SINGULAR LIMIT OF ALLEN–CAHN EQUATION WITH CONSTRAINT AND ITS LAGRANGE MULTIPLIER

. We consider the Allen–Cahn equation with a constraint. Our constraint is provided by the subdiﬀerential of the indicator function on a closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier for our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem.

1. Introduction. In this paper, for each ε ∈ (0, 1] we consider the following Allen-Cahn equation with a constraint: if − 1 < z < 1, (−∞, 0] if z = −1. (1.5) The Allen-Cahn equation was proposed to describe the macroscopic motion of phase boundaries. In the physical context, the function u ε = u ε (t, x) in (P) ε is the nonconserved order parameter that characterizes the physical structure: u ε = 1, −1 < u ε < 1 and u ε = −1 corresponding to pure liquid, mixture and pure solid respectively. The plan of this paper is as follows. In Section 2, we state the main results in this paper. In Section 3 we recall the decomposition result of the subdifferential of convex functions. Also, we prove the main result (Theorem 2.1) concerning the existence-uniqueness of solutions to (P) ε and properties of the Lagrange multiplier λ ε corresponding to the item (i) listed above. In Section 4, we prove Theorem 2.2 corresponding to the items (ii) and (iii) listed above.
Notations and basic assumptions. Throughout this paper, for any reflexive Banach space B, we denote | · | B the norm of B, and denote by B * the dual space of B.
In particular, we put H := L 2 (Ω) with usual real Hilbert space structure, and denote by (·, ·) H the inner product in H. Also, we put V := H 1 (Ω) with the usual norm and denote by ·, · the duality pairing between V * and V . By identifying H with its dual space, we have V ⊂ H ⊂ V * with compact and dense embeddings; then, In the proof of Theorem 2.1, we use some techniques of proper (that is, not identically equal to infinity), l.s.c. (lower semi-continuous), convex functions and their subdifferentials, which are useful in the systematic study of variational inequalities. Therefore, let us outline some notations and definitions. For a proper, l.s.c. and convex function ψ : H → R∪{+∞}, the effective domain D(ψ) is defined by The subdifferential of ψ is a possibly multi-valued operator in H and is defined by z * ∈ ∂ψ(z) if and only if z ∈ D(ψ) and (z * , y − z) H ≤ ψ(y) − ψ(z) for all y ∈ H.
For various properties and related notions of the proper, l.s.c., convex function ψ and its subdifferential ∂ψ, we refer to a monograph by Brézis [6].
Next, let us give an assumption on initial data. Throughout this paper, we assume the following condition (A): Finally, throughout this paper, C i = C i (·), i = 1, 2, 3, · · · , denotes positive (or nonnegative) constants depending only on its arguments.
for all z ∈ K and a.e. t ∈ (0, T ). Now, let us mention the first main result in this paper, which is concerned with the existence and basic property of the solution and the Lagrange multiplier for (P) ε on [0, T ].
In next Section 3, we give the proof of Theorem 2.1. Next, we consider the limiting situation of (P) ε as ε → 0. To this end, we use the following energy functional: Now we state the second main result in this paper, which is concerned with the singular limit of (P) ε as ε → 0: Then, there are a subsequence {ε k } of {ε} with ε k 0 as k → ∞, two functions u ∈ L 2 (0, T ; H) and λ * ∈ L 2 (0, T ; V * ) and a positive constant N 0 , independent of ε ∈ (0, 1], such that u(t, x) takes only the values 1 or −1 for a.e. (t, where Ω |∇u(t)| is the total variation measure of u(t). Moreover, In Section 4 we prove Theorem 2.2 by using a priori estimates of u ε and λ ε .
3. Solvability of (P) ε . In this section we consider (P) ε for each ε ∈ (0, 1]. In fact, we study (P) ε by arguments similar to [15,18], namely by the theory of abstract evolution equations governed by subdifferentials. Now, we define a functional ϕ 0 on H by Clearly, ϕ 0 is proper, l.s.c. and convex on H with ∂ϕ 0 (z) = −∆ N z in H, where ∆ N : D(∆ N ) := z ∈ H 2 (Ω); ∂z ∂ν = 0 a.e. on Γ → H is the Laplacian with homogeneous Neumann boundary condition (cf. [3, Proposition 2.9]). Also, we define the proper, l.s.c. and convex functional I [−1,1] of H by Next, we consider the functional ϕ defined by the form: 1] (z) for any z ∈ H. Clearly, ϕ is proper, l.s.c. and convex on H with the effective domain D(ϕ) = K, where K is the set defined in (A).
Here, we recall the following decomposition result of the subdifferential ∂ϕ. Now, we prove Theorem 2.1 by using Proposition 3.1 and applying the abstract theory of nonlinear evolution equations associated with subdifferential ∂ϕ.
Proof of Theorem 2.1. By the similar arguments as in [15, Section 1], we can show the existence-uniqueness of a solution u ε to (P) ε on [0, T ] for each ε ∈ (0, 1]. In fact, we easily prove the uniqueness of solutions to (P) ε on [0, T ] by the quite standard arguments: monotonicity and Gronwall's inequality. Now, we show the precise proof of the existence of solutions to (P) ε . We easily see that the problem (P) ε can be rewritten in an abstract framework of the form: Therefore, applying the Lipschitz perturbation theory of abstract evolution equations (cf. [7,13,17]), we can show the existence of a solution u ε to (P) ε on [0, T ] for each ε ∈ (0, 1] in the variational sense (cf. Remark 2.1). Also, note from Proposition 3.1 that (CP) ε is equivalent to the following: Namely, there are functions v ε ∈ L 2 (0, T ; H) and λ ε ∈ L 2 (0, T ; H) such that v ε (t) ∈ ∂ϕ 0 (u ε (t)) a.e. in (0, T ), λ ε ∈ ∂I [−1,1] (u ε ) a.e. in Q and (3.3) holds in the following sense: Since ∂ϕ 0 (z) = −∆ N z is the single-valued linear operator in H, we easily see that u ε is a unique solution to (P) ε on [0, T ] and λ ε is the Lagrange multiplier for (P) ε on [0, T ] in the sense of Definition 2.1. Also, by (3.4), the uniqueness of solution u ε to (P) ε and the linearity of ∂ϕ 0 (z) = −∆ N z, we easily show the uniqueness of the Lagrange multiplier λ ε for (P) ε .
We begin by giving the uniform estimate of u ε and λ ε with respect to ε ∈ (0, 1].
Next, we show (4.9). By (4.4) and the Hölder inequality, we have: for all t 1 , t 2 with 0 ≤ t 1 < t 2 ≤ T . Thus, the proof of Corollary 4.1 has been completed. Now, we prove the main Theorem 2.2, which is concerned with the singular limit of (P) ε as ε → 0. Therefore, taking a subsequence if necessary, we see that: Since h is continuous and strictly increasing on [−1, 1] (cf. (4.7)), we can find a unique function u(t, x) such that h * (t, x) = h(u(t, x)), a.e. (t, x) ∈ (0, T ) × Ω (4.14) and which implies that the limit function u of u ε k takes only the values 1 or −1 for a.e. (t, x) ∈ (0, T ) × Ω.