ; H ). Remark 3.1. For any ε ≥ 0, we can see that the convex function Φε , defined in (3.1), corresponds to the convex part of the free-energy, given in (0.4). In addition, the subdifferentials ∂Φε , for ε ≥ 0, are maximal monotone graphs in H × H . So, the well-posedness for the Cauchy problem (CP)ε will be verified, immediately, by applying general theories for evolution equations, e.g., [3,4,25]. In the light of Remark 3.1, the essential points in Main Theorem 1 will be to show a certain association between our system (ACE)ε and the Cauchy problem (CP)ε for any ε ≥ 0. To this end, we need to prepare a class of relaxed convex functions λ {Φε,δ | ε ≥ 0, 0 < δ, λ ≤ 1},

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defined as follows: λ λ U = [u, uΓ ] ∈ H → Φε,δ (U ) = Φε,δ (u, uΓ )     2 κ  2  |∇u| dx + B λ (u) dx f (∇u) +  δ   Ω 2   Ω 1 := |∇Γ (εuΓ )|2 dΓ + + BΓλ (uΓ ) dΓ , if U = [u, uΓ ] ∈ Vε ,   2  Γ Γ   ∞, otherwise,

for ε ≥ 0 and 0 < δ, λ ≤ 1.

(3.2)

In the context, {fδ }0<δ≤1 , {B λ }0<λ≤1 and {BΓλ }0<λ≤1 are sequences of functions, prescribed under the following assumptions. (A4) {fδ }0<δ≤1 ⊂ C 1 (RN ) is a sequence of convex functions and C 1 -regularizations for the Euclidean norm | · | ∈ W 1,∞ (RN ), such that: fδ (0) = 0 and fδ (ω) ≥ 0, for any ω ∈ RN and any 0 < δ ≤ 1,  fδ (ω) → |ω|, for any ω ∈ RN , as δ ↓ 0, fδ → | · | on RN , in the sense of Mosco, and there exists a δ-independent constant C0 > 0, satisfying: |∇fδ (ω)| ≤ C0 (|ω| + 1),

for any 0 < δ ≤ 1 and ω ∈ RN .

(A5) {B λ }0<λ≤1 and {BΓλ }0<λ≤1 are sequences of Moreau–Yosida regularizations of the convex functions B and BΓ , respectively, i.e., {B λ }0<λ≤1 ⊂ C 1 (R), {BΓλ }0<λ≤1 ⊂ C 1 (R), and    1   τ ∈ R → B λ (τ ) := inf |˜ τ − τ |2 + B(˜ τ ) τ˜ ∈ R ,  2λ  1  λ  τ ∈ R → BΓ (τ ) := inf |˜ τ − τ |2 + BΓ (˜ τ ) τ˜ ∈ R , 2λ

for any 0 < λ ≤ 1.

λ Now, the first Key-Lemma is concerned with the representations of the subdifferentials ∂Φε,δ ⊂ H ×H λ of the relaxed convex functions Φε,δ , for ε ≥ 0, 0 < δ, λ ≤ 1.

Key-Lemma 1. Let us fix ε ≥ 0, 0 < δ, λ ≤ 1. Let us put:   2 ∇fδ (∇u) + κ2 ∇u ∈ Ldiv (Ω ) λ   , Dε,δ := [u, uΓ ] ∈ Vε −∆Γ (ε2 uΓ ) + (∇fδ (∇u) + κ2 ∇u) · nΓ Γ ∈ L2 (Γ ) λ and let us define an operator Aλε,δ : Dε,δ ⊂ H → H , by putting: λ [u, uΓ ] ∈ Dε,δ → Aλε,δ [u, uΓ ]   t −div(∇f (∇u) + κ2 ∇u) + β λ (u) δ := −∆Γ (ε2 uΓ ) + [(∇fδ (∇u) + κ2 ∇u) · nΓ ]Γ + βΓλ (uΓ ) λ Then, ∂Φε,δ = Aλε,δ in H × H , i.e. λ λ λ ) = Dε,δ and ∂Φε,δ (u, uΓ ) = Aλε,δ [u, uΓ ] in H , D(∂Φε,δ λ for any [u, uΓ ] ∈ D(∂Φε,δ ).

in H .

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The second Key-Lemma is concerned with the continuous dependence between the convex functions Φε for ε ≥ 0, and the relaxations of those. Key-Lemma 2 (Continuous Dependence of the Convex Functions). Let {εn }∞ n=0 ⊂ [0, ∞),

{δn }∞ n=1 ⊂ (0, 1] and

{λn }∞ n=1 ⊂ (0, 1]

be arbitrary sequences such that: εn → ε0 ,

δn ↓ 0 and

λn ↓ 0,

as n → ∞.

Then, for the sequence {Φελnn,δn }∞ n=1 of convex functions, it holds that: Φελnn,δn → Φε0 on H , in the sense of Mosco,

as n → ∞.

On the basis of Key-Lemmas 1–2, we prove the third Key-Lemma, concerned with representations of the subdifferentials ∂Φε ⊂ H × H of Φε , for ε ≥ 0. Key-Lemma 3. For any ε ≥ 0, the following two items are equivalent. (Key 0) U = [u, uΓ ] ∈ D(∂Φε ) and U ∗ = [u∗ , u∗Γ ] ∈ ∂Φε (U ) = ∂Φε (u, uΓ ) in H . (Key 1) U = [u, uΓ ] ∈ Dε , and there exist νu ∈ L∞ (Ω )N and [ξ, ξΓ ] ∈ H , such that:  νu ∈ Sgn(∇u) and ξ ∈ β(u), a.e. in Ω , ξΓ ∈ βΓ (uΓ ), a.e. on Γ ,  2 νu + κ2 ∇u ∈ Ldiv (Ω ), 2 −∆Γ (ε uΓ ) + [(νu + κ2 ∇u) · nΓ ]Γ ∈ L2 (Γ ),

(3.3)

(3.4)

and 

  u∗ = −div νu + κ2 ∇u + ξ in L2 (Ω ), u∗Γ = −∆Γ (ε2 uΓ ) + [(νu + κ2 ∇u) · nΓ ]Γ + ξΓ

in L2 (Γ ).

(3.5)

The last Key-Lemma 3 is useful to guarantee the association between (ACE)ε and (CP)ε for ε ≥ 0, via the representations of subdifferentials. 4. Proofs of Key-Lemmas In this section, we prove three Key-Lemmas for our Main Theorems. To this end, we first prepare the following lemma. Lemma 4.1. Let (S, B, µ) be a measure space with a σ-algebra B and a finite Radon measure µ. Let X be a (real) Hilbert space. Let Ψ : X → (−∞, ∞] be a proper l.s.c. and convex function, and let {Ψn }∞ n=1 be a sequence of proper l.s.c. and convex functions Ψn : X → (−∞, ∞], n ∈ N, such that: Ψn → Ψ on X, in the sense of Mosco,

as n → ∞.

(4.1)

Then, the following two items hold. (I) There exist two constants c0 , d0 > 0, independent of n, such that: Ψn (z) + c0 |z|X + d0 ≥ 0,

for any z ∈ X and any n ∈ N.

(4.2)

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ˆ n }∞ of proper l.s.c. and convex functions (II) The sequence {Ψ n=1   Ψ (ζ) dµ, if Ψ (ζ) ∈ L1 (S), n n ˆ n (ζ) := ζ ∈ L2 (S; X) → Ψ  S ∞, otherwise,

for n = 1, 2, 3, . . . ,

(4.3)

converges to the convex function   Ψ (ζ) dµ, if Ψ (ζ) ∈ L1 (S), ˆ (ζ) := ζ ∈ L2 (S; X) → Ψ  S ∞, otherwise,

(4.4)

on the Hilbert space L2 (S; X), in the sense of Mosco, as n → ∞. Proof. This lemma can be proved by means of similar demonstration techniques as in [19, Appendix]. However, we report the proof for the reader’s convenience. First, we show the item (I). To this end, let us assume that: Ψnk (yk ) + k 2 (|yk |X + 1) < 0,

∞ ∞ for some {nk }∞ k=1 ⊂ {n}n=1 , {yk }k=1 ⊂ X

(4.5)

to derive a contradiction. Let us fix any z0 ∈ D(Ψ ). Then, by (4.1), we find a sequence {ˆ zn }∞ n=1 ⊂ X, such that: zˆn → z0

in X

and Ψn (ˆ zn ) → Ψ (z0 )

as n → ∞.

(4.6)

Here, we define: zk := εk yk + (1 − εk )ˆ znk in X,

for k = 1, 2, 3, . . . ,

(4.7)

with εk :=

1 ∈ (0, 1), k(1 + |yk |X )

for k = 1, 2, 3, . . . .

(4.8)

Then, it follows from (4.5)–(4.8) that |εk yk | ≤ 1/k, whence: zk → z0 in X

as k → ∞,

and subsequently, it follows from (4.1) that: lim Ψnk (zk ) ≥ lim Ψnk (zk ) ≥ Ψ (z0 ).

k→∞

(4.9)

k→∞

In the meantime, in the light of (4.7)–(4.8), and the convexity of Ψnk , for k ∈ N, lim Ψnk (zk ) ≤ lim εk Ψnk (yk ) + lim (1 − εk )Ψnk (ˆ znk ) k→∞ k→∞   2  ≤ lim εk −k (1 + |yk |X ) + Ψ (z0 )

k→∞

k→∞

= − lim k + Ψ (z0 ) = −∞. k→∞

This contradicts with (4.9). Next, we show the item (II). According to [2, Theorem 3.26], it is sufficient (equivalent) to check the following two conditions:

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ˆ n )−1 ξ → ζ := (IL2 (S;X) + λ∂ Ψ ˆ )−1 ξ in L2 (S; X) as n → ∞, for any λ > 0 and (ii-1) ζn := (IL2 (S;X) + λ∂ Ψ 2 any ξ ∈ L (S; X); ˆ in L2 (S; X) × L2 (S; X) and a sequence {[ζn , ηn ] ∈ Ψ ˆ n }∞ ⊂ L2 (S; X) × (ii-2) there exist [ζ, η] ∈ ∂ Ψ n=1 ˆ n (ζn ) → Ψ ˆ (ζ) as n → ∞. L2 (S; X) such that [ζn , ηn ] → [ζ, η] in L2 (S; X) × L2 (S; X) and Ψ For the verification of (ii-1), let us fix any λ > 0 and any ξ ∈ L2 (S; X). Then, invoking [4, Proposition 2.16], [2, Theorem 3.26], (4.1) and (4.3)–(4.4), we infer that:  (ξ − ζ)(σ) ∈ λ∂Ψ (ζ(σ)) in X, for µ-a.e. σ ∈ S, (4.10) (ξ − ζn )(σ) ∈ λ∂Ψn (ζn (σ)) in X, n = 1, 2, 3, . . . , and ζn (σ) → ζ(σ) in X as n → ∞,

for µ-a.e. σ ∈ S.

(4.11)

Also, by using the sequence {ˆ zn }∞ n=1 ⊂ X as in (4.6), it is seen that:   ˆ n (ζn (σ)) − λΨ ˆ n (ˆ (ξ − ζn )(σ), ζn (σ) − zˆn X ≥ λΨ zn ), for any n ∈ N and µ-a.e. σ ∈ S.

(4.12)

Additionally, by virtue of the item (I), (4.6), (4.12) and the Schwarz and Young inequalities, we can compute that: ˆ n (ˆ ˆ n (ζn (σ)) |ζn (σ)|2X ≤ (ζn (σ), zˆn )X + (ξ(σ), ζn (σ) − zˆn )X + λΨ zn ) − λ Ψ   ˆ n (ˆ ≤ |ζn (σ)|X |ˆ zn |X + |ζn (σ)|X |ξ(σ)|X + |ξ(σ)|X |ˆ zn |X + λΨ zn ) + λ c0 |ζn (σ)|X + d0 5 3 ˆ n (ˆ zn ) + 2|ˆ zn |2X , ≤ |ζn (σ)|2X + |ξ(σ)|2X + λ2 c20 + λd0 + λΨ 4 4 and therefore, ˆ 1, |ζn (σ)|2X ≤ 5|ξ(σ)|2X + M

for µ-a.e. σ ∈ S and any n ∈ N,

(4.13)

where   ˆ 1 := 4 λ(c0 + d0 ) + 1 2 + 4 sup(λΨ ˆ n (ˆ M zn ) + 2|ˆ zn |2X ). n∈N

In view of these, the condition (ii-1) will be obtained as a consequence of (4.11), (4.13) and Lebesgue’s dominated convergence theorem. Finally, for the verification of (ii-2), we consider the class of functions {ζ, ζn |n ∈ N} ⊂ L2 (S; X) as in (ii-1) with fixed λ > 0 and ξ ∈ L2 (S; X), and let us set:  ξ−ζ  η := in L2 (S; X), λ (4.14)  ηn := ξ − ζn in L2 (S; X), n = 1, 2, 3, . . . . λ ˆ λ : L2 (S; X) → R, Ψnλ : X → R and Ψ ˆ nλ : L2 (S; X) → R, n ∈ N, Also, let us denote by Ψ λ : X → R, Ψ ˆ , Ψn and Ψ ˆ n , n ∈ N, respectively. Then, by [3, the Moreau–Yosida regularizations of convex functions Ψ , Ψ Theorem 2.9, p. 48], [4, Proposition 2.11], (ii-1) and (4.14), we immediately have:  ˆ λ (ξ) ∈ ∂ Ψ ˆ (ζ) in L2 (S; X), η = ∂Ψ ˆ λ (ξ) ∈ ∂ Ψ ˆ n (ζn ) in L2 (S; X), n = 1, 2, 3, . . . , ηn = ∂ Ψ n ˆ λ (ξ) = ξ − ζ in L2 (S; X) ˆ nλ (ξ) = ξ − ζn → η = ∂ Ψ ηn = ∂ Ψ λ λ

as n → ∞.

(4.15) (4.16)

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In particular, [4, Proposition 2.16] and (4.16) enable us to say that: ∂Ψnλ (ξ(σ)) → ∂Ψ λ (ξ(σ)) in X as n → ∞,

for µ-a.e. σ ∈ S,

(4.17)

by taking a subsequence if necessary. Besides, from [2, Theorem 3.26] and (4.1), it follows that: Ψnλ (ξ(σ)) → Ψ λ (ξ(σ)) as n → ∞,

for µ-a.e. σ ∈ S.

(4.18)

On account of [3, Theorem 2.9, p. 48], [4, Proposition 2.11] and (4.17)–(4.18), it is inferred that: 1 |(ξ − ζn )(σ)|2X 2λ 1 → Ψ λ (ξ(σ)) − |(ξ − ζ)(σ)|2X 2λ = Ψ (ζ(σ)) as n → ∞, for µ-a.e. σ ∈ S.

Ψn (ζn (σ)) = Ψnλ (ξ(σ)) −

(4.19)

Furthermore, invoking (4.12)–(4.13), the item (I), and using the sequence {ˆ zn }∞ n=1 as in (4.6) and the Schwarz and Young inequalities, we obtain that: |Ψn (ζn (σ))| ≤ Ψn (ζn (σ)) ∨ (c0 |ζn (σ)|X + d0 )      (ξ − ζn )(σ) , zˆn − ζn (σ)  + c0 |ζn (σ)|X + d0 ≤ |Ψn (ˆ zn )| +  λ X 1 1 ≤ |Ψn (ˆ zn )| + |ξ(σ)|X |ˆ zn |X + |ξ(σ)|X |ζn (σ)|X λ λ 1 1 + |ζn (σ)|X |ˆ zn |X + |ζn (σ)|2X + c0 |ζn (σ)|X + d0 λ λ 1 1 λ 3 2 zn |2X + c20 + d0 ≤ |Ψn (ˆ zn )| + |ζn (σ)|X + |ξ(σ)|2X + |ˆ λ λ λ 4  1 3 1 λ 2 2 ˆ 1 + |ξ(σ)|X + |Ψn (ˆ ≤ 5|ξ(σ)|X + M zn )| + |ˆ zn |2X + c20 + d0 λ λ λ 4 16 ˆ 2 , for µ-a.e. σ ∈ S and any n ∈ N, ≤ |ξ(σ)|2X + M λ

(4.20)

where   3 ˆ λ 2 1 2 ˆ M2 := M1 + c0 + d0 + sup Ψn (ˆ zn ) + |ˆ zn |X . λ 4 λ n∈N In the light of (4.19)–(4.20), we can apply Lebesgue’s dominated convergence theorem, to derive that: ˆ n (ζn ) → Ψ ˆ (ζ) Ψ

as n → ∞.

Then, (4.14)–(4.16), (4.21) and the previous (ii-1) imply the validity of (ii-2).

(4.21) 

Now Key-Lemmas 1–3 are proved as follows. λ λ Proof of Key-Lemma 1. First, we show that ∂Φε,δ ⊂ Aλε,δ in H × H . Let us assume that [u, uΓ ] ∈ D(∂Φε,δ ) ∗ ∗ λ and [u , uΓ ] ∈ ∂Φε,δ (u, uΓ ) in H . Besides, let us take arbitrary τ > 0 and [z, zΓ ] ∈ Vε , to compute that:

(u∗ , z)L2 (Ω) + (u∗Γ , zΓ )L2 (Γ )  1 λ λ ≤ Φε,δ (u + τ z, uΓ + τ zΓ ) − Φε,δ (u, uΓ ) τ    κ2 κ2 1 2 2 = fδ (∇(u + τ z)) + |∇(u + τ z)| − fδ (∇u) − |∇u| dx τ Ω 2 2    1 + |∇Γ (ε(uΓ + τ zΓ ))|2 − |∇Γ (εuΓ )|2 dΓ 2τ Γ

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

   λ  1 B λ (u + τ z) − B λ (u) dx + BΓ (uΓ + τ zΓ ) − BΓλ (uΓ ) dΓ τ Γ Ω    → ∇fδ (∇u) + κ2 ∇u · ∇z dx + ∇Γ (εuΓ ) · ∇Γ (εzΓ ) dΓ Ω Γ   + β λ (u)z dx + βΓλ (uΓ )zΓ dΓ as τ ↓ 0. 1 + τ 



47



Ω

Γ

Since the choice of [z, zΓ ] ∈ Vε is arbitrary, we have   ∗   u − β λ (u), z L2 (Ω) + u∗Γ − βΓλ (uΓ ), zΓ L2 (Γ )     2 = ∇fδ (∇u) + κ ∇u · ∇z dx + ∇Γ (εuΓ ) · ∇Γ (εzΓ ) dΓ , Ω

for any [z, zΓ ] ∈ Vε .

(4.22)

Γ

Here, taking any ϕ0 ∈ H01 (Ω ) and putting [z, zΓ ] = [ϕ0 , 0] ∈ Vε in (4.22),    (u∗ − β λ (u), ϕ0 )L2 (Ω) = ∇fδ (∇u) + κ2 ∇u · ∇ϕ0 dx, for any ϕ0 ∈ H01 (Ω ), Ω

which implies   − div ∇fδ (∇u) + κ2 ∇u = u∗ − β λ (u) ∈ L2 (Ω )

in D ′ (Ω ).

(4.23)

Additionally, with Remark 1.2 (Fact 1)–(Fact 3) and (4.22)–(4.23) in mind, we can see that: (u∗Γ − βΓλ (uΓ ), zΓ )L2 (Γ )      ∗  2 λ = ∇fδ (∇u) + κ ∇u · ∇z dx − u − β (u), z L2 (Ω) + ∇Γ (εuΓ ) · ∇Γ (εzΓ ) dΓ Ω Γ     ∇fδ (∇u) + κ2 ∇u · nΓ Γ , zΓ = − 12 1 H

(Γ )

H 2 (Γ )

+ H −1 (Γ ) ⟨−∆Γ (εuΓ ), εzΓ ⟩H 1 (Γ ) ,

for any [z, zΓ ] ∈ Vε .

This identity leads to: − ∆Γ (ε2 uΓ ) +



  ∇fδ (∇u) + κ2 ∇u · nΓ Γ = u∗Γ − βΓλ (uΓ ) ∈ L2 (Γ )

in H −1 (Γ ).

(4.24)

As a consequence of (4.22)–(4.24), we obtain that: λ [u, uΓ ] ∈ Dε,δ

and

[u∗ , u∗Γ ] ∈ Aλε,δ [u, uΓ ]

in H .

(4.25)

λ λ Secondly, we show that Aλε,δ ⊂ ∂Φε,δ in H × H . Let us assume that [u, uΓ ] ∈ Dε,δ and [u∗ , u∗Γ ] ∈ in H , and let us take an arbitrary [z, zΓ ] ∈ Vε . Then, taking into account Remark 1.2 (Fact 1)–(Fact 3), (A4)–(A5) and the convexity of the squared norm, we compute that:

Aλε,δ [u, uΓ ]

([u∗ , u∗Γ ], [z, zΓ ] − [u, uΓ ])H     = −div ∇fδ (∇u) + κ2 ∇u + β λ (u), z − u L2 (Ω)      + −∆Γ (ε2 uΓ ) + ∇fδ (∇u) + κ2 ∇u · nΓ + βΓλ (uΓ ), zΓ − uΓ Γ L2 (Γ )     = ∇fδ (∇u) + κ2 ∇u · ∇(z − u) dx + β λ (u)(z − u) dx Ω Ω + ∇Γ (εuΓ ) · ∇Γ (ε(zΓ − uΓ )) dΓ + βΓλ (uΓ )(zΓ − uΓ ) dΓ Γ Γ      λ  κ2 κ2 ≤ fδ (∇z) − fδ (∇u) + |∇z|2 − |∇u|2 dx + B (z) − B λ (u) dx 2 2  Ω  Ω    λ  1 2 2 + |∇Γ (εzΓ )| − |∇Γ (εuΓ )| dΓ + BΓ (zΓ ) − BΓλ (uΓ ) dΓ 2 Γ Γ λ λ ≤ Φε,δ (z, zΓ ) − Φε,δ (u, uΓ ), for any [z, zΓ ] ∈ Vε .

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48

It implies that: λ [u, uΓ ] ∈ D(∂Φε,δ )

λ [u∗ , u∗Γ ] ∈ ∂Φε,δ (u, uΓ )

and

in H .

(4.26)



Thus, we conclude this lemma by (4.25)–(4.26).

∞

Proof of Key-Lemma 2. First, we verify the condition of lower-bound. Let [ˇ u, u ˇΓ ] ∈ H and {[ˇ un , u ˇΓ ,n ]}n=1 ⊂ H be such that: weakly in H as n → ∞.

[ˇ un , u ˇΓ ,n ] → [ˇ u, u ˇΓ ] Then, we may a subsequence

un , u ˇΓ ,n ) suppose limn→∞ Φελnn,δn (ˇ ∞ {nk }∞ ⊂ {n} , such that: n=1 k=1

(4.27)

< ∞, because the other case is trivial. Hence, there exists

λn ˇ ∗ := lim Φ λn (ˇ Φ ˇΓ ,n ) = lim Φεn k,δn (ˇ unk , u ˇΓ ,nk ) < ∞. εn ,δn un , u k→∞

n→∞

k

(4.28)

k

∞

Here, from (3.2), it can be seen that {[ˇ unk , u ˇΓ ,nk ]}k=1 is bounded in V0 (resp. Vε0 ), if ε0 = 0 (resp. if ε0 > 0). So, by invoking (Fact 6) and (4.27), and taking more subsequences if necessary, we can further suppose that:  [ˇ unk , u ˇΓ ,nk ] → [ˇ u, u ˇΓ ] in H and weakly in V0 , as k → ∞, (4.29) u ˇnk (x) → u ˇ(x) a.e. x ∈ Ω , and in particular, if ε0 > 0, then u ˇΓ ,nk → u ˇΓ

weakly in H 1 (Γ ) as k → ∞.

(4.30)

Additionally, from (A4) and Lemma 4.1, we can infer that: (Fact 8) The sequence of convex functions:  ∞  ω ∈ L2 (Ω )N → fδn (ω) dx ∈ [0, ∞] Ω

n=1

1

converges to the convex function of L -norm: 2

N

ω ∈ L (Ω )

 →

|ω| dx ∈ [0, ∞), Ω

on the Hilbert space L2 (Ω )N , in the sense of Mosco, as n → ∞. In the light of Remark 1.5, (4.27)–(4.30), Fatou’s lemma and the above (Fact 8), the condition of lowerbound is verified as follows:    κ2 ˇ ∗ ≥ lim fδnk (∇ˇ unk ) dx + lim |∇ˇ unk |2 dx + lim B λnk (ˇ unk ) dx Φ 2 k→∞ Ω k→∞ Ω k→∞ Ω   1 λn 2 + lim |∇Γ (εnk u ˇΓ ,nk )| dΓ + lim BΓ k (ˇ uΓ ,nk ) dΓ 2 k→∞ Γ k→∞ Γ ≥ Φε0 (ˇ u, u ˇΓ ). Next, we verify the optimality condition. Let us fix any [ˆ u0 , u ˆΓ ,0 ] ∈ D(Φε0 ), and let us take a sequence 1 {ωi }∞ ⊂ H (Ω ) in the following way: i=1  ∞  u0 }; • if ε0 > 0, then {ωi }i=1 = {ˆ 1 (4.31) ˆ0 in H 1 (Ω ), • if ε0 = 0, then {ωi }∞ ⊂ C (Ω ) is such that ωi → u i=1   and pointwisely a.e. in Ω , as i → ∞.

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49

1 Besides, let us define a sequence {ϕˆi }∞ i=1 ⊂ H (Ω ), by putting:

ϕˆi := TB ωi in H 1 (Ω ),

for i = 1, 2, 3, . . . ,

where the projection function TB is defined by (2.2). Then, in view of (4.31), and taking a subsequence if necessary, we have that   ˆ0 in H 1 (Ω ), and pointwisely a.e. in Ω , as i → ∞ ϕˆi → u 1 (4.32) (ϕˆi )|Γ → u ˆΓ ,0 in H 2 (Γ ), and pointwisely a.e. on Γ , as i → ∞  {(ϕˆ ) }∞ ⊂ H 1 (Γ ), and (ε ϕˆ ) → ε u 1 as i → ∞. i |Γ i=1 0 i |Γ 0 ˆΓ ,0 in H (Γ ), Also, invoking (A4) and Lebesgue’s dominated convergence theorem, we can configure a sequence {ni }∞ i=0 ⊂ N, such that 1 =: n0 < n1 < n2 < n3 < · · · < ni ↑ ∞, as i → ∞, and for any i ∈ N ∪ {0},  1  sup |fδn (∇ϕˆi ) − |∇ϕˆi ||L1 (Ω) < i+1 ,    2 n≥ni         λ  1   < i, sup B n (ϕˆi ) − B(ϕˆi )L1 (Ω) + BΓλn (ϕˆi ) − BΓ (ϕˆi ) 1 (4.33) 2 L (Γ )  n≥n   i    1    sup (ε2n − ε20 ) |∇ϕˆi |2L2 (Ω) < i+1 . 2 n≥ni Based on these, let us define: [ˆ un , u ˆΓ ,n ] := [ϕˆi , (ϕˆi )|Γ ],

if ni ≤ n < ni+1 , for some i ∈ N ∪ {0}.

(4.34)

Then, with condition (a2) in (A1), (4.32) and Lebesgue’s dominated convergence theorem in mind, one can see that:     lim B(ˆ un ) dx = B(ˆ u0 ) dx, and lim BΓ (ˆ uΓ ,n ) dΓ = BΓ (ˆ uΓ ,0 ) dx. (4.35) n→∞

Ω

Ω

n→∞

Γ

Γ

Taking into account (4.31)–(4.35), we obtain that:    λn  un , u ˆΓ ,n ) − Φε0 (ˆ u0 , u ˆΓ ,0 ) Φεn ,δn (ˆ        fδn (∇ˆ un | − |∇ˆ u0 |) dx ≤ un ) − |∇ˆ un | dx +  (|∇ˆ Ω Ω        κ2 1  2 2 |∇ˆ + un | − |∇ˆ u0 |  dx +  |∇Γ (εn u ˆΓ ,n )|2 − |∇Γ (ε0 u ˆΓ ,0 )|2 dΓ  2 2 Γ     Ω       +  B λn (ˆ un ) dx − B(ˆ un ) dx +  B(ˆ un ) dx − B(ˆ u0 ) dx Ω Ω Ω Ω          λ n +  BΓ (ˆ uΓ ,n ) dΓ − BΓ (ˆ uΓ ,n ) dΓ  +  BΓ (ˆ uΓ ,n ) dΓ − BΓ (ˆ uΓ ,0 ) dΓ  Γ Γ Γ  Γ     ≤ fδn (∇ˆ un ) − |∇ˆ un | dx + |∇(ˆ un − u ˆ0 )|L1 (Ω)N Ω

 1 κ2  2 |∇ˆ un | + |∇ˆ u0 |L2 (Ω) |∇(ˆ un − u ˆ0 )|L2 (Ω)N + |ε2n − ε20 | |∇Γ u ˆΓ ,n |L2 (Γ )N 2 2  1 + |∇Γ (ε0 u ˆΓ ,n )| + |∇Γ (ε0 u ˆΓ ,0 )|L2 (Γ ) |∇Γ (ε0 (ˆ uΓ ,n − u ˆΓ ,0 ))|L2 (Γ )N 2    λ    + B n (ˆ un ) − B(ˆ un )L1 (Ω) + BΓλn (ˆ uΓ ,n ) − BΓ (ˆ uΓ ,0 ) 1 L (Γ )           +  B(ˆ un ) dx − B(ˆ u0 ) dx +  BΓ (ˆ uΓ ,n ) dΓ − BΓ (ˆ uΓ ,0 ) dΓ  Ω Ω Γ Γ   1 1 N ˆ ≤ i−1 + 1 + L (Ω ) 2 + Φ∗ |ˆ un − u ˆ0 |H 1 (Ω) + |ε0 (ˆ uΓ ,n − u ˆΓ ,n )|H 1 (Γ ) 2 +

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50

             uΓ ,n ) dΓ − un ) dx − BΓ (ˆ uΓ ,0 ) dΓ  , +  B(ˆ B(ˆ u0 ) dx +  BΓ (ˆ Ω

Γ

Ω

Γ

for any i ∈ N ∪ {0} and n ≥ ni , where ˆ ∗ := sup Φ n∈N



 κ2  |∇ˆ un | + |∇ˆ u0 |L2 (Ω) + 2

  1  |∇Γ (ε0 uΓ ,n )| + |∇Γ (ε0 uΓ ,0 )|L2 (Γ ) . 2

1 1 This implies that the sequence {[ˆ un , u ˆΓ ,n ]}∞ n=1 ⊂ H (Ω ) × H (Γ ) is the required sequence to verify the condition of optimality. 

By a similar demonstration technique, we also see the following corollary. Corollary 4.1. Let {εn }∞ n=1 ⊂ [0, ∞) be arbitrary sequence such that εn → ε0 as n → ∞. Then, for the sequence {Φεn }∞ of convex functions, it holds that: n=1 Φεn → Φε0 on H , in the sense of Mosco,

as n → ∞.

Proof of Key-Lemma 3. Let us fix any ε ≥ 0, and let us define a set-valued map Aε : H → 2H , by putting:   there exist νu ∈ L∞ (Ω )N and [ξ, ξΓ ] ∈ H D(Aε ) := [u, uΓ ] ∈ Vε , (4.36) such that (3.3)–(3.4) hold. and [u, uΓ ] ∈ D(Aε ) ⊂ H → Aε [u, uΓ ]   (3.5) holds, for some νu ∈ L∞ (Ω )N and ∗ ∗ . := [u , uΓ ] ∈ H [ξ, ξΓ ] ∈ H , satisfying (3.3)–(3.4)

(4.37)

Then, the assertion of Key-Lemma 3 can be rephrased as follows: ∂Φε = Aε

in H × H .

(4.38)

This coincidence will be obtained as a consequence of the following Claims 1–2. Claim 1. Aε ⊂ ∂Φε in H × H . Let us assume that [u, uΓ ] ∈ D(Aε ) and [u∗ , u∗Γ ] ∈ Aε [u, uΓ ] in H . Then, from (A1), Remark 1.2 (Fact 1)–(Fact 3) and Remark 1.3, and (4.36)–(4.37), it is inferred that:  ∗ ∗  [u , uΓ ], [z, zΓ ] − [u, uΓ ] H        = −div(νu + κ2 ∇u) + ξ, z − u L2 (Ω) + −∆Γ (ε2 uΓ ) + νu + κ2 ∇u · nΓ Γ + ξΓ , zΓ − uΓ 2 L (Γ )   = (νu + κ2 ∇u) · ∇(z − u) dx + ξ(z − u) dx Ω Ω   + ∇Γ (εuΓ ) · ∇Γ (ε(zΓ − uΓ )) dΓ + ξΓ (zΓ − uΓ ) dΓ Γ Γ     κ2 κ2 2 2 ≤ |∇z| − |∇u| + |∇z| − |∇u| dx + (B(z) − B(u)) dx 2 2 Ω Ω     1 + |∇Γ (εzΓ )|2 − |∇Γ (εuΓ )|2 dΓ + (BΓ (zΓ ) − BΓ (uΓ )) dΓ 2 Γ Γ ≤ Φε (z, zΓ ) − Φε (u, uΓ ), for any [z, zΓ ] ∈ Vε .

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51

Thus, we have: [u, uΓ ] ∈ D(∂Φε )

and

[u∗ , u∗Γ ] ∈ ∂Φε (u, uΓ )

in H ,

and we can say that: Aε ⊂ ∂Φε (u, uΓ )

in H × H .

Claim 2. (Aε + IH )H = H . Since, (Aε + IH )H ⊂ H is trivial, it is sufficient to prove the converse inclusion. Let us take any [w, wΓ ] ∈ H . Then, by applying Minty’s theorem, Key-Lemma 1 and Remark 1.2 (Fact 1)–(Fact 3), we can configure a class of functions {[uλδ , uλΓ ,δ ] | 0 < δ, λ ≤ 1} ⊂ Vε , by setting:  λ λ  [uδ , uΓ ,δ ] := (Aλε,δ + IH )−1 [w, wΓ ], 0 < δ, λ ≤ 1 in H , i.e. λ [w − uλδ , wΓ − uλΓ ,δ ] = ∂Φε,δ (uλδ , uλΓ ,δ ) in H ,

for any 0 < δ, λ ≤ 1,

(4.39)

and we can see that:       λ 2 λ λ λ λ ∇fδ (∇uδ ) + κ ∇uδ · ∇z dx + ∇Γ (εuΓ ,δ ) · ∇Γ (εzΓ ) dΓ + β (uδ )z dx + βΓλ (uλΓ ,δ )zΓ dΓ Ω

Γ

 = Ω

(w − uλδ )z dx +



Ω

(wΓ − uλΓ ,δ )zΓ dΓ ,

Γ

for any [z, zΓ ] ∈ Vε and any 0 < δ, λ ≤ 1.

(4.40)

Γ

In the variational form (4.40), let us first put [z, zΓ ] = [uλδ , uλΓ ,δ ] ∈ Vε in (4.40). Then, with (A1), (A4)–(A5) and Young’s inequality in mind, we deduce that:  2 2  2 1 1  λ λ  [u , u ] + κ2 ∇uλδ L2 (Ω)N + ∇Γ (εuλΓ ,δ )L2 (Γ )N ≤ [w, wΓ ]H , 2 δ Γ ,δ H 2

for any 0 < δ, λ ≤ 1. (4.41)

Next, let us take [z, zΓ ] = [β λ (uλδ ), β λ (uλΓ ,δ )] in Vε . Then, by applying (A1), (A4)–(A5), [1, Theorem 3.99], [5, Lemma 4.4] and Schwarz’s inequality,  λ λ 2     β (uδ ) 2 ≤ w, β λ (uλδ ) L2 (Ω) + wΓ , β λ (uλΓ ,δ ) L2 (Γ ) L (Ω) 2 2 1 1  ≤ |w|2L2 (Ω) + β λ (uλδ )L2 (Ω) + a21 |wΓ |2L2 (Γ ) + 2 β λ (uλΓ ,δ )L2 (Γ ) 4 4a1   1  2 1  2 ≤ |w|2L2 (Ω) + a21 |wΓ |2L2 (Γ ) + β λ (uλδ )L2 (Ω) + 2 a1 |βΓλ (uλΓ ,δ )| + b1 L2 (Γ ) 4 4a1      2 1 1 b2 2 2 ≤ (1 + a21 )[w, wΓ ]H + β λ (uλδ )L2 (Ω) + βΓλ (uλΓ ,δ )L2 (Γ ) + 12 HN −1 (Γ ), 4 2 2a1 so that 2  2 3  λ λ 2 1 b2 β (uδ ) L2 (Ω) − βΓλ (uλΓ ,δ )L2 (Γ ) ≤ (1 + a21 )[w, wΓ ]H + 12 HN −1 (Γ ), 4 2 2a1 for any 0 < δ, λ ≤ 1.

(4.42)

Similarly, putting [z, zΓ ] = [βΓλ (uλδ ), βΓλ (uλΓ ,δ )] ∈ Vε in (4.40), we infer that:   2 1  λ λ 2 3  λ λ 2 1  b2 − β (uδ ) L2 (Ω) + βΓ (uΓ ,δ ) L2 (Γ ) ≤ 1 + 2 [w, wΓ ]H + 0 LN (Ω ), 2 4 a0 2 for any 0 < δ, λ ≤ 1.

(4.43)

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52

Taking the sum of (4.42)–(4.43), it follows that:   2  2 b2 1 1  λ λ b2 β (uδ ), βΓλ (uλΓ ,δ ) H ≤ 2 + 2 + a21 [w, wΓ ]H + 0 LN (Ω ) + 12 HN −1 (Γ ), 4 a0 2 2a1 for any 0 < δ, λ ≤ 1.

(4.44)

On account of (4.41) and (4.44), we find pairs of functions [u, uΓ ] ∈ Vε and [ξ, ξΓ ] ∈ H and sequences δ1 > δ2 > δ3 > · · · > δn ↓ 0, λ1 > λ2 > λ3 > · · · > λn ↓ 0, as n → ∞, such that:  [un , uΓ ,n ] := [uλδ n , uλΓn,δ ] → [u, uΓ ] in H and weakly in Vε , n  n  as n → ∞. (4.45)  β λn (un ), βΓλn (uΓ ,n ) → [ξ, ξΓ ] weakly in H , Here, in the light of Key-Lemma 2, (4.39) and (4.45), we can apply Remark 1.6 (Fact 5) to see that: in H ,

[w − u, wΓ − uΓ ] ∈ ∂Φε (u, uΓ ) and λn Φε,δ (un , uΓ ,n ) → Φε (u, uΓ ) n

as n → ∞.

(4.46)

Also, by (A1), (A5), Remarks 1.5 and 1.6 (Fact 5), we see that: ξ ∈ β(u)

and ξΓ ∈ βΓ (uΓ )

a.e. in Ω ,

a.e. on Γ .

(4.47)

By virtue of (4.45)–(4.46), (A4), Remark 1.5, Lemma 4.1, and (Fact 8), we further compute that:    κ2 κ2 κ2 |∇un |2 dx |∇u|2 dx ≤ lim |∇un |2 dx ≤ lim 2 Ω 2 n→∞ Ω 2 n→∞ Ω     λn ∇Γ (εuΓ ,n )2 dΓ ≤ lim Φε,δn (un , uΓ ,n ) − lim fδn (∇un ) dx − lim n→∞ n→∞ Ω n→∞ Γ    − lim B λn (un ) dx + BΓλn (uΓ ,n ) dΓ n→∞ Ω Γ    2 ≤ Φε (u, uΓ ) − |∇u| dx − ∇Γ (εuΓ ) dΓ Ω

Γ

 −



 B(u) dx +

Ω

BΓ (uΓ ) dΓ

=

Γ

κ2 2



|∇u|2 dx.

(4.48)

Ω

Having in mind (4.45), (4.48) and the above calculation and the uniform convexity of L2 -based topologies, it is deduced that:  un → u in H 1 (Ω ), as n → ∞. (4.49) ∇un → ∇u in L2 (Ω )N , In the meantime, from (A4) and (4.49),   2 |∇fδn (∇un )|L2 (Ω)N ≤ 2C0 sup |∇un |2L2 (Ω)N + LN (Ω ) ,

for any n ∈ N,

n∈N

which enables us to say ∇fδn (∇un ) → νu weakly in L2 (Ω )N as n → ∞,

for some νu ∈ L2 (Ω )N ,

(4.50)

by taking a subsequence if necessary. In view of (4.49)–(4.50), (Fact 8), Remark 1.3, and Remark 1.6 (Fact 5), one can see that: νu ∈ Sgn(∇u)

a.e. in Ω .

(4.51)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

53

Hence, letting n → ∞ in (4.40) yields that:       νu + κ2 ∇u · ∇z dx + ∇Γ (εuΓ ) · ∇Γ (εzΓ ) dΓ + ξz dx + ξΓ zΓ dΓ Ω

Γ

 (w − u)z dx +

=

Ω



Ω

(wΓ − uΓ )zΓ dΓ ,

Γ

for any [z, zΓ ] ∈ Vε .

(4.52)

Γ

In particular, taking any ϕ0 ∈ H01 (Ω ) and putting [z, zΓ ] = [ϕ0 , 0] in Vε ,  (w − u − ξ, ϕ0 )L2 (Ω) = (νu + κ2 ∇u) · ∇ϕ0 dx, for any ϕ0 ∈ H01 (Ω ). Ω

which implies: − div(νu + κ2 ∇u) = w − u − ξ ∈ L2 (Ω )

in D ′ (Ω ).

(4.53)

Furthermore, with Remark 1.2 (Fact 1)–(Fact 3), (4.52)–(4.53) in mind, we can see that: (wΓ − uΓ − ξΓ , zΓ )L2 (Γ )     2 2 = νu + κ ∇u · ∇z dx − (w − u − ξ, z)L (Ω) + ∇Γ (εuΓ ) · ∇Γ (εzΓ ) dΓ Ω Γ     νu + κ2 ∇u · nΓ , zΓ + H −1 (Γ ) ⟨−∆Γ (εuΓ ), εzΓ ⟩H 1 (Γ ) , = − 12 1 H

(Γ )

Γ

H 2 (Γ )

for any [z, zΓ ] ∈ Vε .

This identity leads to: − ∆Γ (ε2 uΓ ) +



  νu + κ2 ∇u · nΓ Γ = wΓ − uΓ − ξΓ ∈ L2 (Γ )

in H −1 (Γ ).

(4.54)

As a consequence of (4.47), (4.51), (4.53)–(4.54), we obtain Claim 2. Now, by using Claims 1–2 and the maximality of ∂Φε in H × H , we can show the coincidence ∂Φε = Aε in H × H , and we conclude this Key-Lemma 3.  5. Proofs of Main Theorems In this section, we will prove two Main Theorems by using the results of the previous sections. Proof of Main Theorem 1. First, we show the item (I-1). In the Cauchy problem (CP)ε , we see from (A3) and (Fact 7) that: Θ = [θ, θΓ ] ∈ L2 (0, T ; H )

and U0 = [u0 , uΓ ,0 ] ∈ D(Φε ).

Hence, by applying the general theories of evolution equations, e.g., [3, Theorem 4.1, p. 158], [4, Theorem 3.6 and Proposition 3.2], [25, Section 2] and [27, Theorem 1.1.2], we immediately have the existence and uniqueness of solution U = [u, uΓ ] ∈ L2 (0, T ; H ) to (CP)ε , such that: 1,2 U ∈ C([0, T ]; H ) ∩ L2 (0, T ; Vε ) ∩ Wloc ((0, T ]; H )

and Φε (U ) ∈ L1 (0, T ) ∩ L∞ loc ((0, T ]).

Also, there exists a positive constant C1 , independent of U0 and Θ, such that:  T √ 2 2 |U |C([0,T ];H )∩L2 (0,T ;Vε ) + Φε (U (t)) dt +  tU ′ L2 (0,T ;H ) + sup tΦε (U (t)) t∈(0,T )

0





≤ C1 1 + |U0 |2H + |Θ|2L2 (0,T ;H ) . Moreover, if U0 ∈ D(Φε ), there exists a positive constant C2 , independent of U0 and Θ, such that:   |U ′ |2L2 (0,T ;H ) + sup Φε (U (t)) ≤ C2 1 + |U0 |2H + |Θ|2L2 (0,T ;H ) + Φε (U0 ) . t∈(0,T )

(5.1)

(5.2)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

54

Now, Key-Lemma 3 guarantees that the solution U = [u, uΓ ] to (CP)ε coincides with that to the system (ACE)ε . Besides, in the light of (3.1) and (A1), the inequalities (2.4) and (2.5) directly follow from (5.1) and (5.2), respectively. Next, we show the item (I-2). For k = 1, 2, let U k := [uk , ukΓ ] be the two solutions to (CP)ε corresponds to forcing term Θ k := [θk , θΓk ] ∈ L2 (0, T ; H ) and initial term U0k = [uk0 , ukΓ ,0 ] ∈ D(Φε ), respectively. Then, we can obtain the inequality (2.6) by using standard method: more precisely, by taking the difference between the two evolution equations, multiplying it by U 1 (t) − U 2 (t), and applying Gronwall’s lemma.  Proof of Main Theorem 2. Main Theorem 2 is proved by referring to the demonstration technique as in [27, Theorem 2.7.1]. Let us set J0 := [ε0 − 1, ε0 + 1] ∩ [0, ∞). On this basis, we divide the proof in the following two steps. (Step 1) The case when {Φε (U0ε )}ε∈J0 is bounded. If {Φε (U0ε )}ε∈J0 is bounded, then the estimate (2.5) implies the following facts:  {U ε }ε∈J0 is bounded in W 1,2 (0, T ; H ) ∩ L∞ (0, T ; V0 ), {εuεΓ }ε∈J0 is bounded in W 1,2 (0, T ; L2 (Γ )) ∩ L∞ (0, T ; H 1 (Γ )). Therefore, applying general theories of compactness, such as Ascoli’s theorem (cf. [36]), we find a sequence 1,2 {εn }∞ (0, T ; H ) ∩ L∞ (0, T ; V0 ), such that: n=1 ⊂ J0 and a limit point U = [u, uΓ ] ∈ W U εn → U

in C([0, T ]; H ), weakly in W 1,2 (0, T ; H ) and weakly-∗ in L∞ (0, T ; V0 ) as n → ∞,

(5.3)

and  ε0 uΓ ∈ W 1,2 (0, T ; L2 (Γ )) ∩ L∞ (0, T ; H 1 (Γ )), εn uεΓn → ε0 uΓ weakly-∗ in L∞ (0, T ; H 1 (Γ )), as n → ∞.

(5.4)

Also, by Corollary 4.1 and Lemma 4.1 with S = (0, T ), we have that ˆε → Φ ˆ ε on L2 (0, T ; H ), in the sense of Mosco, Φ n 0

as n → ∞.

From (5.3), (A2), Remarks 1.4 and 1.6 (Fact 5), it is seen that ˆε [−U ′ − G(U ) + Θ ε0 , U ] ∈ ∂ Φ 0

in L2 (0, T ; H ) × L2 (0, T ; H ),

(5.5)

and ˆ ε (U εn ) → Φ ˆ ε (U ) Φ n 0

as n → ∞.

(5.6)

Note that (2.7), (5.3)–(5.5) and Remark 1.4 (Fact 4) enable us to say that U = [u, uΓ ] is a solution to the Cauchy problem (CP)ε0 . So, due to the uniqueness of solutions, it must hold that: U = [u, uΓ ] = U ε0 = [uε0 , uεΓ0 ]

in L2 (0, T ; H ).

Furthermore, since (A1), (5.3)–(5.4) and (5.7) imply:      κ2 T κ2 T εn 2  |∇u | dxdt ≥ |∇uε0 |2 dxdt, lim    2 2 n→∞ 0 Ω 0 Ω          1 T 1 T ε 2  n  |∇Γ (εuΓ )| dΓ dt ≥ |∇Γ (εuεΓ0 )|2 dΓ dt,  lim 2 2 n→∞ 0 Γ 0 Γ    T      εn ε ε n n  lim |∇u | + B(u ) dx + B (u ) dΓ dt  Γ Γ   Ω Γ n→∞ 0      T      ≥ |∇uε0 | + B(uε0 ) dx + BΓ (uεΓ0 ) dΓ dt, 0

Ω

Γ

(5.7)

(5.8)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

55

one can see that:  T  T   εn 2  |∇u | dxdt = |∇uε0 |2 dxdt, lim n→∞ 0 Ω 0 Ω  T  T   εn 2  lim |∇Γ (εn uΓ )| dΓ dt = |∇Γ (ε0 uεΓ0 )|2 dΓ dt, n→∞

0

Γ

0

(5.9)

Γ

by applying (Fact 0) with (5.6)–(5.8) in mind. Now, taking into account (5.3)–(5.4), (5.7) and (5.9), and applying the uniform convexity of L2 -based topologies and the continuity of the trace operators, we obtain that:  εn ε0 2 1  u → u in L (0, T ; H (Ω )), εn ε0 2 εn uΓ → ε0 uΓ in L (0, T ; H 1 (Γ )), as n → ∞. (5.10)  uεn = uεn → uε0 = uε0 in L2 (0, T ; H 21 (Γ )), Γ Γ |Γ |Γ Therefore, (5.3), (5.7) and (5.10) are sufficient to verify (2.8)–(2.9). (Step 2) The case when {Φε (U0ε )}ε∈J0 is unbounded. Let ρ ∈ (0, 1) be an arbitrary constant. Then, by the assumption for {U0ε }ε≥0 , for any sequence {εn }∞ n=1 ⊂ J0 converging to ε0 we find a large number n1 (ρ) ∈ N, such that: |U0εn − U0ε0 |H ≤ ρ, Also, since

U0ε0

for any n ≥ n1 (ρ)

(5.11)

∈ D(Φε0 ), we find a function W0,ρ ∈ D(Φε0 ), such that: |U0ε0 − W0,ρ |H ≤ ρ.

n Additionally, by Corollary 4.1, there exists a sequence {W0,ρ ∈ D(Φεn )}∞ n=1 ⊂ H , such that:  n W0,ρ → W0,ρ in H , as n → ∞, n Φεn (W0,ρ ) → Φε0 (W0,ρ ),

(5.12)

(5.13)

and in particular, there exists a large number n2 (ρ) ∈ N, with n2 (ρ) ≥ n1 (ρ), such that: n |W0,ρ − W0,ρ |H ≤ ρ,

for any n ≥ n2 (ρ).

(5.14)

From (5.11)–(5.12) and (5.14), it follows that: n n |U0εn − W0,ρ |H ≤ |U0εn − U0ε0 |H + |U0ε0 − W0,ρ |H + |W0,ρ − W0,ρ |H

≤ 3ρ,

for any n ≥ n2 (ρ).

(5.15)

Based on this, let Wρ ∈ W 1,2 (0, T ; H )∩L∞ (0, T ; Vε0 ) be the solution to (CP)ε0 , corresponding to the forcing term Θ ε0 ∈ L2 (0, T ; H ) and the initial data W0,ρ ∈ D(Φε0 ). As well as, for any n ∈ N, let Wρn ∈ W 1,2 (0, T ; H ) ∩ L∞ (0, T ; Vεn ) be the solution to (CP)εn , corresponding to the forcing term Θ εn ∈ L2 (0, T ; H ) n and the initial data W0,ρ ∈ D(Φεn ). Then, by applying the result of the previous (Step 1), we have: Wρn → Wρ

in C([0, T ]; H ) as n → ∞.

Besides, from (2.6) and (5.15)–(5.16), one can see that: lim |U εn − U ε0 |C([0,T ];H )      ≤ lim U εn − Wρn C([0,T ];H ) + Wρn − Wρ C([0,T ];H ) + |Wρ − U ε0 |C([0,T ];H ) n→∞        ε0 n  ≤ C3 lim U0εn − W0,ρ + |W − U | + lim Wρn − Wρ C([0,T ];H ) 0,ρ 0 H H n→∞ n→∞  ≤ 4 C3 ρ.

n→∞

(5.16)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

56

Since ρ ∈ (0, 1) is arbitrary, the above inequality implies that: U εn → U ε0

in C([0, T ]; H ) as n → ∞.

(5.17)

Now, our remaining task will be to verify the convergences (2.8)–(2.9) under the unbounded situation of {Φεn (U0εn )}∞ n=1 . To this end, we first invoke (2.5) and (5.13) to check the existence of a constant K(ρ), depending on ρ ∈ (0, 1), such that: |(Wρ )′ |2L2 (0,T ;H ) + |(Wρn )′ |2L2 (0,T ;H ) + sup Φε0 (Wρ (t)) + sup Φεn (Wρn (t)) ≤ K(ρ), t∈[0,T ]

t∈[0,T ]

for any n ∈ N.

(5.18)

On this basis, let us consider a sequence {Pn }∞ n=1 ⊂ C([0, T ]) of functions, given as:  ς ς ∈ [0, T ] → Pn (ς) := Φεn (Wρn (t)) dt ∈ [0, ∞), for any n ∈ N. 0

Then, by applying a similar method to show (5.6), we have:  ς Pn (ς) → P(ς) := Φε0 (Wρ (t)) dt as n → ∞,

for any ς ∈ [0, T ].

(5.19)

0

Also, from (5.18), it is seen that:  ∞  ∞ d Pn = Φεn (Wρn ) n=1 is bounded in L∞ (0, T ). dς n=1

(5.20)

By (5.19)–(5.20) and Ascoli’s theorem, we may suppose that: Pn → P

in C([0, T ]) as n → ∞,

by taking a subsequence if necessary, and more precisely, we find a large number n∗ (ρ), independent of ς ∈ [0, T ], such that:  ς   ς   n  Φεn (Wρ (t)) dt − n∗ (ρ) ≥ n2 (ρ) and  Φε0 (Wρ (t)) dt < ρ, 0

0

for any n ≥ n∗ (ρ) and any ς ∈ [0, T ].

(5.21)

In the meantime, for the sequence of solutions {U εn }∞ n=1 , it is easily seen that:  ς  εn  (U − Wρn )′ , U εn − Z L2 (0,ς;H ) + Φεn (U εn (t)) dt 0  ς   ≤ Φεn (Z(t)) dt + Θ εn − G(U εn ) − (Wρn )′ , U εn − Z L2 (0,ς;H ) , 0

for any Z ∈ L2 (0, ς; Vεn ) and any n ∈ N. So, putting Z = Wρn , and using (2.6)–(2.7), (5.15), (5.17)–(5.18), (5.21) and (A2), we infer that:  ς  ς 1 εn εn n 2 Φεn (Wρn (t)) dt Φεn (U (t)) dt ≤ |U0 − W0,ρ |H + 2 0 0 + |Θ εn − G(U εn ) − (Wρn )′ |L2 (0,ς;H ) |U εn − Wρn |L2 (0,ς;H )  ς  9 2 ≤ ρ + Φε0 (Wρ (t)) dt + ρ 2 0     ′ n + C3 ς|U0εn − W0,ρ |H |Θ εn − G(U εn )|L2 (0,ς;H ) + | Wρn |L2 (0,ς;H )    ς   11 Φε0 (W0,ρ (t)) dt + 3ρ C3 ς sup |Θ εn − G(U εn )|L2 (0,ς;H ) + K(ρ) , ≤ ρ+ 2 n∈N 0 for any ς ∈ [0, T ] and any n ∈ N. (5.22)

P. Colli et al. / Nonlinear Analysis 158 (2017) 32–59

Here, let us take a constant ς∗ (ρ) ∈ (0, T ), so small to satisfy that:    ς∗ (ρ)  ρ Φε0 (W0,ρ (t)) dt + 3ρ C3 ς∗ (ρ) sup |Θ εn − G(U εn )|L2 (0,ς∗ (ρ);H) + K(ρ) < . 2 n∈N 0

57

(5.23)

Then, having in mind (5.17), (5.22)–(5.23), Corollary 4.1 and Fatou’s lemma, it is observed that:  ς∗ (ρ)  ς∗ (ρ) Φε0 (U ε0 (t)) dt ≤ lim Φεn (U εn (t)) dt n→∞

0

0



ς∗ (ρ)

Φεn (U εn (t)) dt ≤ 6ρ.

≤ sup n∈N

(5.24)

0

1,2 Finally, by (2.4), one can see that the sequence {U εn }∞ (ς∗ (ρ), T ; H ) ∩ L∞ (ς∗ (ρ), T ; n=1 is bounded in W V0 ). So, we can apply a similar argument to obtain (5.6), and we can show that:  T  T Φεn (U εn (t)) dt → Φε0 (U ε0 (t)) dt as n → ∞. (5.25) ς∗ (ρ)

ς∗ (ρ)

In view of (5.17), (5.24)–(5.25), we can say that:  2 U ε0 ∈ L2 (0, T ; V0 ), {U εn }∞ n=1 is bounded in L (0, T ; V0 ), εn ε0 2 U →U weakly in L (0, T ; V0 ) as n → ∞,

(5.26)

and    T  T    εn ε0 lim  Φεn (U (t)) dt − Φε0 (U (t)) dt n→∞  0  0  ς∗ (ρ)  ς∗ (ρ) ≤ sup Φεn (U εn (t)) dt + Φε0 (U ε0 (t)) dt n∈N 0 0    T  T    εn ε0 + lim  Φεn (U (t)) dt − Φε0 (U (t)) dt n→∞  ς (ρ)  ς∗ (ρ) ∗ ≤ 12ρ. Since ρ ∈ (0, 1) is arbitrary, the above inequality implies:  T  εn lim Φεn (U (t)) dt = n→∞

0

T

Φε0 (U ε0 (t)) dt.

(5.27)

0

By virtue of (5.26)–(5.27), we can apply a similar method to derive (5.10), and we obtain that:  εn ε0 2 1  u → u in L (0, T ; H (Ω )), εn ε0 2 εn uΓ → ε0 uΓ in L (0, T ; H 1 (Γ )), as n → ∞.  uεn = uεn → uε0 = uε0 in L2 (0, T ; H 21 (Γ )), Γ Γ |Γ |Γ Hence, (5.17) and (5.28) imply the conclusive convergences (2.8)–(2.9).

(5.28)



Acknowledgments First and second author gratefully acknowledges some financial support from the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia. Fourth author is supported by Grant-in-Aid No. 16K05224, JSPS.

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