Horizontal patterns from finite speed directional quenching
Rafael Monteiro
math.AP
































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LemmaLemma[section]
Theorem[Lemma]Theorem
Proposition[Lemma]Proposition
Corollary[Lemma]Corollary
Remark[Lemma]Remark
Definition[Lemma]Definition
Hypothesis[Lemma]Hypothesis
Assumption[Lemma]Assumption
Observation[Lemma]Observation
Proof
  
  Proof. 
 .4.4

Acknowledgment
 
  Acknowledgments.



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Horizontal patterns from finite speed directional quenching
     Rafael Monteiro

University of Minnesota, School of Mathematics,  
206 Church St. S.E., Minneapolis, MN 55455, USA
    December 30, 2023
=======================================================================================================================







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 our case the interface moves with speed c in such a way that the bistable region grows. According to results from  <cit.>, several patterns exist when c ≳0, and here we investigate their persistence for finite c>0, clarifying the pattern selection mechanism related to the speed c of the quenching front.  




Keywords. Phase separation, directional quenching, Allen-Cahn, Spreading speeds.





§ 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 <cit.> and the survey <cit.>. 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

 
    ∂_tu  = Δ_(ξ,y) u + μ(ξ - ct) u  - u^3,


where Δ_(ξ,y):= ∂_ξ^2 + ∂_y^2 and  μ(ξ - ct ≷ 0) = ∓ 1. Equation (<ref>) 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∈^+ :=[0,∞) and point (ξ,y)∈^2.  The stationary problem in a moving frame (x,t) = (ξ - ct, t) is written 

 
    - c∂_x u = Δ_(x,y) u + μ(x) u  - u^3 .


The most physically relevant scenario to be considered consists of the case c≥ 0. According to results in <cit.>, whenever the speed c of the quenching front is small the equation (<ref>) admits a rich family of patterns , as we now briefly describe. 



  
Pure phase selection: 1->0 fronts.

This is the simplest, one dimensional case, when (<ref>) reads


    - c∂_x u(x) = ∂_x^2 u(x) + μ(x)u(x)  - u(x)^3,      x∈.


The quenching trigger generates a pattern θ^(c)(·) solving (<ref>) and satisfying spatial asymptotic conditions lim_x→ -∞θ^(c)(x)= 1  and  lim_x→ +∞θ^(c)(x)= 0 in the wake and ahead of the quenching front, respectively; see Figure <ref>. 





  
Horizontal patterns: H_kappa, pi <k <=infty.

In this scenario the patterns  sought are truly two-dimensional; furthermore, we can take advantage of the odd nonlinearity to reduce the study of (<ref>) to a  problem in  ×[0,κ]. Whenever κ <∞ the solution Ξ_κ^(c)(·,·) has boundary conditions Ξ_κ^(c)(x,y)|_y=0,κ=0 and  spatial asymptotic conditions  

    lim_x→ +∞Ξ_κ^(c)(x,y)= 0,     lim_x→ -∞|Ξ_κ^(c)(x,y)-u̅(y;κ)|= 0,

where  u̅(y;κ) is a parametrized family of periodic solutions to


    ∂_y^2u̅+u̅-u̅^3=0,     u̅(y;κ)=-u̅(y+κ;κ)=-u̅(-y;κ)≢0,       y∈.

with half-periods  π<κ<∞, and normalization ∂_yu̅(0;κ)>0, u̅(0;κ) = 0. In the limiting case κ = ∞ a single interface is created, describing what we call _∞-pattern (see Fig. <ref>); asymptotically we have


    lim_y→ +∞Ξ_∞^(c)(x,y)= θ^(c)(x),   lim_x→ +∞Ξ_∞^(c)(x,y)= 0,     lim_x→ -∞Ξ_∞^(c)(x,y)=u(y;∞),


where u(·;∞):=tanh(·/√(2)) and θ^(c)(·) is a -front.



We summarize  below the properties of these solutions in the regime c ≳ 0 as given in <cit.>. Their approach is based on a continuation argument from the case in which quenching front has zero speed (c=0), somehow explaining the nature of the smallness on c in their results.

[<cit.>;  problem] For any c ≥ 0 sufficiently small there exists a function θ^(c)(·) ∈𝒞^(1,α)(; [0,1]), ∀α∈ [0,1), that solves the  problem, i.e., θ^(c) solves the boundary value problem  


    {[ ∂_x^2θ^(c)(x) + c∂_xθ^(c)(x) + μ(x) θ^(c)(x) - (θ^(c)(x))^3 = 0;                   0< θ^(c)(x) <1,   θ^(c)(-∞)=1,   θ^(c)(+∞)=0, ].


where the boundary conditions are attained in the limit sense. Furthermore, the mapping x ↦θ^(c)(x) is non-increasing. The solutions found for c=0 can be continued smoothly to c>0 for sufficiently small c. More precisely, there exist families of solutions  θ^(c)(x) to (<ref>) for 0<c<δ_1, satisfying the same limiting conditions as the solutions at c=0 for x→±∞. Moreover, the solutions depend smoothly on c, uniformly in x.
 

[<cit.>; _κ problem, π < κ≤∞] For any c ≥ 0 sufficiently small equation (<ref>) admits a family of solutions (in the sense of distributions) Ξ_κ^(c)(·,·)∈𝒞^(1,α)(^2; ), ∀α∈ [0,1), κ∈ (π,∞]. In the case π < κ <∞ for the solution Ξ_κ^(c)(·, ·) has the symmetries 
Ξ_κ^(c)(x,y)=-Ξ_κ^(c)(x,-y)=-Ξ_κ^(c)(x,y+κ)=-Ξ_κ^(c)(x,y+κ) and satisfies  the asymptotic spatial conditions (<ref>). Moreover, the convergence is exponential, uniformly in y. On the other hand, whenever κ = ∞ the solution Ξ_∞^(c)(·, ·) has the symmetries Ξ_∞^(c)(x,y)=-Ξ_∞^(c)(x,-y) and satisfies the asymptotic spatial conditions (<ref>), where convergence is exponential and  uniform.


In this paper we give a deeper understanding of the range of validity of these continuations in c. 

Besides the patterns described above, oblique and vertical structures with respect to the quenching front were also studied in <cit.>, where they were shown to not exist as solutions to (<ref>) 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 <cit.> 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. <ref>). 




 §.§ 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, <cit.>).


[-patterns] For any fixed c ∈ [0,2) there exists a unique solution θ^(c)(·) ∈𝒞^(1,α)(; (0,1)), ∀α∈ [0,1) satisfying

    {[ ∂_x^2θ^(c)(x) + c ∂_xθ^(c)(x) + μ(x)θ^(c)(x) - (θ^(c))^3(x) = 0;     0< θ^(c)(x) <1,   lim_x→ -∞θ^(c)(x)=1,   lim_x→∞θ^(c)(x)=0. ].

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 ∈. No solution to (<ref>) 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-state u̅(·; κ) play important roles in the analysis. 









[][htb]
  
    < g r a p h i c s >

Existence diagram  for parameters c≥0 (speed of the front) and κ  > π (y-periodicity of the patterns); the dashed curve represents the critical case 𝒫(c,κ) =1 (see Theorem <ref>). 

[ℋ_κ, ℋ_∞ patterns]
Let π <κ≤∞ be a fixed number. 
Define the quantity


    𝒫(c;κ) := {[ c^2/4 + π^2/κ^2,          π<κ <∞;;           c^2/4,           κ = ∞. ].



  * (Existence)  The solutions Ξ_κ^(0)(·, ·) defined in Prop. <ref> when  c=0 can be continued smoothly in c>0 within the range 𝒫(c;κ) < 1 to  solutions  Ξ_κ^(c)(·,·)  solving (<ref>) in the sense of distributions and satisfying the asymptotic spatial condition (<ref>) (resp. (<ref>)) when π <κ <∞ (resp., when κ = ∞). The convergence to their spatial asymptotic states takes place at exponential rate, uniformly. Furthermore, for any κ > π the mapping x↦Ξ_κ^(c)(x,y) is non-increasing for any fixed y ∈[0,κ] and 0 <Ξ_κ^(c)(x,y) <u̅(y;κ) in (x,y) ∈×(0,κ), where u̅(·,·) is given by  <ref>. The solutions have the symmetries Ξ_κ^(c)(x,y)=-Ξ_κ^(c)(x,-y)=-Ξ_κ^(c)(x,y+κ)=-Ξ_κ^(c)(x,y+κ) (resp., Ξ_∞^(c)(x,y)=-Ξ_∞^(c)(x,-y)). 

 
  * (Nonexistence) Whenever κ∈ (π, ∞) (resp., κ =∞) and 𝒫(c;κ) > 1 no solution to (<ref>) satisfying 0 <Ξ_κ^(c)(x,y) <u̅(y;κ) in (x,y) ∈×(0,κ) and the asymptotic spatial condition (<ref>) (resp. (<ref>)) can be found. 





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 (<ref>) 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. 

[Uniqueness results]
 It is worth to point out that the uniqueness result in Theorems <ref> and <ref> allow us to compare the solutions constructed in <cit.> using perturbation methods for c ≳ 0 with those obtained here.




  
Critical cases; S=1 To the best of the author's knowledge, the cases c = 2 (for the -problem), and 𝒫(c;κ) =1 (for the _κ and _∞-problems) are open. See Sec. <ref>.


A summary of our results is given in the Table <ref>.



  



  
Single interfaces with contact angle. The  _∞-patterns obtained in Theorem <ref> 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 (<ref>) could deform this nodal set. More precisely, we consider the equation 


    Δ u +c_x∂_x u + c_y∂_y u +  μ(x) u  - u^3 + α g(x,u)=0 ,     (x,y)∈^2.


where μ(x) = ± 1 when x ≶ 0
and g(x,u)={[ g_l(u),     x<0; g_r(u),     x>0 ].
, for g_{l,r}(·) ∈𝒞^∞(;). Notice that c_y = 0, α =0, the function u(·,·) = Ξ_∞^(c_x)(·,·) solves (<ref>).

With regards to (<ref>), 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. 

[Contact angle]



We say (<ref>) possesses a solution u with contact angle φ_* if u possesses the limits 

    lim_x→ +∞ u(x,y)=0,     lim_x→ -∞ u(x,(φ)x)={[  u_+, φ>φ_*;  u_-, φ<φ_* ]. ,

for all 0<φ<π.  


For instance, when α = 0 and c_y=0 the function Ξ_∞^(c_x)(·,·) solves (<ref>) satisfying the limit (<ref>) for φ^* = π/2 and u_± = ±1.  It was shown in <cit.> shows that  for any c_x≳0 fixed, the function Ξ_∞^(c=c_x)(·,·) can be continued in φ^*  as a solution to (<ref>) 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_∞^(c_x)(x,y) for any x∈, namely,


    ∂_yΞ_∞^(c_x)(x,y) > 0,      (x,y) ∈^2.


where Ξ_∞^(c_x)(·,·) is given in Theorem <ref>. An inconvenient in their analysis is the fact that the patterns Ξ_∞^(c_x) for c_x>0 were obtained through perturbation methods (in <cit.>), hence some qualitative information on the patterns are not immediately available. Nevertheless, the authors managed to prove (<ref>) for c_x ≳0 sufficiently small (see <cit.>).
Our construction readily gives the validity of (<ref>) for all 0<c_x<2, thus we can  make use of the analysis in <cit.> to conclude the following result:

[Unbalanced patterns]
 For 0<c_x < 2, there exists α_0(c_x)>0 such that for all |α|<α_0(c_x) there exist a speed c_y(α) and a solution  u(x,y;α) to (<ref>) 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|_B_R(0) to an arbitrary large ball.








 §.§ Outline


In Section <ref> we focus on the  problem, where we prove Theorem <ref>. Section <ref> is devoted to Theorem <ref> and _κ patterns (π <κ <∞), while the study of the _∞-pattern is left to  Section <ref>.  A brief discussion and further extensions brings the paper to an end in Section <ref>. 



  
Notation. In this paper we write 𝒞^k(X;Y), 𝒞_0^k(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: 𝒟(ℒ) ⊂ X → Y we write Ker( ℒ) :={ u ∈𝒟(ℒ)| ℒu=0 } and Rg( 
ℒ) :={ f ∈ Y |∃ u ∈𝒟(ℒ), Lu =f }. A distribution T ∈
𝒟'(Ω) satisfies  T≥ 0 in the sense of distributions if T(ϕ) ≥ 0 for any ϕ(·) ∈𝒞_0^∞(Ω; [0, ∞)).
We define a 𝒞^∞(;[0,1]) partition of unity {χ^±(·)} of , of the form 


    χ^-(x) + χ^+(x)  =1,    χ^-(x) =1       x≤ -2,    χ^-(x) =0      x≥ -1.

Last, we denote the Implicit Function Theorem by IFT. 



  
Acknowledgments. The author is grateful to Prof. Arnd Scheel for many interesting discussions and insights throughout the writing of this paper. Many thanks also go  to Prof. Yasumasa Nishiura and Prof. Natsuhiko Yoshinaga for sharing their perspective on this work.  The author acknowledges partial support through  NSF grants DMS-1612441 and DMS-1311740.

 

§ ONE DIMENSIONAL DIRECTIONAL QUENCHING: 1-0 PROBLEM, C>0


The construction of the patterns  follows ideas from  <cit.> and <cit.>: initially we solve a family of similar problems in truncated, bounded intervals; later, as we enlarge these intervals and exhaust , we show that these functions  converge to  a solution of problem (<ref>). We begin by setting up a truncated (1 0)^c problem:

    {[ ∂_x^2u(x) + c∂_xu(x) + μ(x) u(x)  - u^3(x)                                       = 0,;                         u(-M)= 1,  u(L)=0,                                            ].

for 0<M,L, with continuity of u and u_x at x=0. It is shown the existence of a unique solution θ^(c)_(-M,L); later on the section we  let M→∞ and, subsequently, L→∞.  Roughly speaking, the  front θ^(c)(·)  will be given by


    θ^(c)(·) = sup_L>0{inf_M>0θ^(c)_(-M,L)(·) }.



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 <ref>.  A substantial part of the techniques and proofs are similar to those in <cit.>; whenever possible we skip details and refer to that article for full proofs. 


 §.§ The 1->0c truncated problem
 

[Existence and uniqueness] The truncated problem (<ref>) has a unique solution θ_(-M,L)^(c)(·) ∈𝒞^(1,α)([-M,L];[0,1]),  ∀α∈ [0,1). Furthermore, θ_(-M,L)^(c)(x) >0 whenever x ∈ (-M,L).

To prove existence of a solution we define an iterative scheme,

    {[ -  U_n+1^”(x) - cU_n+1'(x) +5U_n+1(x)  =  (5 +μ(x) ) U_n(x) -U_n^3(x),     (-M,L);                                                       U_n+1(-M)=1,    U_n+1(L)=0, ].

where we write (·)' = ∂_x(·). Following the reasoning in <cit.>, it is shown that {U_n}_n∈ℕ is pointwisely decreasing and so that θ_(-M,L)^(c)(x) := inf_n ∈ℕU_n(x) is a well defined element of 𝒞^(1,α)([-M,L];[0,1]),  ∀α∈ [0,1). Furthermore, θ_(-M,L)^(c)(x) >0 for x ∈ (-M,L).  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(·). Define the set  ℐ = {x ∈ [-M,L]   | θ(x) ≠θ(x) }; 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[Recall that distributions of finite order (say, order k) can be extended to the space of 𝒞_0^k functions (cf. <cit.>; see also <cit.>).] e^c xθ(x) and e^c xθ(x) on the interval (a,b):
 
 
    ∫_a^b e^c  x(-θ”θ +θθ”)(x) dx - c∫_a^b e^c  x(θ' θ - θθ')(x)dx = ∫_a^be^c  x(θ^2 -  θ^2) (x)θ(x) θ(x)dx,

 
 where (·)' denotes ∂_x. Integration by parts gives
 
 
    e^c  x(-θ'θ +θθ')(x)|_a^b = ∫_a^be^c  x(θ^2 -  θ^2)(x) θ(x) θ(x)dx.

 
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 θθ 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, 


    ℰ[u](x) = {[       u(x),             x ∈ (-M, L);          1,                  x ≤ -M;          0,                  x ≥ L. ].


By construction, ℰ[θ_(-M,L)^(c)(x)] is a continuous function, and this extension is one of the main tools to make (<ref>) meaningful and rigorous. The proof of the next result  follows the results in <cit.>.

[Properties of solutions to the truncated problem] The following properties of ℰ[θ_(-M,L)^(c)](·) hold.


  * (Monotonicity of ℰ)	
 We have 0 ≤ℰ[θ_(-M,L)^(c)](·) ≤ 1. Furthermore, for w defined on a subset A, (-M,L) ⊂ A ⊂ (-∞,L) with  0 ≤ w(·) ≤ 1 and 0 ≤ w(·) ≤θ_(-M,L)^(c)(·) in (-M,L), we have
 
0 ≤ℰ[w](·) ≤ℰ[θ_(-M,L)^(c)](·)      ℝ.                                                                                                                                                                                                                          
 
 
  * (Monotonicity in M)
Let  0 ≤ M <M and L ≥ 0 be fixed. Then ℰ[θ_(-M,L)^(c)](x) ≤ℰ[θ_(-M,L)^(c)](x).


  * (Monotonicity in L)
Let  0 ≤ L <L and M ≥ 0 be fixed. Then ℰ[θ_(-M,L)^(c)](x) ≤ℰ[θ_(-M,L)^(c)](x).

  * (Monotonicity in x) For every fixed M and L the mapping x↦ℰ[θ_(-M,L)^(c)](x) is non-increasing.


  * (Continuous dependence of θ_(-M,L)^(c)(·) on L,M) Let 0<M< ∞, 0 <L< ∞. The mappings L↦ℰ[ θ_(-M,L)^(c) ](·) and M↦ℰ[ θ_(-M,L)^(c) ](·) 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 <ref>. Define 
 
    θ_(-∞,L)^(c)(x) := inf_M >0ℰ[θ_(-M,L)^(c)](x)  = lim_M →∞ℰ[θ_(-M,L)^(c)](x),

where the last equality is a consequence of Lemma <ref>(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_(-∞,L)^(c)(x) ≡ 0 when c>2.  Although zero is a (trivial) solution to the -truncated problem, one might wonder about the usefulness of the minimax construction we developed in (<ref>), for it seems to be not good  enough to obtain nontrivial solutions to (<ref>) in (-∞,L). It turns out that the limitation is not on the method, but on the nature of the problem: no solution to (<ref>) exists  when c>2, as we will show afterwards in Lem. <ref>.  

[Dichotomy c ≷ 2] For any fixed c>0 we verify


    θ_(-∞, L)^(c)(·) ≠ 0,      c<2;        θ_(-∞, L)^(c)(·) ≡ 0,       c>2.


Furthermore, whenever 0 ≤ c<2 and L>0, the family θ_(-∞,L)^(c)(·) has the following properties: 

 
  *  The mapping L↦θ_(-∞,L)^(c)(·) is continuous in the sup norm on 0≤ L≤∞.
 
  * (Monotonicity)
 The functions x↦θ_(-∞,L)^(c)(x) are defined for every x ∈ℝ.
  The mapping  L ↦θ_(-∞,L)^(c)(x) is non-decreasing for any fixed x. Furthermore, the mapping  x ↦θ_(-∞,L)^(c)(x) is non-increasing  for any fixed L.

 
  * The function θ_(-∞,L)^(c)(x) solves (<ref>) on  (-∞, L). Furthermore, lim_x → -∞θ_(-∞,L)^(c)(x) = 1.
      
 

 We deal first with the case c<2: from ODE theory (cf. <cit.>) there exists a solution w(·)∈𝒞^∞(;[-1,1]) to ∂_x^2w + c∂_xw + w - w^3=0  satisfying lim_x→-∞w(x) =1 and so that w(x) is oscillatory as x → +∞ whenever 0 ≤ c<2; see Fig. <ref>. 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 (<ref>) on the interval [-M,0] we conclude that w(x) ≤θ_(-M,0)^(c)(x) hence  w(x) ≤ℰ[θ_(-M,0)^(c)](x)≤ℰ[θ_(-M,L)^(c)](x) for M>0, thanks to Lem. <ref> and to the  monotonicity Lemma <ref>. Taking the infimum in M>0 we conclude that θ_(-∞,L)^(c)(·)≠ 0, which proves the first part of the result. As a byproduct we obtain (ii) using a squeezing property, for


    1 = lim_x → -∞w(x) ≤lim inf_x→ -∞θ^(c)_(-∞,L)(x)≤ 1.
 

Item (i) is a direct consequence of Lem. <ref>. To show that lim_x→∞θ^(c)(x)=0 we use the function w̅(x) := (x + x_0)/√(2)  appropriately shifted so that w̅(0) =1; notice that w̅ satisfies ∂_x^2w̅ - w̅ - (w̅)^3 =0 and that is it monotonic, i.e., ∂_xw(·) ≤ 0. Hence, w(·) is a supersolution on any interval [0,L] and classical comparison principles imply that 


    θ^(c)_(0,L)(x) ≤w̅(x),      x ∈[0,L].


Thus,  θ_(-∞,L)^(c)(x) ≤w̅(x) using Lem. <ref>. The result is obtained using that θ^(c)(x) ≥ 0 and w̅(x) → 0 exponentially fast as x →∞. 

In order to prove the strict monotonicity of the solution, we use Prop. <ref>: 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_(-∞,L)^(c)(·) ≡ 0. We argue by  contradiction: assume the existence of a solution θ_(-∞,L)^(c)(·) satisfying (<ref>) in (-∞,L) and so that θ_(-∞,L)^(c)(x) >0 when  x ∈ (-∞, L). There exists a d ∈ (2, c) and a solution v(·) to ∂_x^2v + d∂_xv + v - v^3=0  satisfying v(-∞) =1, v(∞) =0, v(·) >0, ∂_x v(·) <0. Define m(x) = - x/2+ √(x^2/4 +2). Results from the asymptotic theory of ODEs  (cf. <cit.>) show that


    v(x) = 1 + 𝒪(e^m(d)x),     θ_(-∞,L)^(c)(x) = 1 + 𝒪(e^m(c)x),      x→ -∞.

As m(c) <m(d), there exists a R> sufficiently large such that  v(x) ≥θ_(-∞,L)^(c)(x) for all x ≤ -R. Positivity of v(·) implies  that  v(x)≥θ_(-∞,L)^(c)(x) whenever x ≥ L. As θ_(-∞,L)^(c)(·) is monotone, there exists an ϵ >0 such that 

    θ_(-∞,L)^(c)(x) ≤ 1 - ϵ,       x ∈ [-R,L].
 In conclusion, we can make use of monotonicity of v(·) 
to obtain a τ∈ such that  w(x + τ) ≥θ_(-∞,L)^(c)(x) on x ∈ [-R,L], hence  guaranteeing that v(x + τ) ≥θ_(-∞,L)^(c)(x) for all x∈. For such a v(·) we define z(x) := v(x + τ) - θ_(-∞,L)^(c)(x) ≥ 0. Using the properties of v(·) and the assumption  θ_(-∞,L)^(c)(·)≢0 we can find a shift τ in such a way that z(·) vanishes in at least one point. Properties of both θ_(-∞,L)^(c)(·) and v(·) imply that


    ∂_x^2 z(x) + c ∂_x z(x) + μ(x)z(x) + f[θ_(-∞,L)^(c)(x),w(x)] z(x) = (c-d)∂_x w(x) + [μ(x) - 1]w(x)≤ 0,

in x∈ (-∞,L), where f[a,b]:= a^3 - b^3/a-b whenever a≠ b 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_(-∞,L)^(c)(·), which is a contradiction, for v(·) and θ_(-∞,L)^(c)(·) satisfy different equations. It finishes the proof. 



[Existence/non-existence;    problem] There exists a solution θ^(c)(·) ∈𝒞^(1,α)()
 


[of Theorem <ref>] We begin by proving existence when c<2, following the ideas in <cit.> to which we refer to for further details: define θ^(c)(x) = sup_L>0θ_(-∞,L)^(c)(x) = lim_L→∞θ_(-∞,L)^(c)(x). The asymptotic behavior at x → -∞ is a consequence of Prop. <ref>-<ref>, since  the mapping L ↦θ_(-∞,L)^(c)(·) 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(·), ∂_xw(·) < 0, satisfying 


    ∂_x^2w(x) + d∂_xw(x) + w(x) - (w(x))^3 = 0,     lim_x→ -∞w(x) = 1,     lim_x→ -∞w(x) = 0.


A direct computation shows that ∂_x^2w(x) + c∂_xw(x) + μ(x)w(x) - (w(x))^3 = (μ(x) -1)w +(c-d)∂_x w≤ 0. 
A similar analysis to that in (<ref>) shows that w(x) ≥θ^(c)(x) for all x → -∞. In order to understand and compare θ^(c)(x) and w(x) as x →∞ we use an analysis similar to that of <cit.>: we have that  w(·), θ^(c)(·) >0, and both  satisfy

    ∂_x^2w + d∂_xw - w≤ 0 ≤∂_x^2θ^(c)+c∂_xθ^(c) - θ^(c).


We conclude from <cit.> the existence of positive constants M and K such that 

    w(x)≥ K exp(-d/2 - √(d^2/4 -1)),  θ^(c)(x)≤ M exp(-c/2 - √(c^2/4 +1)).

Reasoning as in the proof of Prop. <ref> one obtain an R>0 so that 


    w(x) - θ^(c)(x) >0,        x∈ (-∞,-R]∪ [R, ∞).

Now, as both w(·) and θ^(c)(·) are bounded and non increasing, satisfying respectively the asymptotic properties (<ref>) and (<ref>), we conclude that we can shift w(·) so that w(x - τ) ≥θ^(c)(x) for all x ∈.
Now we take the infimum of τ such that w(x - τ) ≥θ^(c)(x) 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 (<ref>),


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:

[Exponential convergence;  problem] For any θ^(c)(·) solving (<ref>) when c<2 there exists a C, δ >0 independent of x such that


    lim_x→∞|θ^(c)(x)| ≤ Ce^-δ |x|,     lim_x→∞|θ^(c)(x) - 1 | ≤ Ce^-δ |x|.




To finalize this section we  show that the solutions obtained in <cit.> for small c≥0 through perturbation methods agree with those constructed here. In passing we show that their continuity in the parameter c.

[Uniqueness of the  continuation in c;    fronts] There exists a unique continuation θ^(c)(·) solving the problem (<ref>) whenever c ∈ [0, 2). Moreover, the mappings c↦θ^(c)(·): [0,2) → L^∞(;) is continuous.



The proof consists in showing that for any d∈[0,2) the linearization of the equation (<ref>) at θ^(d)(·) is invertible, i.e., 


    ℒ_θ^(d)[v] = ∂_x^2 v + d∂_x v + μ(x) v  - 3(θ^(d))^2 v ,   𝒟(ℒ_θ^(d)) = H_2().


is a boundedly invertible operator from H^2 to L^2. Indeed, assume the latter to be true. Then, plugging θ^(d) + u in (<ref>) we rewrite it as


    ℒ_θ^(d)[u] = 𝒩[θ^(d),u] + (d-c)∂_x θ^(d).


As 𝒩[θ^(d),u] =𝒪(|u|^2) the term  on the right hand side is in L^2() and we can apply the IFT to obtain existence and  the uniqueness  of solutions in a neighborhood of (θ^(d)(·), d). Furthermore, the mapping d↦θ^(d)(·) is continuous in L^∞(), thanks to the Sobolev embedding H^2() ↪ L^∞() (cf. <cit.>).


Keeping the previous discussion in mind, we devote the rest of the proof to showing the invertibility of operator ℒ_θ^(c). 

Initially, we write the conjugate operator  ℒ_θ^(c)[·] := e^d x/2ℒ_θ^(c)[e^-d x/2 ·]:


    ℒ_θ^(c)[] = ∂_x^2 + (μ(x)-d^2/4)  - 3(θ^(d))^2  ,   ṽ∈𝒟(ℒ_θ^(d)) = e^d x/2H^2() =: H_d^2().

 
 Defining ||u||_H_d^2 = ||e^d x/2u||_H^2, we see that the isometry ℐ: H_d^2() → H^2(), ℐ[u]= e^d x/2u  implies that ℒ_θ^(d) is invertible on H^2() if and only if  ℒ_θ^(d) is invertible on H_c^2(). For a moment, consider the operator ℒ_1[·] := ℒ_θ^(d)[·] with domain  𝒟(ℒ_1)= H^2(); the analysis in <cit.> shows that this  is a self-adjoint, Fredholm operator of index 0, with essential spectrum contained in {z∈| 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 ∈| 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, 
 
 
    |∇ u_0(x,y)|+|u_0(x,y )| ≤  C e^- δ|x| ,    (x, y)  ∈ℝ× [0,κ]    a.e.       u_0 ∈Ker(ℒ_Ξ).




In fact, we know that we can take δ =d/2, thanks to the results in <cit.>. From the self-adjoint properties of ℒ_1 we derive u_0 is a ground state associated to its maximal eigenvalue λ_0∈, therefore it satisfies u_0(·) ≥ 0  almost everywhere (cf. <cit.>).  We can write the eigenvalue equation ℒ_1[u_0] = λ_0 u_0 as



    ℒ_1[u_0] = ∂_x^2 u_0(x) + (μ(x)-d^2/4)u_0(x)  - 3e^-d x(θ^(d)(x))^2 u_0(x) = λ_0 u_0(x)


Setting  θ^(d)(·) =e^d  x/2θ^(d)(·) and using the properties of the function θ^(d)(·), we have


    ∂_x^2 θ^(d)(x) + (μ(x) - d^2/4) θ^(d)(x) - e^-d x (θ^(d))^3(x) = 0;


asymptotic theory of ODEs (cf. <cit.>) implies that lim_|x|→∞θ^(d)(x) =0. Now, multiply (<ref>) by θ^(d)(·) and (<ref>) by u_0(·) subtract both equations and integrate in  to find


    ∫_ (θ^(d)(x) ∂_x^2u_0(x) - u_0(x)∂_x^2θ^(d)(x) dx -2 ∫_ e^-d x(θ^(d)(x))^2θ^(d)(x) u_0(x) dx = λ_0 ∫_θ^(d) u_0  dx
 
Integration by parts shows that the first integral vanishes, thanks to the decay estimates for θ^(d)(·) and u_0(·). We are left with  


    -2 ∫_ e^-d x(θ^(d)(x))^2θ^(d)(x) u_0(x) dx = -2 ∫_(θ^(d)(x))^2θ^(d)(x) u_0(x) dx= λ_0 ∫_ w u_0  dx.

We observe that the spatial localization of u_0(·) as asserted in (<ref>) and the fact that θ^(d)(x) = 𝒪(e^-d x/2) as x → -∞ imply that both integrals are finite. Sign considerations of both θ^(d)(·) and  u_0(·) show that the right-hand 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. 

An intermediate step is necessary in order to go back to the operator ℒ_θ^(d): first, define the family of weighted Sobolev spaces H_(δ,δ)^2() =  e^δ<x>H^2(), where <x> := √(1 + x^2). The action of the operator ℒ_1 on these spaces can be studied  by the operators ℒ_1^(δ)[·] = e^-δ<x>ℒ_1[e^δ<x>·], defined as H^2() → L^2() mappings; we point out that the mapping δ↦ℒ_1^(δ) is continuous in the operator norm. Standard Fourier analysis shows that the far field operators ℒ_1^(δ;±∞) = lim_x→±∞ℒ_1^(δ) are boundedly invertible operators from  H^2() to L^2() for any |δ|≤d/2, hence  for any δ in this range the operators ℒ_1^(δ) are Fredholm (cf. <cit.>). Continuity in δ implies that these operators also have index 0. Therefore, proving invertibility is equivalent to showing that the kernel is trivial. In that regard, observe that we have a  scale of Banach spaces, i.e., H_(δ',δ')^2() ⊂ H_(δ,δ)^2() whenever δ' > δ. Hence, one can invoke <cit.> or <cit.> to derive the persistence of elements in the kernel, namely, the equality Ker(ℒ_1^(δ)) = Ker(ℒ_1) ={0} holds for any |δ|≤d/2; being Fredholm operators of index 0 this property is equivalent to their invertibility.

We finally go back to the family of operators ℒ_θ^(d)[·]. Clearly,  e^d/2xH^2()  ⊂ H_(d,d)^2() = 𝒟(Ker(ℒ_1^(δ))|_δ = d/2). Since the latter set is trivial, the same is also true of the kernel of ℒ_1 taken with domain e^d/2xH^2(), 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.



[of Prop. <ref>]
Combine the proofs of Lem. <ref>, <ref> and  <ref>.




§ TWO-DIMENSIONAL QUENCHED PATTERNS – PERIODIC HORIZONTAL INTERFACES: H-KAPPA PATTERNS, PI<KAPPA<INFTY

In this section we prove Theorem <ref> 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 (<ref>) to the stripe (x,y) ∈×[0,κ]; any solution U(·,·) in ×[0,κ]  is extended to the whole plane ^2 by successive reflections U(x,-y)=-U(x,y) and U(x,κ+y)=-U(κ-y). The method of proof  is similar to the one used in the  problem, although the  construction of subsolutions is more involved; we mostly follow the arguments in <cit.> by truncating ℋ_κ-problem (with parameter π <κ < ∞) to a strip 𝒮_(-M,L) := (-M,L)×(0,κ), with  Dirichlet boundary conditions:

  
    {[ Δ_x,y U +c∂_x U+  μ(x) U - U^3                            =0,               (x,y)∈𝒮_(-M,L) ,;                     U-h_(-M,L)                            =0,               (x,y)∈∂𝒮_(-M,L), ].

where h_(-M,L)(x,y) := θ_(-M,L)^(c)(x)·(y;κ), for θ_(-M,L)^(c)(·) solution to the 1-D problem  <ref> and (·, κ) given in (<ref>). We obtain a unique solution Ξ_(-M,L)^(c;κ)∈𝒞^(1,α)(𝒮_(-M,L);[0,1]) by an  iteration scheme


    {[ -  ΔΞ_n+1 - c∂_xΞ_n+1 + 5 Ξ_n+1 =  (5 +μ(x)) Ξ_n - Ξ_n^3;                          (Ξ_n+1 - h_(-M,L))|_∂𝒮_(-M,L)=0 ].

where Ξ_0(·) is chosen in the class 


    Ψ_ℋ_κ   := {Ξ∈𝒞^(1,α)(𝒮_(-M,L)) | (Ξ_0 - h_(-M,L))|_∂𝒮_(-M,L)≥ 0, Ξ(x)∈[0,1]  x,y }
         ∩{ΔΞ + μ(x)Ξ- Ξ^3 ≤ 0,   ΔΞ + μ(x)Ξ - Ξ^3 ≢0  }.


Throughout this section, we fix κ∈ (π, ∞) and suppress the dependence of Ξ and u̅ on κ for ease of notation. As in the previous section, proofs that are similar to those in <cit.> are only outlined and details are referred to that paper. 
 

[Existence and uniqueness; truncated _κ problem] Problem (<ref>) has a unique solution 
Ξ_(-M,L)^(c)(·,·) ∈𝒞^(1,α)([-M,L]× [0,κ]; [0,1]), ∀α∈ [0,1).
 The existence is obtained as in  <cit.> using the iterative scheme (<ref>); uniqueness follows as in <cit.> and integration by parts, as in Lem. (<ref>). The stated regularity is derived using classical results in elliptic theory, as shown in <cit.>.

 We define extension operators in order to compare solutions for different values of M,L, namely,  

    ℰ[Ξ_(-M,L)^(c)](x,y) = {[      Ξ_(-M,L)^(c)(x,y),         (x,y) ∈𝒮_(-M,L);                   (y) , (x,y) ∈ (-∞, -M)× [0,κ];                     0 ,  (x,y) ∈ (L, ∞)× [0,κ], ].
 
where (·) = (·;κ) is given in (<ref>). 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. <ref> and Prop. <ref> are obtained as  in <cit.>: 

[Comparison principles; ℋ_κ-problem] Let Ξ_(-M,L)^(c)(·,·) be the solution from Proposition <ref>.


  * (2D supersolutions) 
 If v satisfies, in the sense of distributions,

    Δ v  + c∂_x v +  μ(x) v- v^3 ≤ 0, (x,y)∈𝒮_(-M,L),   (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,      0 ≤ v ≤ 1,

then v ≥Ξ_(-M,L)^(c) in 𝒮_(-M,L). In particular, v ≥Ξ_(-M,L)^(c) in 𝒮_(-M,L) for any solution of 

    Δ v  + c∂_x v + μ(x) v- v^3 =0, (x,y)∈𝒮_(-M,L),   (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,      0 ≤ v ≤ 1.



  * (2D subsolutions) 
If v satisfies, in the sense of distributions,

    Δ v  + c∂_x v + μ(x) v- v^3 ≥ 0, (x,y)∈𝒮_(-M,L),   (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,      0 ≤ v ≤ 1,

then v ≤Ξ_(-M,L)^(c) in 𝒮_(-M,L). In particular, v ≤Ξ_(-M,L)^(c) in 𝒮_(-M,L) for any solution of 

    Δ v  + c∂_x v +  μ(x) v- v^3 =0, (x,y)∈𝒮_(-M,L),   (v - h_(-M,L))|_∂𝒮_(-M,L)≤ 0,      0 ≤ v ≤ 1.



  * 
The functions u̅(·;κ) and θ_(-M,L)^(c)(x) given by (<ref>) and Theorem <ref>, respectively, are supersolutions to (<ref>) in 𝒮_(-M,L). Consequently, for any L, M >0 we have that 
 
 
    Ξ_(-M,L)^(c)(x,y)≤min{θ_(-M,L)^(c)(x), u̅(y)}.

 


[Properties of the extension operator, ℋ_κ-problem] The following properties of  ℰ[·] hold.


  * (Monotonicity of ℰ)
  We have 0 ≤ℰ[Ξ_(-M,L)^(c)](·, ·) ≤ 1. Furthermore, if w is only defined in a subset A ⊂ℝ^2 so that  𝒮_(-M,L)⊂ A ⊂Ω_(-∞,L), 0 ≤ w(·) ≤ 1 and w(·) ≤Ξ_(-M,L)^(c)(·, ·),  then 
0 ≤ℰ[w](·) ≤ℰ[Ξ_(-M,L)^(c)(·, ·)](·)      ℝ^2.                                                                                                                                                                                                                                        
 
  * (Monotonicity in M)
Let  0 ≤ M <M and L ≥ 0 be fixed. Then M <Mℰ[Ξ_(-M,L)](x,y) ≤ℰ[Ξ_(-M,L)^(c)](x,y).

 
  * (Monotonicity in L) 
Let  0 ≤ L <L and M ≥ 0 be fixed. Then L <  Lℰ[Ξ_(-M,L)^(c)](x,y) ≤ℰ[Ξ_(-M,L)](x,y).

 
  * (Monotonicity in x)
Let  L,M,ybe fixed. Then
the mapping  x →ℰ[Ξ_(-M,L)^(c)](x,y) is non-increasing. 





 §.§ Passing to the limit


We are now ready to pass to the limit M=∞. Define 

    Ξ_(-∞,L)^(c)(x,y) := inf_M >0 ℰ[Ξ_(-M,L)^(c)](x,y) =lim_M → +∞ℰ[Ξ_(-M,L)^(c)](x,y),

where the last equality holds due to monotonicity of the mapping M ↦Ξ_(-M,L)^(c)(x,y), Proposition <ref>(ii).


In this section we verify monotonicity properties and limits at spatial infinity of the limits Ξ_(-∞,L)(x,y) constructed in (<ref>).
 

 [Monotonicity of Ξ_(-∞,L)(x,y)]
 The following properties hold:
   

  * The function Ξ_(-∞,L)^(c)(·,·) solves the problem (<ref>) in  𝒮_(-∞,L):= {(x,y) ∈ℝ^2 | x <L, y ∈ [0,κ] }. 
 
  * The function Ξ_(-∞,L)^(c)(·,·)  is non-increasing in x and non-decreasing in L. 
 
  *  The inequality Ξ_(-∞,L)(x,y)≤min{θ_(-∞,L)^(c)(x), u̅(y)}
 
 holds for all (x,y) ∈𝒮_(-M,L), where  θ_(-∞,L)^(c)(·) is given by (<ref>) and u̅(·) by Prop. <ref>.  In particular, we have  sup_L>0(Ξ_(-∞,L)(x,y) )≤min{θ^(c)(x), θ_(-∞,0)^(c)(y)}.
 

 

 

Assertion (i) and (ii) follow as in <cit.>, being a consequence of a comparison method and monotonicity  of Ξ_(-M,L)^(c)(·) and Ξ_(-∞,L)^(c)(·) in their arguments. We derive the inequalities in (iii) by passing to the limit L=∞ and using the fact that for all x∈ the mapping L ↦θ_(-∞,L)^(c)(x) is non-decreasing. 
 

 
  One can readily conclude from Lem. <ref> that Ξ_(-∞,L)^(c)≡ 0 whenever c >2, L>0. In fact, the non-existence of nontrivial solutions happens in a wider range for the  parameter c, as we show next.
 


 



 

 §.§ Existence for the H-problem: case c2/4 + pi2/kappa2 <1 


 In order to prove the existence of solutions we construct appropriate subsolutions with the help of the next lemmas:
[Properties of the family of periodic solutions u̅(·,κ)] Let u̅(·;κ) be a solution to (<ref>) and κ > π.  The following two properties hold:



  * 	The quantity M := sup_y∈ [0,κ]u̅(y;κ) satisfies M < √((1 - π^2/κ^2));
 
  *  For any 0≤α≤  M we have  v(y):= αsin(π y/κ) ≤u̅(y;κ), for all y ∈ [0,κ].




To prove the estimate in (i) we use the elliptic integral that gives the relation between amplitude and spatial period given in <cit.>[Lemma 4.1, equation (4.4)], <cit.>, 

    κ := κ(M) = 2 √(2)∫_0^1 dv/√([(1 - v^2)(2 - M^2(1 + v^2))]) = 2 √(2)γ/M∫_0^1 dv/√([(1 - v^2)(1 - (γ v)^2)]),

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:

    κ(M) >  2 √(2)γ/M∫_0^1 dv/√([(1 - v^2)(1 - γ^2)]) =  2 √(2)γ/M√(1 - γ^2)∫_0^1 dv/√((1 - v^2))=  π√(2)γ/M√(1 - γ^2).

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 (<ref>), whose Hamiltonian is ℋ(u, u_y) = u_y^2 + u^2 - u^4/2,  cf. <cit.>. Indeed, considering u̅(·) a periodic orbit with period κ and maximal amplitude M, we readily obtain that ℋ(u̅, u̅_y) =   M^2(2 - M^2)/2.
Let  v(y) := αsin(π  y/κ) and z(x) := u̅(y) - v(y). Whenever 0 ≤α≤ M one can see that 


    ℋ(v, v_y)≤ℋ(u̅, u̅_y).


As u̅_y(0) >v_y(0) and (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 (<ref>),  reversibility of the solutions u̅ 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 


    ∂_y^2z(y) + (1 - f[u̅,v](y) - A)z(y) ≤∂_y^2z(y) + (1 - f[u̅,v](y) )z(y)=0,


where f[a,b]:= a^3 - b^3/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 (<ref>) prevents it from happening. Therefore z(y) ≥ 0, hence v(y) ≤u̅(y) for  y ∈[0,κ/2], and by symmetry, for y ∈[0,κ]. 

[Sub and supersolutions] Choose d ∈ (c,2)  so that c^2/4 + π^2/κ^2 < d^2/4< 1 and let 
w(·)∈𝒞^∞(;[-1,1]) be a solution to  ∂_x^2w + d∂_xw + w - w^3=0  satisfying w(-∞) =1 and so that 0 = w(0) < w(x) <1 for x <0 (see. Prop. <ref> and Fig. <ref>). Let M>0, α and u̅(·) be given as in Lem. <ref>. Define 


    V(x,y) := e^-(c-d)  x/2w(x)· v_κ(y),        v_κ(y) := αsin(π  y /κ).


 Then, 
 

    V(x,y) ≤Ξ_(-∞,L)^(c)(x,y)       [-M,0]×[0,κ].


Furthermore, whenever  θ^(c)(·) is given by (<ref>) and u̅(·) by Prop. <ref>, the following inequality holds

    Ξ_(-∞,L)^(c)(x,y)≤min{θ^(c)(x), u̅(y)}.
  


 Inequality (<ref>) is readily derived  from Lem. <ref>, taking infimum in M>0, using the definition of Ξ_(-∞,L)^(c)(·,·) (see (<ref>)) and  the monotonicity of the mapping  L ↦θ_(-∞,L)^(c)(·).

In order to obtain (<ref>) we use Lem. <ref>, showing that V(·,·)≤Ξ_(-M,0)^(c)(·,·) in  𝒮_(-M,0) = [-M,0]×[0,κ] for any fixed M>0. Thanks to Lemma <ref> it is straightforward to show that   V(x,y) ≤Ξ_(-M,0)^(c)(x,y) on the boundary ∂𝒮_(-M,0). Now we show that V satisfies


    Δ_x,yV + c∂_x V + V - V^3 ≥ 0,

in 𝒮_(-M,0) = [-M,0]×[0,κ]. Indeed, a direct calculation shows that 
 
 
    Δ_x,yV + c∂_x V + V - V^3    = V[d^2/4 - c^2/4  - π^2/κ^2] + e^-(c-d) x/2w^3(x)[v_κ(y) - e^-(c-d) xv_κ^3(y)]  ≥ 0,


since V(·,·) is non-negative, x≤ 0 and d >c. It follows that V(x,y) ≤Ξ_(-M,0)^(c)(x,y).  We obtain (<ref>) by invoking the  monotonicity of the mapping L ↦Ξ_(-M,L)^(c)(·,·) and using the definition (<ref>). 



[Existence; ℋ_κ-problem, π <κ <∞]
 Equation (<ref>) has  a solution Ξ^(c)(·,·)∈𝒞^(1,α)(× [0,κ];), for any 0≤α<1, where the latter is defined as
 
 
    Ξ^(c)(x,y) := lim_L→∞Ξ_(-∞,L)^(c)(x,y)





Most of the proof goes as in the paper <cit.>. The monotonicity properties of the functions Ξ^(c)_(-∞,L)(·,·) show that the definition (<ref>) makes sense and Lem. <ref> shows that Ξ_(-∞,L)^(c)(·,·)≢0 whenever c^2/4 +π^2/κ^2<1. As 0≤Ξ_(-∞,L)^(c)(·,·) ≤Ξ_κ^(c)(·,·) we conclude that Ξ_κ^(c)(·,·) is also nontrivial. Using Lebesgue dominated convergence we conclude that 

exists in the pointwise sense and that this sequence converges in L_loc^1 hence in the sense of distribution, solving the equation (<ref>) in the domain × [0,κ]. Now it remains to show that the asymptotic limits are satisfied, namely, that 


    lim_x → -∞Ξ_κ^(c)(x,y) = u̅(y),     lim_x → +∞Ξ_κ^(c)(x,y) = 0.

The limit on the right follows easily from inequality (<ref>), for lim_x→∞θ^(c)(x) =0.
The proof of the limit on the left is more involved, and our analysis has some similarities to those of <cit.> and <cit.>. 

Indeed, monotonicity results derived in Lem. <ref> allow us to conclude that 


    v_L(y) := lim_x→∞Ξ_(-∞,L)^(c)(x,y)≤lim inf_x→∞Ξ_κ^(c)(x,y)≤lim sup_x→∞Ξ_κ^(c)(x,y)≤u̅(y),


where the first limit is known to exists thanks to the monotonicity in x of Ξ_(-∞,L)^(c)(·, ·). According to Lem. <ref>  we know that Ξ_(-∞,L)^(c)(·, ·) solves the equation (<ref>). Thus, v_L(·)   satisfies 

    ∂_y^2v_L(y) + v_L(y) - (v_L(y))^3=0

in the sense of distributions in [0,κ], hence in the classical sense.  As v_L(y)|_y = 0, κ =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. <ref> implies that  αsin(π  y/κ)≤ v_L(y)≤(y). The same inequality also implies that τ = 2κ, i.e., u̅(·) and v_L(·) have the same period therefore and obey the same normalization, therefore v_L(·) ≡u̅(·), and the result follows from the equality of  (<ref>).


Unlike in the previous case, it is not directly clear that the convergence lim_x→ -∞Ξ_κ^(c)(x,y) = u̅(y) has exponential rate of convergence. Our next result implies that.


[Exponential convergence; ℋ_κ-problem, π <κ <∞] There exists a C, δ >0 independent of x and y such that

    lim_x→∞|Ξ_∞^(c)(x,y)| ≤ Ce^-δ |x|,lim_x→ -∞|Ξ_∞^(c)(x,y) - u̅(x)|≤ Ce^-δ |x|.


 
Initially we show exponential rate of convergence to the far-field as |x|→∞. The result follows if we show that ∂_x(Ξ_∞^(c))∈ e^-δ<x>H^2(×[0,κ]) for some δ >0 and <x> :=√(1 +x^2). Indeed, as we know from Lem. <ref>, lim_x→ -∞Ξ_∞^(c)(x,y) = u̅(y); using the Sobolev embedding H^2(×[0,κ])↪  L^∞(×[0,κ]) we have


    |Ξ_∞^(c)(x,y) - u̅(y)| ≤|∫_∞^x ∂_x(Ξ_∞^(c))(s,y)ds| ≲∫_∞^x e^δ sds ≲ e^δ x,      x≤ 0,


which gives  the result. The proof requires several tools of Fredholm theory for elliptic operators. The linearization of the equations (<ref>) at Ξ_∞^(c)(·,·) gives


    ℒ_Ξ[v] :=  Δ v + c∂_x v + [μ(x)  - 3 (Ξ_∞^(c)(x,y))^2]v,

 with domain of definition 𝒟(ℒ_Ξ) = H^2(ℝ× [0,κ]) ∩ H_0^1(ℝ× [0,κ]). 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 <cit.> apply, showing that the operator ℒ_Ξ is Fredholm of index 0, with essential spectrum strictly negative in H^2(ℝ× [0,κ]) ∩ H_0^1(ℝ× [0,κ]). According to <cit.>, we know that elements in the kernel of ℒ_Ξ are spatially exponentially localized, namely, 


    |∇ u_0(x,y)|+|u_0(x,y )| ≤  C e^- δ|x| ,    (x, y)  ∈ℝ× [0,κ]    a.e.       u_0 ∈Ker(ℒ_Ξ)


 Recall the partition of unity χ^±(·) defined in (<ref>). We know that ∂_x(χ^-(x)Ξ^(c)(x,y)) ∈𝒞^∞(×[0,κ];) solves a problem of the form 
ℒ_Ξ[v] = f,
where f∈ L^2(×[0,κ) is spatially localized. Writing H^2(×[0,κ]) = Ker(ℒ_Ξ) ⊕𝒳, we can assume with no loss of generality that v∈𝒳, thanks to property (<ref>) 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 <cit.> to conclude that 
e^δ<x>v∈ H^2(×[0,κ]) ∩ H_0^1(×[0,κ]) 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. <ref> that the operator ℒ_Ξ in (<ref>) is boundedly invertible from H^2(×[0,κ]) to  L^2(×[0,κ]). Once more, using the IFT, we conclude the following result. 
 
[Uniqueness of the  continuation in c;  ℋ_κ problem, π < κ < ∞] 
Recall the definition of 𝒫(c;κ) given in (<ref>). For any fixed κ∈ (π, ∞) the following properties hold:

 
  * there exists a unique solution Ξ_κ^(c)(·,·) to the ℋ_κ^(c) problem;
 
  * the mapping   c↦Ξ_κ^(c)(·):{c≥ 0|  𝒫(c;κ) <1}→ L^∞(×[0,κ];) is continuous.


 The analysis is analogous to that of Lem. <ref> and is outline below, where we point out the necessary modifications. Fix κ∈ (π, ∞). Initially we define the linearized operator about the solutions Ξ_κ^c, obtaining the linearized operator

    ℒ_Ξ^(c)[v] = ∂_x^2 v + c∂_x v + μ(x) v  - 3(Ξ^(c))^2 v ,   𝒟(ℒ_θ^(c)) = H_2(×[0,κ])∩ H_0^1(×[0,κ]).


Writing v = e^-c x/2 u we rewrite the above operator in a “self-adjoint” form,


    ℒ_Ξ^(c)[v] = ∂_x^2 u + (μ(x) - c^2/4) u  - 3(Ξ^(c))^2 u ,   𝒟(ℒ_θ^(c)) = e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])).

Notice that the mapping u(·) ↦ e^c x/2u(·) is an isometry between the spaces H_2(×[0,κ])∩ H_0^1(×[0,κ]) and e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])), therefore it suffices to  study the invertibility of ℒ_Ξ^(c) only. At this point we define ℒ_Ξ^(c)^(δ) as the action of the operator ℒ_Ξ^(c) on the scale of Banach spaces e^δ<x>H^2(×[0,κ]). Continuity of the Fredholm index on the weight δ shows in the range |δ|≤c/2 we have that ℒ_Ξ^(c)^(δ) are Fredholm operators with index zero, with the same kernel. As ℒ_Ξ^(c)^(δ)|_δ =0 is invertible one can argue as in Lem. <ref> to obtain the bounded invertibility of ℒ_Ξ^(c) in e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])). The results are then obtained from an application of the IFT.






 §.§ Non-existence for the H-kappa-problem: case c2/4 + pi2/kappa2 >1 


In this section we prove the non-existence of patterns when c^2/4 +π^2/κ^2>1. The method is standard: 
roughly speaking, assuming an existing solution Ξ_κ^(c)(·,·), we can obtain a supersolution V(·,·) that is above Ξ_κ^(c)(·,·) and which, under certain conditions, can touch the solution in at least one point. The function z(·,·):= Ξ_κ^(c)(·,·) - V(·,·) solves an elliptic problem and has an interior maximum, which contradicts maximum principle and Hopf's Lemma. We make these words more precise in what follows. We begin with the main ingredients in the construction of the subsolution.


[Non-existence; c^2/4 + π^2/κ^2>1]
 Recall (see Fig. <ref>) the existence of a solution w_d(·)>0 to the problem 


    ∂_x^2w_d + d∂_xw_d + w_d - (w_d)^3,     lim_x→ -∞w_d(x) = 1,   lim_x→∞w_d(x) = 0,


where d>max{2,c} is chosen in such a way that  d^2/4 - c^2/4 - π^2/κ^2<0. Likewise, we write  w_c(·) for a function with similar spatial asymptotic properties and solving  ∂_x^2w_c + c∂_xw_c + w_c - (w_c)^3=0. Define 


    V(x,y) := w_c(x) + e^-(c-d) x/2w_d(x)v_κ(y),        v_κ(y) := αsin(π  y /κ).


Then V(·,·) is a supersolution. Furthermore, V(·,·)≤Ξ_κ^(c)(·,·). 


Recall that 𝒫(c,κ) := c^2/4 + π^2/κ^2 >1. The first step on our proof is the construction of a supersolution V ≥Ξ_κ^(c). 
 Initially, we fix w_c(·) in such a way that w_c(x) ≥ 0 on x≤ 0. Now, choosing d>max{2,c} in such a way that  d^2/4 - c^2/4 - π^2/κ^2<0 we fix a translated version of w_d in such a way that 


    d^2/4 - c^2/4 - π^2/κ^2 + w_d^2(x) - 3 w_c^2(x) ≤ 0,      x ≤ 0.


 In fact, we can exploit the translation invariance of solutions to the ODE (<ref>) and the fact that x↦ w_{c,d}(x) is non-increasing for  x≤ 0 to conclude that  (<ref>) also holds for  translated version of w_d(·) and w_c(·) of the form w_d(·) ↦ w_d(· - τ) and w_c(·) ↦ w_c(· - τ), whenever τ≥ 0. 
 
 Set V(x,y) = e^-(c-d) x/2w^(d)(x)v_κ(y) and write V(x,y) = w_c(x) + V(x,y). We claim that V is a supersolution on x≤ 0 (where μ(x) ≡ 1). Indeed, a tedious but straightforward computation shows that


    Δ_x,y  V + c∂_xV + V - V^3    =  V[d^2/4 - c^2/4 - π^2/κ^2] + e^-(c-d) x/2w_d^3(x)v_κ(y) + w_c^3 - [w_c(x) + e^-(c-d) x/2w_d(x)v_κ(y)]^3
        =   V[d^2/4 - c^2/4 - π^2/κ^2 +  w_d^2(x) - 3 w_c^3(x)] + J_2


where J_2= J_2(w_c, w_d, v_κ) consists of non-positive terms only. Thus,


    Δ_x,y  V + c∂_xV + μ(x)V - V^3    =   V[d^2/4 - c^2/4 - π^2/κ^2 +  w_d^2 - 3 w_c^3] + J_2 ≤ 0,      x≤ 0,


thanks to (<ref>), which concludes the proof that V is a supersolution in x≤ 0.

Our second step consists in proving that V ≥Ξ^(c). Unfortunately, we cannot play with the translation of both w_d and w_c separately without destroying the inequality in (<ref>), so we pursue another route. The asymptotic properties of w_c imply that V(x,y) ≥ w_c(x) ≥Ξ^(c)(x,y) as x → -∞. As w_c(·) can be chosen so that w_c(0) = 0 we need to understand the term V(x,y) for x <0, x finite. Inspecting the calculations that lead to (<ref>), one can see that the parameter α≥ 0 in  v_κ(y):= αsin(π y/κ) plays no crucial role. Standard elliptic regularity estimates show that ∂_yΞ^(c)(·,·) is a  bounded function on ×[0,κ]. Consequently, we can take α>0  sufficiently large so that 


    w_d(0) v_κ(y) = α w_d(0) sin(π  y/κ)≥Ξ^(c)(0,y);


thanks to  w_d(·) >0. Furthermore, the inequality (<ref>) persists  when we use a translated version of w_d(·), for x ↦  w_d(x) is non-increasing, namely, w_d(-τ) v_κ(y) ≥ w_d(0) v_κ(y)≥Ξ^(c)(0,y), for all τ >0. Thus, for a fixed M̃>0 we have that 


    V(x,y) ≥Ξ^(c)(x,y),     x∈ [-M̃,0]× [0,κ].


Translating w_c(·) in such a way that w_c(x) ≥Ξ^(c)(x,y) for x ≤ -M̃ and y ∈ [0,κ] we get that V(x,y)≥Ξ^(c)(x,y) for all (x,y)∈ (-∞,0]×[0,κ]. The third and last part consists of varying the parameter   α >0 continuously in order to satisfy V ≥Ξ^(c) with an equality in at least one point. Now the proof goes as in Prop. (<ref>): define Z(x,y) := Ξ(x,y) - V̅(x,y); clearly Z(x,y)≤ 0 in x≤ 0, y ∈ [0,κ].   The function Z satisfies


    Δ_x,y Z + c∂_x Z + μ(x)Z - f[Ξ,V̅](Z) ≥ 0,      x≤ 0.


where f[a,b]:= a^3 - b^3/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,V̅] >0; thus we write the above equation as 


    Δ_x,y Z + c∂_x Z + μ(x)Z - (f[Ξ,V̅] + M)(Z)≥ - M Z ≥ 0


As Z has a maximum point in ×[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:
[of Theorem <ref>; case π <κ < ∞] Combine the above discussion of the non-existence of solutions with the results of Lemmas <ref>, <ref>, and  <ref>






§ TWO-DIMENSIONAL QUENCHED PATTERNS – SINGLE HORIZONTAL INTERFACES; H INFTY PROBLEM


In this section, we shall prove Theorem <ref> in the case κ =∞. To be consistent with the notation introduced in Section <ref> we exploit the fact that the nonlinearity in (<ref>) is odd to solve the problem in the half space × (-∞,0]. Further symmetries of the equation are also exploited: we solve an equivalent ℋ_∞-problem, seeking for a solution  Ξ_∞^(c)(·,·) to (<ref>) in ^2, satisfying

    lim_x→ -∞Ξ_∞^(c)(x,y)= -tanh(y/√(2)),  lim_y→±∞Ξ_∞^(c)(x,y)=∓θ^(c)(x),    lim_x→∞Ξ_∞^(c)(x,y)=0,

where θ^(c)(·) is the one-dimensional solution to the -problem. Notice that the odd symmetry solutions to (<ref>) with respect to  (y, Ξ_∞^(c)(x,y))↦(-y,-Ξ_∞^(c)(x,-y)) readily gives the pattern with the properties stated in Theorem <ref>. Moreover, the Dirichlet boundary conditions at y=0 removes the non-uniqueness of solutions induced by y-translation invariance. 

 Thanks to the results of Sec. <ref> related to the 1-d problem the following observation is readily available. 
[Restriction to the case c<2]
 It is clear from (<ref>) 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 Theorem<ref>. The strategy goes as in Sections <ref> and  <ref>: 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 


    {[ Δ_x,y u +c∂_xu + μ(x) u - u^3                             =                            0,               (x,y)∈Ω_(-M,L),;                             u                             =                     g_(-M,L),              (x,y)∈∂Ω_(-M,L), ].


where g_(-M,L)(x,y):=θ_(-M,L)^(c)(x)·θ_(-M,0)^(c)(y),   θ_(-M,L)^(c)(·) the solutions to the truncated one-dimensional problem  (<ref>) on the interval (-M, L). Similarly to the results of Lem. <ref> and <cit.>, we construct unique solutions to this truncated problem using an iterative scheme. The solution Θ_(-M,L)^(c)(·,·): Ω_(-M,L)→ [0,1] is shown to be unique; Furthermore, exploiting that 0≤ u≤ 1 and  Agmon-Douglis-Nirenberg regularity we readily conclude that u and derivatives are Hölder continuous across x=0, and, in fact, Θ_(-M,L)^(c)∈𝒞^(1,α)(Ω_(-M,L)),  for all 0≤α <1.

Following the method in Sec. <ref>, we extend these functions to the whole plane ^2:


    ℰ[Θ_(-M,L)^(c)](x,y) = {[                     Θ_(-M,L)^(c)(x,y),                       (x,y) ∈Ω_(-M,L),; ℰ[θ_(-M,L)^(c)](x)·ℰ[θ_(-M,0)^(c)](y),                  (x,y) ∈ℝ^2 ∖Ω_(-M,L). ].
 
                          
We summarize the main properties of the functions ℰ[Θ_(-M,L)^(c)](·,·) in the following Proposition, whose proof is similar to that of <cit.>:
[Properties of the extension operator, ℋ_∞-problem] The following properties of  ℰ[·] hold.


  * (Monotonicity of ℰ)
  We have 0 ≤ℰ[Θ_(-M,L)^(c)](·, ·) ≤ 1. Furthermore, if w is only defined in a subset A ⊂ℝ^2 so that  Ω_(-M,L)⊂ A ⊂Ω_(-∞,L), 0 ≤ w(·) ≤ 1 and w(·) ≤Θ_(-M,L)^(c)(·, ·),  then 
0 ≤ℰ[w](·) ≤ℰ[Θ_(-M,L)^(c)(·, ·)](·)      ℝ^2.                                                                                                                                                                                                                                        
 
  * (Monotonicity in M)
Let  0 ≤ M <M and L ≥ 0 be fixed. Then M <Mℰ[U_(-M,L)](x,y) ≤ℰ[Θ_(-M,L)^(c)](x,y).

 
  * (Monotonicity in L) 
Let  0 ≤ L <L and M ≥ 0 be fixed. Then L <  Lℰ[Θ_(-M,L)^(c)](x,y) ≤ℰ[U_(-M,L)](x,y).

 
  * (Monotonicity in x)
Let  L,M,ybe fixed. Then
the mapping  x →ℰ[Θ_(-M,L)^(c)](x,y) is non-increasing. 


  * (Monotonicity in y)
Let  L,M,xbe fixed. Then
the mapping  y →ℰ[Θ_(-M,L)^(c)](x,y) is non-increasing. 





[Existence; ℋ_∞ problem] There exists a solution Ξ_∞^(c)(·, ·) satisfying  (<ref>) in ^2 satisfying the limits (<ref>). Furthermore, ∂_y Ξ_∞^(c)(·, ·) ≤ 0. 


We obtain a solution in (x,y) ∈×(-∞,0), extending it to the whole plane ^2 as a function satisfying Ξ^(c)(x,-y) = -Ξ^(c)(x,y). The result follows the construction in Lem. <ref>, first defining Θ_(-∞,L)(·,·) :=inf_M>0Θ_(-∞,L)(·,·) and then defining  
 
 
    Ξ_∞^(c)(·,·) = sup_L>0Θ_(-∞,L)(·,·).

 Further properties of Ξ_∞^(c)(·, ·) are derived as in Lem. <ref> (see also <cit.>). The inequality ∂_y Ξ_∞^(c)(·, ·) ≤ 0 is a consequence of  Prop. <ref>, because the mapping y ↦Ξ_∞^(c)(x, y) is non-increasing.  


The next result provides a crucial ingredient in the construction of interfaces with contact angle:
[Monotonicity in the y-direction]
 For all x∈ the mapping y ↦Ξ_∞^(c)(x,y) is strictly monotonic. 


 It suffices to show that ∂_yΞ_∞^(c)(x,·)≠ 0 a.e., thanks to the regularity of Ξ_∞^(c)(·,·). The inequality ∂_yΞ_∞^(c)≤ 0 is obtained without difficulty by making use of Prop. <ref> and the limiting construction in Lem. <ref>. The strict inequality ∂_yΞ_∞^(c) < 0 a.e. then follows using a Harnack inequality (cf. <cit.>; see also <cit.>).  


[Exponential convergence; ℋ_∞ problem] The limits in (<ref>) take place at exponential rate, i.e., there exists a C, δ >0 independent of x and y such that

    lim_x→∞|Ξ_∞^(c)(x,y)| ≤ Ce^-δ |x|,   lim_x→∞|Ξ_∞^(c)(x,y) - tanh(y/√(2))| ≤ Ce^-δ |x| ,lim_y→±∞|Ξ_∞^(c)(x,y) ±θ^(c)(x)|≤ Ce^-δ |y|.



 The proof is similar to that of Lem. <ref>, with small differences. As before, we first trace the essential spectrum of  the linearized operator (<ref>) at Ξ_∞^(c)(·,·) gives


    ℒ_Ξ[v] :=  Δ v + c∂_x v + [μ(x)  - 3 (Ξ_∞^(c)(x,y))^2]v,

 with domain of definition 𝒟(ℒ_Ξ) = H_odd^2(ℝ^2):= {w ∈ H^2(ℝ^2); w(x,y) = -w(x,-y)}. We remark that <cit.> still applies: indeed, we can define the asymptotic operators
 
   
    ℳ^-[v] := ∂_y^2v(·)  + [1- 3tanh^2(y/√(2))]v(·),      ℳ^+[v] :=  ∂_y^2v(·)  -v(·),

with domain  𝒟(ℳ^±) =  H_odd^2([0,κ])∩ H_0^1([0,κ]). We claim that these operators are invertible: indeed,  coercivity implies that Ker(ℳ^-) = {0}; The same holds in the case of ℳ^+, for in H^2 its kernel is given by  {∂_ytanh(·/√(2))} which is a simple eigenfunction, since it has no nodal points (see <cit.>).  Therefore, in the space of odd functions, we have that both operators ℳ^± are invertible.  In our next step, we argue as in <cit.>, describing limiting operators associated with ℒ_Ξ[·]:



    ℒ_Ξ^(x→ +∞)[v]    = (∂_x^2 + c∂_x + ℳ^+)[v], 
    ℒ_Ξ^(x→ -∞)[v]    = (∂_x^2 + c∂_x + ℳ^-)[v],
    ℒ_Ξ^(y→ +∞)[v]   = (Δ_x,y + c∂_x + μ(x) - 3(θ^(c))^2)[v].



Fourier transforming the operators (<ref>)-(<ref>) in x, and (<ref>) in y, shows that these operators are boundedly invertible. Observe that v(x,y) = ∂_x(χ^-(x)Ξ_∞^(c)(x,y)) (resp., v(x,y) = ∂_x(χ^+(x)Ξ_∞^(c)(x,y))) satisfies  ℒ_Ξ^(x→ -∞)[v] = f (resp., ℒ_Ξ^(x→ +∞)[v] = f) where the left hand side is spatially localized, it follows from the same reasoning as in Lem. <ref> that e^δ<x>v ∈ H^2(^2)∩ H_0^1(^2). A similar result holds for 
 v = ∂_y (χ^+(y)Ξ), which solves  ℒ_Ξ^(y→ +∞)[v] = f for f spatially localized in y, i.e., f∈ e^δ <y>L^2(^2), where <y> = √(1 +y^2). The rest of the analysis is similar to that used in the proof of Lem. <ref>.


[Uniqueness up to translation in the y-direction] Whenever c≥ 0 the solutions constructed in Theorem <ref> and the solutions constructed in <cit.> by continuation are the same up to translation in the y-direction.

 The machinery given in <cit.> can be used  to derive a simple proof: first notice that  ∂_y Ξ_∞∈ Kerℒ_Ξ_∞[·]. As ∂_yΞ has a sign one can use <cit.> to conclude that ∂_yΞ^∞=C∂_yΘ 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.


 The mapping c↦Ξ_∞^(c)(·):[0,2) → L^∞(^2;) is continuous.


As before, we exploit the symmetries of the problem to reduce the analysis to the half-space ×(-∞,0]. A similar analysis to that of Lem. <ref> shows that the linearization of the equation (<ref>) at Ξ_∞^(c) given in (<ref>) is an invertible operator from H^2(×(-∞,0])∩ H_0^1(×(-∞,0]) (i.e., restricted to odd solutions in y) to L^2(×(-∞,0]). However, the result does not readily follows, because


    Ξ_∞^(c) - Ξ_∞^(d)∉H^2(×(-∞,0])∩ H_0^1(×(-∞,0]).


 We circumvent this issue with a far-field core decomposition, which we now explain: for a fixed d∈[0,2),  we use an Ansatz of the form  w(x,y)+ Ξ_∞^(c) + χ^-(y) [θ^(d)(x) - θ^(c)(x)], where w ∈ H^2(×(-∞,0];); notice that Ξ_∞^(c) + χ^-(y) [θ^(d)(x) - θ^(c)(x)] converges exponentially fast to its far-field and asymptotically solves the PDE (<ref>). We rewrite problem (<ref>) as
 
 
    ℒ_Ξ_∞[w] : = Δ_x,y w + c∂_x w + μ(x) w - 3(Ξ_∞^(c))^2 w = 𝒩(w, Ξ_∞^(c),θ^(d)(x), θ^(c)(x) )

 
A reasoning similar to that in the proof of Lem. <ref> shows that ℒ_Ξ_∞ is a bounded, invertible operator from H^2(×(-∞,0])∩ H_0^1(×(-∞,0]) to L^2(×(-∞,0]). Furthermore, the right hand side is in L^2(×(-∞,0]), thanks to the Sobolev embedding H^2(^2) ↪ L^∞(^2). The conclusion then follows from the IFT.




 [of Theorem <ref>; case κ = ∞] Combine the results of Lemmas <ref>, <ref>, <ref>, and Prop. <ref>.
   



§ 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  <cit.>, 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 <cit.> for a general discussion about wavenumber selection.




  
Critical cases; S=1. The behavior of the patterns in this critical scenario possibly requires a different approach, since one can see in the proofs of Theorem <ref> and <ref> 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 _κ-patterns? Although it was shown in <cit.> that oblique patterns do not exist in (<ref>), these patterns can still exist in the case of the unbalanced equation (<ref>). It is worth to mention that  the result of <cit.> 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   <cit.> for physics motivation and a more detailed discussion on the chirality of helicoidal patterns in the context of recurrent precipitation. 





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