The shape of a free boundary driven by a line of fast diffusion

We complete the description, initiated in [6], of a free boundary travelling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dicatated by the velocity of the wave imposed by the line. This helps understanding some rather counter-intuitive numerical simulations of [8].

We ask for the global shape of the free boundary Γ. Before that, we ask about the existence of a solution (c, Γ, u) to (1.1), and of a solution (c, Γ, u, v) to (1.2), this indeed deserves some thought, as the condition at −∞ is rather strong. Systems (1.1) and (1.2) arise from a class of models proposed by H. Berestycki, L. Rossi and the second author to model the speed-up of biological invasions by lines of fast diffusion, see for instance [3] or [4]. The two-dimensional lower half-plane (that was called "'the field"' in the afore-mentionned references), in which reaction-diffusion phenomena occur, interacts with the x axis ("'the road") which has a much faster diffusion D of its own. In Model (1.2), u(x) the density of individuals on the road, and v(t, x, y) the density of individuals in the field. The road yields the fraction µu to the field, and retrieves the fraction νv in exchange; the converse process occurs for the field. Model (1.1) is obtained from (1.2) by forcing µu = v on the road, so that the sole unknown is v(x, y), and the exchange term is replaced by v y µ (and v has been renamed u). In the sequel, Model (1.2) will be called the "model with two species" (that is, the density on the road and in the field may be different), while Model (1.1) will be called "Model with one species". Also note that, in both models, the unknown functions are travelling waves of an evolution problem where the term c∂ x u (resp. c∂ x v) is replaced by ∂ t u (resp. ∂ t v). This is explained in more detail in [6], where the study of (1.1) and (1.2) was initiated.
-Γ ∩ {y = 0} is nonempty, assume (0, 0) ∈ Γ. There is λ > 0 and ε 0 > 0 such that The main question of this work, namely, how Γ looks like, is addressed in the following theorem. Both models with one species and model with two species are concerned. Let c 0 be the speed of the basic travelling wave φ 0 (x): (1.7) In what follows, we stress the dependence on D of the velocity, free boundary and solution of the PDE by denoting them c C , Γ D , u D , v D .
In other words, Γ D has an asymptotic direction, which is a line making the angle α D with the horizontal. We have a more precise version of Theorem 1.3: Thus there is a straight line making the angle α D with the horizontal that is asymptotic to Γ D at infinity, and the distance between the two shrinks exponentially fast.

Underlying question, comments, organisation of the paper
Let us be more specific about the question that we wish to explore here. We want to account for a loss of monotonicity phenomenon for u (Model (1.1)) or v (Model (1.2)) in the y variable, a study that was initiated in [6]. This phenomenon was discovered numerically by A.-C. Coulon in her PhD thesis [8]. There, she provided simuations for the evolution system (notice that the propagation takes place upwards): (1.10) The figures below account for some of her results; the parameters are The top figure represents the levels set 0.5 of v at times 10, 20, 30, 40, while the bottom figure represents the shape of v(40, x, y).
The surprise is the location of the leading edge of the invasion front: rather than being located on the road, as one would have expected (especially for large D), it sits a little further in the field. This entails a counter-intuitive loss of monotonicity. As working directly with the reaction-diffusion (1.10) has not resulted in a significant outcome so far, the idea was to replace the reaction-diffusion model by a free boundary problem, that may be seen as an extreme instance of reaction-diffusion. Using this idea, a first explanation of the location of the leading edge is provided in [6]. The conclusions are included in Theorems 1.1 and 1.2.
The goal of the paper is, as already announced, to account for the global shape of the free boundary. We claim that it will provide a good theoretical explanation of A.-C. Coulon's numerics. Indeed it may be expected (although it is not a totally trivial statement) that the solution of (1.10) will converge to a travelling wave. As the underlying situation is that of an invasion, it is reasonable to assume that no individuals (if we refer, as was the initial motivation, to a biological invasion) are present ahead of the front. This is why we impose the seemingly stringent, but reasonable from the modelling point of view, condition at −∞. Theorem 1.3 shows that it entails a very specific behaviour. Let us explain the consequences of our results on the understanding of the model. Theorems 1.1 to 1.3 put together depict a free boundary whose leading edge is in the lower half plane and which, after a possibly nonempty but finite set of turns, becomes asymptotic to a line that goes to the right of the lower half plane. This is in good qualitative agreement with the upper two-thirds of the picture presented here, the lower third accounting for the fact (still to be described in mathematically rigorous terms) that the free boundary bends in order to connect to a front propagating downwards, which is logical as we start from a solution that is compactly supported. Let us, however, point out that the analogy should not be pushed further than what is reasonable, as the logistic nonlinearity displays -and this is especially true for the model (1.10) -more counter-intuitive oddities of its own, see [5]. Some additional mathematical comments are in order. The first one is that we have put some effort in proving existence theorems. The reason is that we could not entirely rely on our previous study [6] for that: although the leading edge was analysed, we had chosen to wipe out the additional difficulties coming from the study in the whole lower half-plane, by studying a model in a strip of finite length with Neumann conditions at the bottom. While the study of the leading edge is purely local, and will not need any more development, the condtion lim x→−∞ u(x, y) = 1 (resp. lim x→−∞ u(x, y) = 1) uniformly in y ∈ R − requires some additional care, that is presented in Section 3. Theorem 1.3 is hardly incidental. It is in fact a general feature in reaction-diffusion equations in the plane. The heuristic explanation is the following: looking down very far in the lower half plane, we may think that the free boundary Γ D propagates like the 1D wave in its normal direction, that is, V n = c 0 . On the other hand, it propagates with speed c D horizontally: this imposes the angle α D . For a rigorous proof of that, we take the inspiration from previous works on conical-shaped waves for reaction-diffusion equations in the plane. A first systematic study may be found in [2], while the stability of these objects is studied in [11]. Further qualitative properties are derived in [12]. Solutions of the one phase free boundary problem for c∂ x u − ∆u = 0 are classified by Hamel and Monneau in [10]. One of their results will play an important role in the proof of Theorem 1.3, this will be explained in detail in Section 4. Although this work is clearly aimed at understanding the situation for large D (the case D ≤ d poses interesting technical questions in for the one species model) we have not really provided a systematic study of the limit D → +∞. This will be the object of a forthcoming paper. Another interesting question is whether the free boundary has turning points. While the simulations cleary point at convexity properties of the sub-level sets, we do not, at this stage, have real hints of what may be true. The paper is organised as follows. In Section 2, we provide some universal bounds on the velocities, in therms of the diffusion on the road D. In Section 3, we construct the wave for the one species model and prove Theorem 1.2. In Section 4, we indicate the necessary modifications for the two-species model. In Section 5, we prove the exponential convergence of the level sets.
2 The one species model: Bounds on the velocity in a truncated problem Solutions to (1.1) will be constructed through a suitable approximation in a finitely wide cylinder; we set Σ L = R × (−L, 0), from (a trivial modification of) [6], for L > 1, there is a solution (c L D , Γ L D , u L D ) to the auxiliary problem In biological terms, this means that the boundary {y = −L} is lethal for the individuals. The limit at x = +∞, namely the function (1 − y − L) + , is of course not chosen at random, it solves the one-dimensional free boundary problem To ensure the maximum chance to retrieve, in the end, a solution that converges to 1 uniformly in y as x → −∞, we have imposed the Dirichlet condition u = 1 at the bottom of the cylinder.

Exponential solutions
At this point, we need to make a short recollection of what the exponential solutions of the linear problem are. System (2.1), linearised around 0, reads It has exponential solutions that decay to 0 as x → −∞, i.e. solutions of the form ψ L D (x, y) = e αx h(y) (we have to put here the dependence with L and D. If L = +∞ we may choose the function h as an exponential in y. We have so that the exponents α L D and β L D satisfy Three types of limits will be considered. Case 1. The limit D L 1, c bounded. We expect β L D to go to 0 as D → +∞, so that Then (2.3) yields estimates of the form It is to be noted that these equivalents may be pushed up to c = o( √ D).
We have here a first occurrence of the critical order of magnitude c ∝ √ D. Case 3. The limit L D 1, c 1. We expect that Lβ L D will go to infinity, so that cotanh(β L D ) → 1. In this setting, we have the estimates (2.5).

Universal upper bound
In the construction of a travelling wave for (1.1), the first task is to bound the velocity from above. We will prove straight away the upper bound that will serve us in the later section, namely that c L D cannot exceed a (possibly large) multiple of √ D.
Theorem 2.1 There is K > 0, independent of D and L, such that The function u L D vanishes on the curve Γ 2L D whose equation is In particular, for y = −L we have Translate the solution u L D (x, y) of (1.1) so that the leftmost point of Γ L D is located at the vertical of the origin. Then, from (2.7), Γ 2L D lies entirely to the left of Γ L D . By the maximum principle we have an impossibility. If not, we have x L D <x L D and this time we have the last inequality because of the Hopf lemma. This prevents the Wentzell relation from holding at (x L D , 0). So, the only possibility left is y L D < 0. By the strong maximum principle, contradicting the free boundary relation for u L D . This proves the theorem.

Universal lower bound
uniformly with respect to L ≥ L 0 .
The proof of Theorem 2.2 will be by contradiction. From now on, and until this has been proved wrong, we assume that, c L D is bounded, both with respect to L and D. Recall that Γ L D , the free boundary, is an analytic graph {x = ϕ L D (y)}. Moreover, from [6], we may always assume that it intersects the line {x = 0}, so that we may always assume ϕ L D (0) = 0. Definē x L D as the last x such that, for all y ∈ (−L, 0), then (x, y) / ∈ Γ L D . Our main step is to prove that the front goes far to the left of the domain, this is expressed by the following lemma.
For every x < 0, let y L D (x) be the first y such that x = ϕ L D (y). We will prove that which implies Lemma 2.3. Assume (2.9) does not hold, and consider δ 0 > 0 such that, for a sequence (D n , L n ) n , going to +∞, and a sequence (ε n ) n going to 0 as n → +∞, we have Obviously, we must assume the boundedness from below of the sequence (ε n √ D n ) n . For every n set X n = (−ε n D n , −δ 0 /2),X n = (−ε n D n , 0).
For ε 0 , to be chosen small in due time, the segment (X n , X n ) is at distance at least δ 0 /2 from the free boundary. By nondegeneracy (recall the boundedness of (c Ln Dn ) n ), there is q 0 > 0 universal such that, for all n: On the other hand, recall that ∂ y u Ln Dn (x, 0) ≤ 1, this follows from [6]. So, the equation for u Ln Dn (x, 0) reads, simply This yields, for n large enough, the existence of δ 0 > 0, universal, such that This comes from the Hopf boundary lemma. So now, we now write the equation for u Ln Dn (x, 0) as Integrating this equation on (−ε 0 √ D n , 0) and invoking (2.11) allows us to find a small Proof of Theorem 2.2. Recall that we have still assumed the boundedness of (c Ln Dn ) n . In order to allevite the notations a little, we omit the index n . Integration of (2.1) over Σ L yields This expression has to be handled with care, because each integral, taken separately, diverges.
We will see, however, that there is much less nonsense in (2.12) than it carries at first sight. The curve Γ L D has an upper branch, that we call Γ L D,+ , and that connects (0, 0) to X L D ; the latter point being a turning point. The lower branch, called Γ L D,− , connects X L D to (+∞, −L + 1), in other words it is asymptotic to the line {y = −L + 1} as x goes to +∞. We decompose (2.12) into Let us remark that ∂ y u L D (x, −L) ≥ −1 for all x ∈ R. Indeed, the function u L D (x, y) being decreasing in x, it is larger than (1 − L − y) + . So, we have u L D (x, −L) ≥ −1. Now, the graph Γ L D has a discrete set of turning points, due to its analyticity. Away from these turning points, for x ≥ x L D , there is a finite number of y s: The point (x, y 1 ) has already been counted in the integration over Γ L D,+ , and so does not need to be counted again. Notice then that, if x is not the abscissa of a turning point, then n D (or n D − 2) is even, because of the configuration of Γ L D .
it suffices to consider the sole point (x, y n D (x) ) in the computation of III, however at this point we also have ∂ ν u L D = 1. Therefore we end up with III ≥ 0. Of course the integral giving III converges, but we do not even have to bother to prove it. From Lemma 2.3, we have for some universal C > 0. We already saw that III was nonnegative, so let us deal with II. By nondegeneracy, there is A > 0 universal such that

This yields
for a universal q, as soon as we choose D L 1. This is an obvious contradiction. Now, note that, because of the Dirichlet condition at y = −L, the sequence (c L D ) L , for fixed D, is increasing. This ends the proof of the theorem.
3 The one species model: construction and properties of the free boundary

Global solutions of the free boundary problem in the plane
In this short section we recall a result of Hamel and Monneau that we will use to analyse the behaviour of the free boundary at infinity. Consider a solution (c > 0, Γ, u) of the free boundary problem in the whole plane The result is a classification of the solutions of (3.1) having certain additional properties. We rephrase it here to avoid any confusion, since the function u in [10] corresponds to 1 − u in our notations. Then c ≥ c 0 and, if we set either u is the tilted one-dimensional solution φ 0 (y cos α ± x sin α), or u is a conical front with angle α to the horizontal, that is, the unique solution u α (x, y) of (3.1) such that Γ is asymptotic to the cone ∂C α , with Note that the fact that u α is unique is not exactly trivial, it is given by Theorem 1.3 of [10]. Let us already notice that Properties 1 and 2 of this theorem are satisfied by the solution (c L D , Γ L D , u L D ) of (2.1): Property 1 is clear, and Property 2 is readily granted by the monotonicity in x. Property 3 will be a little more involved to check.

Construction of a solution in the whole half-plane
From then on, we fix D > 0 large enough so that Theorem 2.2 holds. Pick L 0 such that c L 0 D > c 0 , this implies c L D ≥ c L 0 D > c 0 . Notice that we have almost all the elements for the proof of Theorem 1.1, we just need, in addition, to control where the free boundary meets the line {y = 0}. Since we have this freedom, we assume Γ L D to intersect the line {y = 0} at the origin. LetX L D = (x L D ,ȳ L D ) the point of Γ L D that is furthest to the left, our sole real task will be to prove that X L D cannot escape too far as L → +∞. Indeed, we notice that the property which, from the uniform boundedness of c L D in Theorem 2.1, yields the boundedness of the family (x L D ) L≥L 0 . Proof of Theorem 1.1. We send L to infinity, a sequence (c Ln D , Γ Ln D , u Ln D ) n will converge to a solution (c D , Γ D , u D ) of (1.1). Because of Lemma 3.2, the free boundary Γ D meets the line {y = 0} at a point (−x D , 0) and the expansion (1.4) is granted by Theorem 1.4 of [6]. Notice also that the uniform limit at −∞ is also granted because of (3.6). So, to finish the proof of the theorem, it remains to prove that, for all y < 0, the positivity set of u only extends to a finite range. Such were it not the case, the limit  simply because c L D ≤ c D . The only use of this result that we are going to make in this section is that there exists D 0 > 0 such that c D > c 0 for all D ≥ D 0 , which is the first part of Theorem 1.3. And so, we drop the indexes D in the rest of this section, for the simple reason that the dependence with respect to D will not appear anymore. To prove the second part of Theorem 1.3, we apply Theorem 3.1 to any sequence of translates of u:

The tail at infinity
u n (x, y) = u(ϕ(y n ) + x, y n + y), to infer that any possible limit of u n is the one-dimensional wave, tilted in the correct direction. Properties 1 and 2 of the theorem being readily true, we concentrate on Property 3. The main step will be to prove that φ is in fact globally Lipschitz in (any sub-plane of) the half plane, once this is done a suitably designed Hamel-Monneau type [10] subsolution, placed under u, will give the property.

Proposition 3.3
The function ϕ is globally Lipschitz in (−∞, y 0 ], for any y 0 < 0. Proof. Note that the lemma is trivially false if we inisist in making y vary on the whole half-line. Also, as will be clear from the proof, the value of y 0 will play no role as soon as it remains a little away from 0. So, we will assume for definiteness y 0 = −1. The main step of the proposition consists in proving that no point of Γ ∩ {y ≤ −1} may have a horizontal tangent. Assume that there is such a point X 0 = (ϕ(y 0 ), 0), y 0 ≤ −1. Translate u and Γ so that it becomes the origin, still denoting them by Γ and u. In a neighbourhood of 0 of size, say, ρ > 0, Γ may be written (recall that it is an analytic curve) as We have ψ(0) = ψ (0) = 0. Two cases have to be distinguished. The first one is ψ ≡ 0 in [−ρ, ρ]. By analyticity and Cauchy-Kovalevskaya's Theorem, this implies that ψ is defined and equal to 0 on the whole line; actuallly this case may happen only if X 0 is a point at infinity, that is, u is a limit of translations of infinite size. But then we have u(x, y) = y + or u(x, y) = y − , depending on whether the positivity set of u is above or below Γ. This is in contradiction with the boundedness of u. The other case is ψ nonconstant in [−ρ, ρ], so that u is nonconstant either in B ρ . By analyticity again, u has an expansion of the following type, in a neighbourhood of the origin: with n ≥ 2, the functions P and R being smooth in their arguments. Because ∂ x u it maximal at the origin, we have u xy (0, 0) = 0 from the Hopf Lemma. Consider the situation where we have, for instance u y (0, 0) = −1, u xy (0, 0) = β > 0. We have therefore .
This implies, because n ≥ 2: Thus the free boundary relation cannot be satisfied in B ρ , except at the origin. We note that these situations exhaust what can happen on Γ. Indeed, if there is a sequence (X n ) n of Γ, going to infinity, such that the tangent to Γ at X n makes an angle α n with the horizontal, with lim n→+∞ α n = 0, the usual translation and compactness argument yields a solution of the free boundary problem (3.1), where the tangent at the origin is horizontal. Once again we are in one of he above two cases, that are impossible. The last step is to check a property that will imply Property 3 for any limiting translation of (Γ, u) of the form (3.7). This is the goal of the next proposition. Proof. We use the notations of Theorem 3.1. Assume for definiteness that the leftmost point of Γ is located at X l = (0, y l ) with y l < 0. PickX ∈ Ω, call M > 0 its distance to Γ, and also set X 0 = (x 0 , −N ), both M and N will be assumed to be large, independently of one another. From Proposition 3.3 we claim the existence of a cone C β (the notation is given by (3.4)), the angle β > 0 depending on the Lipschitz constant of ϕ, but independent of M and N , and a pointX M,N ∈ Ω, such that Let Γ M,N be the graph It meets the line of fast diffusion at a pointx M,N that satisfies an estimate of the form with a universal γ > 0, possibly different from that of (3.10).
Pick now any small ε > 0 and consider the function We claim that u ≤ u in Γ M,N ∩ {y ≤ N }. Indeed, recall that we have Therefore, u ≤ u, so that we have, if we still callX the original pointX translated byX M,N : which is the sought for estimate.
Proof of Theorem 1.2. Proposition 3.3 proves that any limit of a sequence of translations (u n ) n in (3.4) satisfies the assumptions of Theorem 3.1. Therefore it is either a conical front u α (Case 1), a tilted one-dimensional wave φ 0 (y cos α − x sin α) (Case 2), or a tilted wave φ 0 (y cos α + x sin α) (Case 3). We wish to prove that only Case 3 survives. Let us consider the set T of turning points of Γ, that is, the set of all points (ϕ(y), y) such that ϕ (y) = 0. From the analyticity of Γ, T is discrete: If we manage to prove that it is finite we are done because this excludes Case 1 trivially, and Case 2 because u(x, y) → 1 as x → −∞, uniformly in y. So, assume that T is infinite. We claim the existence of two sequences of T , decreasing to −∞, (y i n ) n , i ∈ {1, 2}, such that • we have y 1 n+1 ≤ y 2 n ≤ y 1 n for all n, • we have lim n→+∞ (y 2 n − y 1 n+1 ) = lim n→+∞ (y 1 n − y 2 n ) = +∞, such that, if we set x i n = ϕ(y i n ) we have lim n→+∞ u(x i n + x, y i n , y) = u α (x, y), uniformly on compact sets in (x, y). Indeed, there is N 0 ∈ N such that T may be organised in disjoint clusters each Λ n having at most N 0 elements, and lim n→+∞ min (y,ϕ(y))∈Λn This is because of the fact that u α has a finite number of turning points and that the convergence of u to its limits implies the convergence of the free boundaries in C 1 norms. We claim that (x 1 n , y 1 n ) and (x 2 n , y 2 n ) cannot be in two consecutive clusters, because of the orientation of C α . So, if Λ i n is the cluster of (x i n , y i n ), let Λ 3 n be a cluster in-between. Let (x 3 n , y 3 n ) be the leftmost point of Λ 3 n . From the definition of Λ 3 n , there is R n → +∞ such that (x 3 n , y 3 n ) is the leftmost point of Γ in B Rn (x 3 n , y 3 n ). Let u ∞ (x, y) be a limit of translations of u with the sequence of points (x 3 n , y 3 n ), it does not converge to any of the functions prescribed by Theorem 3.1, which is a contradiction. This finishes the proof of Theorem 1.2.

The model with two species
The first thing one must understand is (1.2) in the truncated cylinder Σ L , namely

Exponential solutions
System (4.1), linearised around 0, reads The solutions that decay to 0 as x → −∞ are looked for under the form so that the exponents α L D and β L D satisfy Once again we consider types of limits. Case 1. The limit D L 1, c bounded. We expect β L D to go to 0 as D → +∞, so that Then (4.3) yields estimates of the form And, once again, the estimate may be pushed up to c = o( √ D).

Estimates on the velocity
Let (c L D , Γ L D , v L D ) a solution to (4.1), we may infer its existence from a once again slight modification of Theorem 1.1 of [6]. We assume that it meets the line {y = 0} at the point (0, 0). Theorem 4.1 There is a constant K > 0, independent of L and D, such that Moreover there is L 0 > 0 such that we have Proof. That of (4.6) is similar to that of Theorems 2.1, one compares (u L D , v L D ) to As for (4.7), the only point to prove is that, under the assumption that (c L 0 D ) D is bounded, the width L 0 being large but fixed, then the leftmost point of Γ L D , denoted by ( Once this is proved, the proof of the theorem proceeds much as that of Theorem 2.2, using in particular estimates (4.4) for the linear exponentials. So, assume (4.8) to be false, that. is, there are two diverging positive sequences (D n ) n and (x n ) n , and a positive constant d 0 such that By nondegeneracy, there is γ > 0 such that Choose another sequence (x n ) n such that as soon as n is large enough. This implies u Dn L 0 (−x n ) < 0, contradiction.
From then on, the rest of the study of the two species model parallels exactly that of the one species model.

Exponential convergence
In this section, we assume that all the requirements on the coeffiients are fulfilled, and we drop the index D for the velocity c, the free boundary Γ and the solution u. Theorem 1.4 is proved by deriving a differential inequlity for ϕ, exploiting Theorem 1.3 and the fact that ϕ as a limit at infinity. This allows indeed to write the free boundary problem for u in a suitable perturbative form, and translate the double Dirichlet and Neumann boundary condition into the sought for differential inequality. In the whole section, the considerations will be rigorously identical for the one species model or the two species model, except the global estimate in Proposition 5.2 below, where the computations are slightly different -but left to the reader. Thus we will concentrate on the one species model (1.1). Recall from [1] that Γ is an analytic graph. Theorem 1.3 readily implies This entails the following improvement of Proposition 3.4.
On the line we have, on the same pattern: and if ε small enough.
And so, as soon as ρ ∈ (0, c D ) and ε > 0 is small enough, the function u := (1 −ū) + is a sub-solution to the equations for u in the region {u > 0}, moreover ∂{ū > 0} coincides Γ sufficiently far in the lower half plane. The maximum principle implies the proposition. Proof of Theorem 1.4. From now on, translate the origin so that we are in the following situation: with φ (x) < 0 and lim x→+∞ φ (x) = −tanα. Actually, the translation may be adjusted so that φ (x) is as colse as we wish to −tanα, this will be quantified later. The following chain of transformations is then made.
1. Rotate the coordinates by the angle α, so as to obtain the new set (x, Y ) given by X = x cos α − y sin α, Y = x sin α + y cos α.
In this new system the free boundary may be written as Y = ψ(X), with, by Proposition 5.1: lim 2. Straighten the free boundary by setting In this new coordinate system, u solves the over-determined problem

A standard compactness/uniqueness argument shows that
uniformly in Y < 0. As we will not need any additional change of coordinates, let us, for notational simplicity, revert to the initial notation x := X , y := Y .
The function u(x, y) is this looked for under a perturbation of φ 0 (y): u(x, y) = φ 0 (y) + u(x, y). The free boundary condition writes .
The full PDE forũ will not be written, as we need another change of unknowns.
The system for (h, v) is thus the function R[h](x, y) being: While the expression of R[h](x, y) is utterly unpleasant, its structure is quite simple: it is quadratic in h and its derivatives up to order 2, which are known to vanish at infinity. Finally, set S[h, v](x, y) = h 2 v yy − h v y + 2hv xy + hc cos αv y .
By elliptic regularity, all derivatives of v go to 0 as x → +∞, uniformly in y. Problem (5.3) now reads We will use the estimate with θ(x, y) a positive function that tends to 0, uniformly in y, as x → +∞. It only remains to apply Proposition A1 of the appendix together with (5.4)), with the data From Proposition 5.2, estimate (A1) holds, so that the function h solves the differential inequality |h (x) − h(x)c cos α| ≤θ(x)(|h (x)| + |h(x)|), the functionθ(x) going to 0 as x → +∞. This implies the exponential estimate.
Appendix: compatibility relations for over-determined equations with right handside the goal of this section is to prove that the right handside of the solution of.an elliptic equation in the lower half plane, both with Dirichlet and Neumann condition, has to satisfy an integral equation.
In other words, the right handside of (A3) equals that of (A2) as soon as x ≥ 2.
A second observation is that the general Dirichlet problem is well-posed as soon as the Dirichlet datum u 0 (y) and the right handside F (x, y) satisfy the estimate (A1), with any ρ < c 0 . Indeed, the classical change of unknown u(x, y) = e ρy v(x, y) changes (A3) into an elliptic equation the the zero order term ρc 0 − ρ 2 . Thus it is enough to assume tht g and k are compactly supported in y, in order to perform Fourier transforms. An easy density argument then allows to treat general data satisfying (A1). Set w(x, y) = e (c cos αx+c 0 y)/2 v(x, y), the equation for w is now (A4) −∆w + β 2 w = F (x)e 0 (y) + G(x, y) (x ∈ R, y ≤ 0) w(x, 0) = w y (x, 0) = 0.