Kobayashi--Warren--Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the $\omega$-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the $\omega$-limit set belong to thee space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constants solutions.


Introduction
Let 0 < T < ∞ and N ∈ N be fixed constants. Let Ω ⊂ R N be a bounded domain with a smooth boundary Γ if N > 1. Let n Γ be the unit outer normal on Γ. Let Q := (0, T ) × Ω be the product set of the time-interval (0, T ) and the spatial domain Ω, and let Σ := (0, T ) × Γ.
In this paper, we consider a coupled system, denoted by (S). Our system consists of the following two initial-boundary problems.
For parabolic systems kindred to (S), various mathematical methods have been developed in the literature. For instance, when both boundary conditions for η and θ are of zero-Neumann type, the study of the corresponding system was treated in the recent works [24, 25, 27-30, 32, 33]. Also, when γ ≡ 0, Ito-Kenmochi-Yamazaki [11][12][13] and Kenmochi-Yamazaki [16] studied relaxed versions of (S) that include an additional term −∆θ of parabolic regularization in the initial-boundary problem (0.2). However, in the original (non-relaxed) problem (0.2), we first need to address a problem of ambiguity in the Dirichlet boundary condition. Actually, in rigorous mathematics, the condition θ = γ on Σ, as in (0.2), does not make sense, because the singular diffusion −div α(η) Dθ |Dθ| generally allows spatial discontinuity for the solution θ, including the jumps between the trace of θ and the boundary data γ. For the problem of ambiguity, we adopt the method to define a weak formulation of the Dirichlet boundary condition in the singular problem (0.2), by means of the subdifferential of the L 1 -integral term Γ α(η)|θ − γ| dΓ, as in (0.3). The idea of the weak formulation is based on the general theory, established in [3,4,23].
In the case that γ is not constant, the nonhomogeneous boundary data will bring various equilibrium patterns of grain boundaries, as solutions to the steady-state problem for our system (S). Therefore, it could be expected that the structural observations for steady-state solutions would provide fruitful information for more advanced researches, such as the stability analysis of polycrystalline structure, the determination of possible (ideal) target profiles in the control problems of grain boundary motions, and so on.
In view of these, we set the objective of this paper to deal with the following issues: issue ♯1) the mathematical solvability of the system (S) with nonhomogeneous Dirichlet boundary data; issue ♯2) the derivation of the steady-state problem as an asymptotic governing system for the large-time behavior; issue ♯3) the structural observation for steady-state solutions.
Besides, we largely divide the content of this paper into two parts: The former part deals with issues ♯1) and ♯2), and the conclusions are the main results of this paper, in short: (Main Theorem 1) the existence of time-global solutions to the system (S) with the dissipation in time of the free-energy; (Main Theorem 2) the semicontinuous association between the ω-limit set of the solution, as time tending to ∞, and the steady-state problem for our system (S); and some auxiliary results, in short: (Theorem 1) the solvability and energy-dissipation property for the approximating systems of time-discretization type; (Theorem 2) the energy-inequality for a priori estimates of approximating solutions.
The precise statement of the above results is contained in Sections 2 (Main Theorems) and 3 (Theorems), and proved in Sections 5 and 6, respectively, on the basis of the preliminaries prepared in Section 1, and some auxiliary Lemmas obtained in Section 4.
The latter part is assigned to the last Section 7, and is devoted to issue ♯3). Under the following concrete setting: (0.4) the results of structural observations will be reported as conclusions of this study. They concern with: ♯3-a) the exact profiles and SBV-regularity of one-dimensional steady-state solutions; ♯3-b) the sufficient conditions for the existence of radially symmetric two-dimensional steady-state solutions;

Preliminaries
We begin with some notations used throughout this paper. We denote by |x| and x · y the Euclidean norm of x ∈ R N and the standard scalar product of x, y ∈ R N , respectively, as usual, i.e., |x| := x 2 1 + · · · + x 2 N and x · y := x 1 y 1 + · · · + x N y N for all x = [x 1 , . . . , x N ], y = [y 1 , . . . , y N ] ∈ R N .
Abstract notations. (cf. [5,Chapter II]) For an abstract Banach space X, we denote by | · | X the norm of X, and by ·, · X the duality pairing between X and its dual X * . In particular, when X is a Hilbert space, we denote by ( · , · ) X the inner product of X. Besides, for two Banach spaces X and Y , let L (X; Y ) be the space of bounded linear operators from X into Y .
For any proper functional Ψ : X → (−∞, ∞] on a Banach space X, we denote by D(Ψ) the effective domain of Ψ.
For any proper lower semi-continuous (l.s.c., in short) and convex function Φ defined on a Hilbert space H, we denote by ∂Φ the subdifferential of Φ. The subdifferential ∂Φ corresponds to a weak differential of Φ, and it is a maximal monotone graph in the product space H 2 (= H × H). More precisely, for each v 0 ∈ H, the value ∂Φ(v 0 ) of the subdifferential at v 0 is defined as a set of all elements v * 0 ∈ H which satisfy the following variational inequality: The set D(∂Φ) := {z ∈ H | ∂Φ(z) = ∅} is called the domain of ∂Φ. We often use the notation " , by identifying the operator ∂Φ with its graph in H 2 .
Notations of basic elliptic operators. Let F : H 1 (Ω) −→ H 1 (Ω) * be the duality mapping, defined as subject to the Neumann-zero boundary condition. Notations in basic measure theory. (cf. [2,4]) We denote by L N the N-dimensional Lebesgue measure, and we denote by H N −1 the (N − 1)-dimensional Hausdorff measure. In particular, the measure theoretical phrases, such as "a.e.", "dt" and "dx", and so on, are all with respect to the Lebesgue measure in each corresponding dimension. Also, in the observations on a Lipschitz surface S, the phrase "a.e." is with respect to the Hausdorff measure in each corresponding Hausdorff dimension. Let U ⊂ R N be any open set. We denote by M (U) (resp. M loc (U)) the space of all finite Radon measures (resp. the space of all Radon measures) on U. Recall that the space M (U) (resp. M loc (U)) is the dual of the Banach space C 0 (U) (resp. dual of the locally convex space C c (U)), for any open set U ⊂ R N .
We set: B := ξ ∈ R N |ξ| < 1 and S N −1 := ∂B, and for any ν ∈ S N −1 , we set: For a function u ∈ L 1 loc (U) and a point x 0 ∈ U, we denote by iff. there exists a value [ũ](x 0 ) ∈ R such that: The above [ũ](x 0 ) ∈ R N is called the approximate limit of the function u ∈ L 1 loc (U) at the point x 0 ∈ U, and the real value [ũ](x 0 ) exists, iff. for any ν ∈ S N −1 , a real value [ũ] ν (x 0 ) ∈ R such that: exists, and it coincides with the value As is well-known (cf. [4, Section 10.3]), any function u ∈ L 1 loc (U) admits an approximate limit [ũ](x) at a.e. x ∈ U, and in this regard, the function x ∈ U → [ũ](x) is called the continuous representative of u. The measurable set J u ⊂ U defined as: is called the jump set of u. Additionally, the unit vector ν Besides, we set Notations in BV-theory. (cf. [2,4,8,10]) Let U ⊂ R N be an open set. A function u ∈ L 1 (U) (resp. u ∈ L 1 loc (U)) is called a function of bounded variation, or a BV-function, (resp. a function of locally bounded variation or a BV loc -function) on U, iff. its distributional differential Du is a finite Radon measure on U (resp. a Radon measure on U), namely Du ∈ M (U) (resp. Du ∈ M loc (U)). We denote by BV (U) (resp. BV loc (U)) the space of all BV-functions (resp. all BV loc -functions) on U. For any u ∈ BV (U), the Radon measure Du is called the variation measure of u, and its total variation |Du| is called the total variation measure of u. Additionally, the value |Du|(U), for any u ∈ BV (U), can be calculated as follows: The space BV (U) is a Banach space, endowed with the following norm: |u| BV (U ) := |u| L 1 (U ) + |Du|(U), for any u ∈ BV (U).
Also, BV (U) is a complete metric space, endowed with the following distance: The topology provided by this distance is called the strict topology of BV (U). For any u ∈ BV (U), it is known that (cf. [4,Sections 10.3 and 10.4]), the sets S u and Now, by Radon-Nikodým's theorem [2,Theorem 1.28], the measure Du for u ∈ BV (U) is decomposed in the absolutely continuous part D a u for L d and the singular part D s u, i.e.
Furthermore, (cf. [2,4]), this decomposition is precisely expressed as follows: In this context, ∇u ∈ L 1 (U) denotes the Radon-Nikodým density of D a u for L d . The part and it provides an exact expression of the variation at the discontinuities of u. The part D s u ⌊ (U \ S u ) is called the Cantor part of Du and it is denoted by D c u. We denote by SBV the space of special functions of bounded variation; i.e. those BV functions such that D c u = 0. We note that a function defined as provides a precise representative of u ∈ BV (U), H N −1 -a.e. on U.
In particular, if U is bounded and the boundary ∂U is Lipschitz, then the space BV (U) is continuously embedded into L N/(N −1) (U) and compactly embedded into L q (U) for any Moreover, the trace operator tr ∂U : BV (U) → L 1 (∂U) is continuous with respect to the strict topology of BV (U). Namely, tr ∂U u n → tr ∂U u in L 1 (∂U) as n → ∞, if u ∈ BV (U), {u n } ∞ n=1 ⊂ BV (U) and u n → u in the strict topology of BV (U) as n → ∞. Extensions of functions. (cf. [2,4]) Let µ be a positive measure on R N , and let B ⊂ R N be a µ-measurable Borel set. For a µ-measurable function u : B → R, we denote by [u] ex an extension of u over R N , i.e. [u] ex : R d → R is a measurable function such that [u] ex = u µ-a.e. in B. In general, the choices of extensions are not necessarily unique. -for any 1 < q < ∞, ex U (W 1,q (U)) ⊂ W 1,q (R N ), and the restriction ex U | W 1,q (U ) :

Remark 3 (Harmonic extension)
Under the same notations as in Remark 2, we denote by hm ∂U the operator of harmonic extension hm ∂U : H 1 2 (∂U) → H 1 (U), which maps any ̺ ∈ H 1 2 (∂U) to a (unique) minimizer [̺] hm ∈ H 1 (U) of the following proper l.s.c. and convex function on L 2 (U): Specific notations. (cf. [1,23,24]) Throughout this paper, Ω ⊂ R N is a bounded domain with Γ being its boundary. We assume sufficient smoothness (at least C 1 -regularity) for Γ when N > 1. Also, we fix a large open ball B Ω ⊂ R N such that B Ω ⊃ Ω.

Remark 4
In the construction of the sequence {u i } ∞ i=1 ⊂ H 1 (Ω) as in (1.10), the key point is to prepare an auxiliary sequence and such a sequence is easily obtained by referring to some appropriate general theories, e.g. [9,23]. In addition, if we suppose γ ∈ H 1 (Γ), then we can take the auxiliary se- , and thereby, we can also suppose for the sequence as in (1.10).

Main Theorems
We begin with the assumptions for the Main Theorem.
Next, for simplicity of descriptions, we prescribe some additional notations.
Additional notations. We define a functional Ψ 0 : L 2 (Ω) −→ (−∞, ∞], by letting: Then, the free-energy F γ = F γ (η, θ) given in (0.3) can be described in the following simple formula: Based on these, the solution to the system (S) is defined as follows.

Remark 6
In the light of (1.5), the variational identity in Definition 1 (S1), can be rewritten into the following weak formulation: where F : H 1 (Ω) → H 1 (Ω) * is the duality map between H 1 (Ω) and H 1 (Ω) * , and s > N/2 is a large constant such that the embedding H s (Ω) ⊂ C(Ω) holds true. Thus, we note that the inhomogeneous boundary condition in (0.1) is implicitly built in the expres- In the meantime, the variational inequality in Definition 1 (S2) is equivalent to the following evolution equation: which is governed by the L 2 -subdifferential ∂Φ γ (α(η(t)); · ) of the time-dependent convex function Φ γ (α(η(t)); · ). From this reformulation, we can see that the mathematical meaning of the singular diffusion −div(α(η) Dθ |Dθ| ) in (0.2) is given in terms of the timedependent subdifferential ∂Φ γ (α(η(t)); · ) of the weighted total variation, including the Dirichlet type boundary condition. Now, the Main Theorems of this paper are stated as follows.
fulfills the following inequality: for a.e. s > 0 and any t ≥ s.
Remark 7 Property (S5), though not used in the paper, is characteristic of solutions to parabolic PDEs or parabolic systems of PDEs. In this special case, [η 0 , θ 0 ] ∈ D 0 ∩ (H 1 (Ω) × BV (Ω)), we will call the solution a strong solution, in view of the smoothing effect as in (S0) and (S5).

Approximating problem
In this section, the approximating problems to the system (S) are prescribed, and some key-properties of the approximating solutions are verified.
Definition 2 We say that a collection of functions {|·| ν } ν∈(0,1) is a suitable approximation to the Euclidean norm if the following properties hold.
Also, we note that the class of possible regularizations verifying (AP1) and (AP2) covers a number of standard type regularizations, while it is a restricted version of the class adopted in [25]. For instance, • Hyperbola type, i.e. ξ ∈ R N → |ξ| ν := |ξ| 2 + ν 2 − ν, for ν ∈ (0, 1), Such flexibility is the reason for the "suitability" as in Definition 2, and indeed, the essential thing in the proof of Main Theorem is not in the precise expression of {|·| ν } ν∈(0,1) , but in its suitability.
(S ν ): In the context, the initial data [η 0 , θ 0 ] satisfy On account of Remark 9, the exact statement of the approximating problem (AP ν h ) is prescribed as follows.
On this basis, we call the above sequence (Ω) 2 a solution to the approximating problem (AP ν h ), or an approximating solution in short.

Remark 10
We note that the range D rx 0 of approximating solutions coincides with the effective domains of the relaxed convex functions D(Φ ν γ (β; · )) in D 0 , for all ν ∈ (0, 1), 0 ≤ β ∈ L 2 (Ω), and γ ∈ H 1 2 (Γ). Furthermore, invoking (A3) and Remark 4, we can see that D rx 0 is dense in D 0 , in the topology of L 2 (Ω) 2 . In fact, for any [η,θ] ∈ D 0 , we can construct an approximating sequence Note that we can take the above sequence {θ ⊥ n } ∞ n=1 by referring to Remark 4. Now, for the mathematical analysis of the approximating problem, we need to prove the following theorems, concerned with energy-inequalities for the approximating solutions.

Auxiliary lemmas
As an advanced version of the above (Fact 5), we next prove the following Lemma.
Then, we immediately observe that this sequence {v n } ∞ n=1 satisfies the assertion (4.2). Meanwhile, by (4.1), the compactness of embedding H 1 2 (Γ) ⊂ L 1 (Γ) leads to: the remaining assertion (4.3) can be verified as follows: Thus, we conclude this Lemma. In what follows, we set, for any open set Ω ⊂ R N .

Proofs of Theorems 1 and 2
In this section, Theorems 1 and 2, concerned with the key-properties of approximating solutions, will be proved via some auxiliary Lemmas.
The first result deals with solvability of the elliptic problem (3.4). The proof is analogous to that of [24, Lemmas 9-11] and we omit it.
In the meantime, the following inequality , for any t ∈ (0, ∞) and ℓ ∈ N, and (6.22) imply that , for any ℓ ∈ N.

Proof of Main Theorem 2
Remark 13 Observe that, due to the nonincreasing property of J γ , the condition "for a.e. 0 < s < t < ∞" in (6.28) can be rephrased as "for all 0 < s ≤ t < ∞". Moreover, taking into account Remark 12 and (S0) in Definition 1, one can deduce from (6.28) that for all 0 < s ≤ t < ∞.

Asymptotic behavior
In this Section we study some structural observations of the steady state solutions in the precise setting of (0.4). The qualitative properties of the steady states differ in the case of a one dimensional domain or in the two dimensional case.
Proof. We fix r 0 and we let g(x) := f (r 0 , x) and a(x) := b(r 0 , x). We will show that, if r 0 γ(R) ≥ 1, then there is only one critical point (and therefore a global positive maximum since g(x) → −∞ as x → +∞) of g. In fact, (noting that a ′′ = a − a ′ x ) Observe that g ′ (r 0 ) = 0 and that g ′′ (r 0 ) ≥ 0 if, and only if r 0 γ(R) ≥ 1. Moreover, in the case r 0 γ(R) = 1, g ′ (r 0 ) = g ′′ (r 0 ) = 0 and g ′′′ (r 0 ) > 0. At a critical point, However, y = a ′ a satisfies y ′ = 1 − y x − y 2 , y(r 0 ) = 1, and therefore it is a concave increasing function from 0 to 1 in [r 0 , +∞[ with initial slope 1. On the other hand: is easily seen to be increasing, with initial slope (at r 0 ) less than or equal to 1 (strictly less if r 0 γ(R) > 1 and tending to ∞. Moreover, it is concave at r 0 and at +∞ and it only has two inflexion points in ]r 0 , +∞[. Therefore, both functions y and h intersect only once (see (a) and (b) in Figure 1). Therefore, for (7.8) to hold it suffices to show that Adding the two equalities in (7.8), we obtain 0 < d 1 = − C 1 + C 2 + r 2 r 1 (C 3 + C 4 ) γ(R) − (C 1 + r 1 r 2 C 2 + r 2 r 1 C 3 + C 4 ) We observe that this is not always the case since the following inequality is always true (and therefore γ(R) needs to be big enough): (7.10) To show this last inequality we use the fact that C 1 + r 1 r 2 C 2 + r 2 r 1 C 3 +C 4 = C 1 +C 2 + r 2 r 1 2 − r 2 r 1 C 3 +C 4 ≤ C 1 +C 2 +C 3 +C 4 ≤ 0.
Then, (7.10) is seen to be equivalent to which is easy to show to hold. Observe that we have not imposed that r 1 > r 0 . In case r 1 = r 0 , then (7.9) will suffice for existence of solution. However, in the case that r 0 < r 1 , then there is an extra compatibility condition as in the previous examples: Defining w(r) := r 1 α(d) rα(η(r)) , the new compatibility condition is 1 − w(r 0 ) ≥ 0. (7.11) We conclude by showing some examples of the difficulty of the casuistic. We first show an example (Fig.3) in which (7.9) is violated for any admissible range of boundary data and therefore no solution exists. In the second example (Fig.4), we obtain two different threshold values for γ(R) for which (7.9), resp. (7.11), is satisfied. We point out that, in the particular case of Fig. 4, the only possible situation for an admissible (big enough) boundary datum is r 0 = r 1 . In this case, we find a threshold point ̺ 1 (≈ 3.5) such that (7.9) does not hold for all γ(R) ∈ [0, ̺ 1 ), and the range [0, ̺ 1 ) contains the physically admissible range [0, π) of the crystalline orientation angle. In this case, we find two threshold points ̺ 2 ∈ (1, 2) and ̺ 3 ∈ (7, 8) such that (7.9) (resp. (7.11)) holds iff. γ(R) ≥ ̺ 2 (resp. γ(R) ≥ ̺ 3 ).