Horizontal patterns from finite speed directional quenching

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in the case presented here the interface moves with speed $c$ in such a way that the bistable region grows, which is the most physically relevant scenario. According to previous results in the literature (Phase separation patterns from directional quenching, Journal of Nonlinear Science, 2017, by Monteiro, R. and Scheel, A.), several patterns exist when $c \gtrsim0$, and here we investigate their persistence for finite $c>0$, clarifying the pattern selection mechanism related to the speed $c$ of the quenching front.


Introduction
In the theory of reaction diffusion, the interplay between stable and unstable mechanisms can give rise to spatial patterns, i.e., stationary non-homogeneous structures. In the presence of controllable external parameters the existence and persistence of these patterns are worth to investigate, both for their mathematical interest and industrial applications; see [ZWH + 16] and the survey [FW12]. In this paper we are interested in a directional quenching scenario, where a planar interface (also called the quenching front) moves with constant speed c, across which a phase separation process takes place: ahead of the interface the system is monostable, while in its wake it is bistable. We study this phenomenon in the scalar model ∂ t u = ∆ (ξ,y) u + µ(ξ − ct)u − u 3 , (1.1) where ∆ (ξ,y) := ∂ 2 ξ + ∂ 2 y and µ(ξ − ct ≷ 0) = ∓1. Equation (1.1) is a particular case of the Allen-Cahn model, which describe the behavior of a heterogeneous, binary mixture: the unknown u(ξ, y; t) denotes the relative concentration of one of the two metallic components of the alloy at time t ∈ R + := [0, ∞) and point (ξ, y) ∈ R 2 . The stationary problem in a moving frame (x, t) = (ξ − ct, t) is written (1. 2) The most physically relevant scenario to be considered consists of the case c 0. According to results in [MS17b], whenever the speed c of the quenching front is small the equation (1.2) admits a rich family of patterns , as we now briefly describe.
In this paper we give a deeper understanding of the range of validity of these continuations in c. Remark 1.3 Besides the patterns described above, oblique and vertical structures with respect to the quenching front were also studied in [MS17b], where they were shown to not exist as solutions to (1.2) when c > 0. Therefore, the only patterns of relevance for us are the ones described above. Throughout this paper we add sub and superscripts to the patterns found in [MS17b] under the inconvenience of disagreeing with that paper's notation; this is done because the classification of patterns has become more involved and richer. In this way our notation highlights the dependence of the solutions on the quenching speed c and on the κ-periodicity in the y-direction as x → −∞ (in the multidimensional case; see Fig. 1.2).

Main results
Initially we study the problem in one dimensional setting; although less physically relevant, it stands as one of the cornerstone of the construction of the 2D patterns -H κ and H ∞ (see for instance, [MS17b, Sec. 2 and 3]).
The convergence takes place at exponential rate. Furthermore, the solution solution θ (c) (·) is strictly positive and satisfies (θ (c) ) ′ (x) < 0 for all x ∈ R. No solution to (1.7) exists when c > 2.
In the higher dimensional setting, both the magnitude of the speed c of the quenching front and the y-period κ of the end-stateū(·; κ) play important roles in the analysis. for parameters c 0 (speed of the front) and κ > π (yperiodicity of the patterns); the dashed curve represents the critical case P(c, κ) = 1 (see Theorem 1.5).
One can see from the previous result that whenever c 0 the region P(c; κ) < 1 (resp. P(c; κ) > 1) corresponds to c < 1 − π 2 κ 2 (resp., c > 1 − π 2 κ 2 ) , namely, the linear spreading speed obtained from the linearization of (1.2) about u ≡ 0 on the region x 0. Overall, we point out that the dependence of the critical spreading speed curve on the parameter κ is a true manifestation of the multidimensionality of the H κ -patterns; the quantity P(·; ·) describes the maximal speed of spreading of the bistable region and its dependence on the y-period of the pattern away from the quenching interface.
Remark 1.6 (Uniqueness results) It is worth to point out that the uniqueness result in Theorems 1.4 and 1.5 allow us to compare the solutions constructed in [MS17b] using perturbation methods for c 0 with those obtained here.
A summary of our results is given in the Table 1. Single interfaces with contact angle. The H ∞ -patterns obtained in Theorem 1.5 are odd functions with respect to the y variable, and as so they satisfy Ξ (c) ∞ (x, 0) = 0. Thus, the patterns present a nodal set at the negative part of the x-axis that forms a right angle ϕ = π 2 with respect to the quenching front located at a line x − ct = 0 (parallel to the y axis). This observation motivates the question of how extra terms added to equation (1.2) could deform this nodal set. More precisely, we consider the equation ∞ (·, ·) solves (1.9). With regards to (1.9), we remark that two mechanisms are in play: the growth of the bistable region and new, "unbalancing terms", that break the odd symmetry of the solutions.
For instance, when α = 0 and c y = 0 the function Ξ (cx) ∞ (·, ·) solves (1.9) satisfying the limit (1.10) for ϕ * = π 2 and u ± = ±1. It was shown in [MS17a] shows that for any c x 0 fixed, the function Ξ (c=cx) ∞ (·, ·) can be continued in ϕ * as a solution to (1.9) for all ϕ * − π 2 sufficiently small. One of the most important properties used in their proof concerns to the strict monotonicity of the mapping y → Ξ (cx) ∞ (x, y) for any x ∈ R, namely, ∞ (·, ·) is given in Theorem 1.5(i). An inconvenient in their analysis is the fact that the patterns Ξ for c x > 0 were obtained through perturbation methods (in [MS17b,§5]), hence some qualitative information on the patterns are not immediately available. Nevertheless, the authors managed to prove (1.11) for c x 0 sufficiently small (see [MS17a,Prop. 4.1]).Our construction readily gives the validity of (1.11) for all 0 < c x < 2, thus we can make use of the analysis in [MS17a] to conclude the following result: there exist a speed c y (α) and a solution u(x, y; α) to (1.9) with contact angle ϕ(α). We have that u(x, y; π 2 ) = Ξ (c) (·, ·; ∞). Moreover, c y (α) and ϕ(α) are smooth with ϕ(0) = π/2, c y (0) = 0, and u(x, y; α) is smooth in α in a locally uniform topology, that is, considering the restriction u| BR(0) to an arbitrary large ball.

Outline
In Section 2 we focus on the (1 0) (c) problem, where we prove Theorem 1.4. Section 3 is devoted to Theorem 1.5 and H κ patterns (π < κ < ∞), while the study of the H ∞ -pattern is left to Section 4. A brief discussion and further extensions brings the paper to an end in Section 5.
Notation. In this paper we write C k (X; Y ), C k 0 (X; Y ) and C (k,α) (X; Y ) denote respectively, the space of k-times continuously differentiable functions, the space of k times continuously differentiable functions with compact support in X, the space of (k, α)-Hölder continuously differentiable functions from X to Y . We denote the Sobolev spaces over an open set Ω by H k (Ω). The inner product of elements in a Hilbert space H is written as , H . Norms on a Banach space B are denoted as || · || B . For a given operator L : We define a C ∞ (R; [0, 1]) partition of unity {χ ± (·)} of R, of the form (1.12) Last, we denote the Implicit Function Theorem by IFT.
2 One dimensional directional quenching: (1 0) c problem, c > 0 The construction of the patterns (1 0) (c) follows ideas from [MS17b] and [KS03]: initially we solve a family of similar problems in truncated, bounded intervals; later, as we enlarge these intervals and exhaust R, we show that these functions converge to a solution of problem (1.3). We begin by setting up a truncated (1 0) c problem: ; later on the section we let M → ∞ and, subsequently, L → ∞. Roughly speaking, the (1 0) (c) front θ (c) (·) will be given by (2. 2) The qualitative properties of the function θ (c) (·) are proved in this section, where we also show that θ (c) (x) converges to 1 and 0 as x → ∞ and x → −∞, respectively. We finalize with a proof of Theorem 1.4. A substantial part of the techniques and proofs are similar to those in [MS17b]; whenever possible we skip details and refer to that article for full proofs.
Proof. To prove existence of a solution we define an iterative scheme, where we write (·) ′ = ∂ x (·). Following the reasoning in [MS17b, Sec. 2.2], it is shown that {U n } n∈N is pointwisely decreasing and so that θ The uniqueness proof is a bit different due to the transport term c∂ x and we give it here for completeness: assume the existence of two solutions, θ(·), θ(·) so that θ(·) ≡ θ(·).
this set is open due to continuity of θ, θ. As an open subset of the real line, we can assume without loss of generality that I = (a, b) where θ(x) > θ(x), x ∈ (a, b), and θ(x) = θ(x), x ∈ {a, b}. Now, since both θ and θ are solutions, we can integrate against test functions 1 e c x θ(x) and The term on the left hand side is non-positive, since θ > θ in (a, b), θ(x) = θ(x) for x ∈ {a, b}. On the right hand side, the term θ θ is strictly positive, thanks to the strict positivity of solutions in (−M, L). Using that θ > θ in (a, b) we conclude that the integral on the right hand side is positive. This contradiction proves the result.
In order to compare the families of solutions as M, L vary, we construct trivial extensions of functions u defined on an interval (−M, L) given by the operator E , is a continuous function, and this extension is one of the main tools to make (2.2) meaningful and rigorous. The proof of the next result follows the results in [MS17b,§2].
are continuous in the sup norm.

Passing to the limit
We are now ready to pass to the limit M = ∞ as a first step towards the proof of Proposition 1.1. Define where the last equality is a consequence of Lemma 2.2(i). The following proposition highlights the role of the front speed c: roughly speaking it says that stretching procedure M → ∞ we designed "looses mass" whenever c > 2, i.e., θ Although zero is a (trivial) solution to the (1 0) (c) -truncated problem, one might wonder about the usefulness of the minimax construction we developed in (2.2), for it seems to be not good enough to obtain nontrivial solutions to (2.1) in (−∞, L). It turns out that the limitation is not on the method, but on the nature of the problem: no solution to (1.7) exists when c > 2, as we will show afterwards in Lem. 2.4.

]) there exists a solution
w(x) = 1 and so that w(x) is oscillatory as x → +∞ whenever 0 c < 2; see Fig. 2.1. Translation invariance of solutions to this ODE allow us to assume without loss of generality that 0 = w(0) < w(x) < 1 for x < 0.
Applying classical comparison principles to the problem (2.1) on the interval for M > 0, thanks to Lem. 2.3(i) and to the monotonicity Lemma 2.3(ii). Taking the infimum in M > 0 we conclude that θ (c) (−∞,L) (·) = 0, which proves the first part of the result. As a byproduct we obtain (ii) using a squeezing property, for √ 2 appropriately shifted so thatw(0) = 1; notice thatw satisfies ∂ 2 xw −w − (w) 3 = 0 and that is it monotonic, i.e., ∂ x w(·) 0. Hence, w(·) is a supersolution on any interval [0, L] and classical comparison principles imply that The result is obtained using that θ (c) (x) 0 andw(x) → 0 exponentially fast as x → ∞. In order to prove the strict monotonicity of the solution, we use Prop. 2.2(iv): the mapping x → θ (c) (x) is monotonic in x as the sup of monotonic functions, i.e., ∂ x θ (c) 0. One obtains ∂ x θ (c) < 0 by applying Hopf lemma and the maximum principle (notice that the discontinuity of the control parameter µ(·) plays no role here since, by classical regularity theory, we know that θ (c) (·) is in fact smooth away from the quenching front).
We now study the case c > 2, showing that θ (c) (−∞,L) (·) ≡ 0. We argue by contradiction: assume the existence of a solution θ In conclusion, we can make use of monotonicity of v(·) to obtain a τ ∈ R such that w(x + τ ) Using the properties of v(·) and the assumption θ (c) (−∞,L) (·) ≡ 0 we can find a shift τ in such a way that z(·) vanishes in at least one point. Properties of both θ (c) (−∞,L) (·) and v(·) imply that and 3a 2 otherwise. One conclude that z(·) is a supersolution. As z(·) 0 has an interior minimum point, the maximum principle allied to the Hopf Lemma implies that z ≡ 0. However the latter is equivalent to v(·) ≡ θ (c) (−∞,L) (·), which is a contradiction, for v(·) and θ (c) (−∞,L) (·) satisfy different equations. It finishes the proof. Proof.
[of Theorem 1.4] We begin by proving existence when c < 2, following the ideas in [MS17b, Sec. 2] to which we refer to for further details: define is monotonic. Next, we turn to the proof of nonexistence when c > 2: choosing d so that 2 < d < c we can find a strictly positive function w(·), ∂ x w(·) < 0, satisfying In order to understand and compare θ (c) (x) and w(x) as x → ∞ we use an analysis similar to that of [BDNZ09]: we have that w(·), θ (c) (·) > 0, and both satisfy We conclude from [BDNZ09, Prop. 4.1 and Prop. 4.2] the existence of positive constants M and K such that Reasoning as in the proof of Prop. 2.3 one obtain an R > 0 so that Now, as both w(·) and θ (c) (·) are bounded and non increasing, satisfying respectively the asymptotic properties (2.6) and (2.1), we conclude that we can shift w(·) so that w( holds with an equality in at least one point (clearly, the asymptotics in w(·) and θ (c) (·) shows that τ < +∞). Now, defining z(·) = w(· − τ ) − θ (c) (·) we get that z solves an inequality as that of (2.5), thanks to both the non increasing property of w(·) and its positivity. We conclude from the maximum principle and the Hopf's lemma that z ≡ 0, which is an absurd due to the asymptotic behavior of w(·) and θ (c) (·), and this contradiction gives the result.
From the properties of the subsolution used in the previous proof we readily derive the next result: To finalize this section we show that the solutions obtained in [MS17b] for small c 0 through perturbation methods agree with those constructed here. In passing we show that their continuity in the parameter c.
Proof. The proof consists in showing that for any d ∈ [0, 2) the linearization of the equation (1.2) at θ (d) (·) is invertible, i.e., is a boundedly invertible operator from H 2 to L 2 . Indeed, assume the latter to be true. Then, plugging θ (d) + u in (1.2) we rewrite it as As N [θ (d) , u] = O(|u| 2 ) the term on the right hand side is in L 2 (R) and we can apply the IFT to obtain existence and the uniqueness of solutions in a neighborhood of (θ (d) (·), d).
. For a moment, consider the operator L 1 [·] := L θ (d) [·] with domain D L 1 = H 2 (R); the analysis in [MS17b,§5] shows that this is a self-adjoint, Fredholm operator of index 0, with essential spectrum contained in {z ∈ C|Re(z) < 0}. In order to show invertibility we show that this operator has a trivial kernel, which is proved as follows: the properties of the operator L 1 imply that the σ L 1 ∩ {x ∈ R|x 0} is either empty or consists of point spectrum only. It is straightforward to show that this set is bounded, therefore assume that there exists a λ 0 0 maximal eigenvalue, with corresponding eigenfunction u 0 . In the referred paper it was also proven that u 0 ∈ Ker L 1 is spatially localized, namely, whenever u 0 ∈ Ker (L Ξ ) . (2.10) In fact, we know that we can take δ = d 2 , thanks to the results in [MS17a,§4]. From the self-adjoint properties of L 1 we derive u 0 is a ground state associated to its maximal eigenvalue λ 0 ∈ R, therefore it satisfies u 0 (·) 0 almost everywhere (cf. [RS78, XII.12]). We can write the eigenvalue equation and using the properties of the function θ (d) (·), we have by θ (d) (·) and (2.12) by u 0 (·) subtract both equations and integrate in R to find Integration by parts shows that the first integral vanishes, thanks to the decay estimates for θ (d) (·) and u 0 (·).
We are left with We observe that the spatial localization of u 0 (·) as asserted in (2.10) and the fact that ) as x → −∞ imply that both integrals are finite. Sign considerations of both θ (d) (·) and u 0 (·) show that the righthand side is non-positive while the left-hand is nonnegative (since λ 0 0), therefore the integral on the left is zero. Now, invoking the strict positivity of the pattern θ (d) (·) (or equivalently, that of θ (d) (·)) we conclude that u 0 (·) ≡ 0 almost everywhere, which contradicts the fact that u 0 (·) is an eigenfunction. Therefore, no eigenvalue can be found on {z ∈ C|Re(z) 0}; in other words, the operator L 1 is boundedly invertible. Since the latter set is trivial, the same is also true of the kernel of L 1 taken with domain e d 2 x H 2 (R), which corresponds to the operator L θ (d) [·]. As the latter is a Fredholm operator with index 0, this property immediately implies bounded invertibility, and we conclude the proof. 3 Two-dimensional quenched patterns -periodic horizontal interfaces: H κ patterns, π < κ < ∞ In this section we prove Theorem 1.5 in the case π < κ < ∞. As mentioned before, it is important in our approach that the nonlinearity is odd so that we can restrict the study of equation ( where Ξ 0 (·) is chosen in the class Throughout this section, we fix κ ∈ (π, ∞) and suppress the dependence of Ξ andū on κ for ease of notation. As in the previous section, proofs that are similar to those in [MS17b] are only outlined and details are referred to that paper.  We define extension operators in order to compare solutions for different values of M, L, namely, whereū(·) =ū(·; κ) is given in (1.5). We use the same symbols for the one-and two-dimensional extension operators, slightly abusing notation, distinguishing between the two through the domain of definition of the function E is applied to. The proofs of the following Lem then v Ξ

Passing to the limit
We are now ready to pass to the limit M = ∞. Define where the last equality holds due to monotonicity of the mapping M → Ξ   3.2 Existence for the H κ -problem: case c 2 4 + π 2 κ 2 < 1 In order to prove the existence of solutions we construct appropriate subsolutions with the help of the next lemmas: Lemma 3.6 (Properties of the family of periodic solutionsū(·, κ)) Letū(·; κ) be a solution to (1.5) and κ > π. The following two properties hold: Proof. To prove the estimate in (i) we use the elliptic integral that gives the relation between amplitude and spatial period given in [MS17b] [Lemma 4.1, equation (4.4)], [Hal80,§V], for γ 2 = M 2 2−M 2 . Notice that 0 γ < 1. We find a lower bound to the integral on the right hand side: Squaring both sides and plugging γ we obtain κ 2 > π 2 1 − M 2 ⇐⇒ M 2 < 1 − π 2 κ 2 , which finishes the proof of (i). In order to prove (ii) we exploits the structure of this ODE in (1.5), whose Hamiltonian is H(u, u y ) = u 2 y + u 2 − (3.7) Asū y (0) > v y (0) andū(0) = v(0) = 0 it is clear that z(x) > 0 for and x > 0 sufficiently small. By translation invariance of the solutions to the ODE (1.5), reversibility of the solutionsū with respect to x → −x, and the fact that the mapping y → sin(y + π/2) is even it suffices to show that z(y) 0 for 0 y κ 2 . Assume that there exists a 0 < x 0 < κ 2 such that z(x 0 ) = 0. As z(·) 0 solves the elliptic differential equation, we can find and A > 0 sufficiently large so that a−b whenever a = b and 3a 2 otherwise. We conclude using Hopf's lemma that ∂ y z(x 0 ) < 0, which is absurd, since the inequality (3.7) prevents it from happening. Therefore z(y) 0, hence v(y) ū(y) for y ∈ 0, κ 2 , and by symmetry, for y ∈ [0, κ] .
Lemma 3.8 (Existence; H κ -problem, π < κ < ∞) Equation (1.2) has a solution Ξ (c) (·, ·) ∈ C (1,α) (R × [0, κ]; R), for any 0 α < 1, where the latter is defined as Proof. Most κ (·, ·) is also nontrivial. Using Lebesgue dominated convergence we conclude that exists in the pointwise sense and that this sequence converges in L 1 loc hence in the sense of distribution, solving the equation (1.2) in the domain R × [0, κ]. Now it remains to show that the asymptotic limits are satisfied, namely, that The limit on the right follows easily from inequality (3.9), for lim = 0 we conclude that either v ≡ 0 or v is a periodic solution with period τ so that κ τ ∈ N. We can readily exclude the first possibility, since Lem. 3.6(ii) implies that α sin π y κ v L (y) ū(y). The same inequality also implies that τ = 2κ, i.e.,ū(·) and v L (·) have the same period therefore and obey the same normalization, therefore v L (·) ≡ū(·), and the result follows from the equality of (3.11).
Unlike in the previous case, it is not directly clear that the convergence lim x→−∞ Ξ (c) κ (x, y) =ū(y) has exponential rate of convergence. Our next result implies that.
Lemma 3.9 (Exponential convergence; H κ -problem, π < κ < ∞) There exists a C, δ > 0 independent of x and y such that ∞ (x, y)| Ce −δ|x| , and lim (3.12) Proof. Initially we show exponential rate of convergence to the far-field as |x| → ∞. The result follows if we for some δ > 0 and < x >:= √ 1 + x 2 . Indeed, as we know from Lem. 3.8, lim x ∞ e δs ds e δx , for x 0, which gives the result. The proof requires several tools of Fredholm theory for elliptic operators. The linearization of the equations (1.2) at Ξ (c) . Although this operator is nonself-adjoint, the limits as |x| → ∞ of Ξ (c) are the same for all c ∈ [0, 2), therefore the results of [MS17b, Lem. 5.1 and 5.2] apply, showing that the operator L Ξ is Fredholm of index 0, with essential spectrum strictly negative in . According to [MS17b,Lem. 5.3], we know that elements in the kernel of L Ξ are spatially exponentially localized, namely, whenever u 0 ∈ Ker (L Ξ ) (3.14) Recall the partition of unity χ ± (·) defined in (1.12). We know that ∂ we can assume with no loss of generality that v ∈ X , thanks to property (3.14) for elements in the kernel. However, as the operator L Ξ : X → Rg (L Ξ ) is boundedly invertible, we can apply the same reasoning used in [MS17b,Cor. 5.5] to conclude that for all δ > 0 sufficiently small. A similar analysis can be done by considering w(x, y) = ∂ y (χ + (y)Ξ (c) (x, y)), whence exponential rate of convergence to the far field as y → ∞ is derived. It finishes the proof.
In fact, one can show by following the steps in the proof of Lem. 2.6 that the operator L Ξ in (3.13) is boundedly invertible from H 2 (R × [0, κ]) to L 2 (R × [0, κ]). Once more, using the IFT, we conclude the following result. Proof. The analysis is analogous to that of Lem. 2.6 and is outline below, where we point out the necessary modifications. Fix κ ∈ (π, ∞). Initially we define the linearized operator about the solutions Ξ c κ , obtaining the linearized operator 2 u we rewrite the above operator in a "self-adjoint" form, Notice that the mapping u(·) → e a−b whenever a = b and 3a 2 otherwise. We can apply classical maximum principles, since for M > 0 chosen large enough in order to give M + f [Ξ,V ] > 0; thus we write the above equation as As Z has a maximum point in R × [0, κ] we get that Z ≡ 0. This contradiction leads to the non-existence of solutions in the case c 2 4 + π 2 κ 2 > 1.
Finally, we put all these auxiliary results together and prove the main result of this section: Proof.
[of Theorem 1.5; case π < κ < ∞] Combine the above discussion of the non-existence of solutions with the results of Lemmas 3.8, 3.9, and 3.10 4 Two-dimensional quenched patterns -single horizontal interfaces; H ∞ problem In this section, we shall prove Theorem 1.5 in the case κ = ∞. To be consistent with the notation introduced in Section 2 we exploit the fact that the nonlinearity in (1.2) is odd to solve the problem in the half space R × (−∞, 0]. Further symmetries of the equation are also exploited: we solve an equivalent H ∞ -problem, seeking for a solution Ξ Thanks to the results of Sec. 2 related to the 1-d problem the following observation is readily available. Observation 4.1 (Restriction to the case c < 2) It is clear from (4.1) that the above problem is meaningless when c > 2 for the patterns θ (c) do not exist. We can readily say that no solution to this problem exists when c > 2, immediately restricting our study to the range 0 c < 2.
In fact, in this section we prove that for all quenching fronts speeds in the range c ∈ [0, 2) there exists a unique H ∞ -pattern (up to translations in the y direction), which corresponds to the statement of Theorem1.5. The strategy goes as in Sections 2 and 3: first, by reducing the problem to a half plane and truncating it, restricting the problem to a rectangle Ω (−M,L) := (−M, L) × (−M, 0). Then we let M → ∞ and, subsequently, we let L → ∞. For the sake of simplicity in this section we will omit any sub-index ∞.
The truncated H ∞ -problem is set up as  L) ), for all 0 α < 1. Following the method in Sec. 3, we extend these functions to the whole plane R 2 : We summarize the main properties of the functions E Θ
Lemma 4.6 (Uniqueness up to translation in the y-direction) Whenever c 0 the solutions constructed in Theorem 1.5 and the solutions constructed in [MS17b] by continuation are the same up to translation in the y-direction.

Proof.
The machinery given in [MS17a] can be used to derive a simple proof: first notice that ∂ y Ξ ∞ ∈ KerL Ξ∞ [·]. As ∂ y Ξ has a sign one can use [MS17a,Lem. 4.9] to conclude that ∂ y Ξ ∞ = C∂ y Θ for a C constant. Upon integration in y and using the fact that both solutions converge to the same limit as y → ±∞ and satisfy Θ(x, 0) = Ξ ∞ (x, 0) = 0 we get that C ≡ 1, and the result follows.
Our last result concerns the continuity of the patterns Ξ (c) ∞ (·, ·) in c.

Discussion
Among several possible directions of further investigation, we would like to mention the following: Metastability of patterns. As addressed by the numerical studies of [FW12, Sec. VI], defining the parameter regions of metastability for creation of patterns (either perpendicular or parallel to the quenching front) is a challenging and interesting direction of investigation. From a broader perspective, a numerical, if possible analytical, description of parameter curves on the boundary of different morphological states would be valuable in applications.
Selection mechanisms. What are the crucial mechanisms involved in the wavenumber selection in the wake of the front? How relevant are the nonlinearity and the speed of the quenching front in this selection? We refer to [Nis02, §3.3] for a general discussion about wavenumber selection.
Critical cases; P(c; κ) = 1. The behavior of the patterns in this critical scenario possibly requires a different approach, since one can see in the proofs of Theorem 1.4 and 1.5 that the speed of the quenching front has to be away from the critical case. The result would be interesting and add valuable knowledge in the classification of patterns obtained from directional quenching.
Non-planar quenching fronts and oblique stripes. Is it possible to control the contact angle of the H κpatterns? Although it was shown in [MS17b] that oblique patterns do not exist in (1.2), these patterns can still exist in the case of the unbalanced equation (1.9). It is worth to mention that the result of [MS17a] describes a family of solutions displaying one single (almost) horizontal interface whose contact angle with the quenching front can be varied by modification of the chemical potential parameters across the quenching interface. We refer to [TLMR13] for physics motivation and a more detailed discussion on the chirality of helicoidal patterns in the context of recurrent precipitation.