Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.

Model (1.1) is known as the Peierls-Nabarro model for dislocations in crystals where W is called the stacking fault energy. Here φ(x) is a phase transition representation of a dislocation localized at the position x = 0. For a physical introduction to the model, see for instance [25]; for a recent reference, see [37]; we also refer the reader to the paper of Nabarro [33] which presents an historical tour on the model.
We now consider a scalar function v solution of the following evolution problem associated to the Peierls-Nabarro model: where σ ε is the exterior stress acting in the material. This equation has been for instance considered in [32] (see also [14] for a similar model). We assume that this exterior stress is given for ε > 0 small, by where σ is a given suitable function. Then we can consider the following rescaling v ε (t, x) = v t ε 2 , x ε .
We now look for scalar functions v ε satisfying When v ε has N transition layers as ε goes to zero, we will see that we can associate a particle to each transition layer such that the dynamics of v ε is driven by the following ODE system for particles (x i (t)) i=1,...,N : , on (0, +∞),

Statement of the main result
To state precisely our main result, we make the following assumptions: (A) i) Assumption on the potential                            W ∈ C 2,β (R) for some 0 < β < 1, ii) Assumption on the exterior stress            σ ∈ BUC ([0, +∞) × R) and for some 0 < β < 1 and K > 0, we have We recall that BUC is the space of bounded uniformly continuous functions.
Remark 1.2. We recall here that the semi-continuous envelopes of a given any function u are defined as follows The proof of Theorem 1.1 is done by constructing suitable sub and supersolutions essentially based on the following ansatz of the solution: where ψ is a suitable corrector.

Brief review of the literature
As we have mentioned, the diffusion equation (1.3) is the evolution equation for the Peierls-Nabarro model, which describes the dynamics of dislocation defects in crystal. For general references, we refer the reader to [3] (see also the paper [36]). The Peierls-Nabarro model can also be derived from Frenkel-Kontorova models (see [19]).
In the time-independent setting, our model is related to a pioneering work of Alberti-Boucchitté-Seppecher. They have shown in [1] that if we consider the energy functional for Ω ⊂ R n and a double well potential W given by then, in particular, the rescalings E ε [u]/ |log ε| Gamma-converge when ε → 0 to 2 π H 0 (S u 0 ) if n = 2 (thus ∂Ω is one dimensional) and the minimizers u ε converge to u 0 , where u 0 = 0, 1 on ∂Ω, ∆u 0 = 0 in Ω, and the singular set is S u 0 = {z ∈ ∂Ω : u 0 jumps at z} .
A generalization to the case of elasticity equations with application to dislocations has been done by Garroni and Müller in [22].
For the time-dependent case, the (anisotropic) mean curvature motion has been obtained at the limit (in the framework of viscosity solutions) by Imbert and Souganidis [26]. Related to this result, let us also mention the work [12].
It is important to mention that a result similar to Theorem 1.1 has been obtained for the non linear heat equation (after a suitable rescaling) Here the interaction force between particles is also related to the behavior of the layer solution associated to the non linear heat equation, and hence is of exponential type. For such results using an invariant manifold approach, we refer the reader to Carr-Pego [10], Fusco-Hale [21], and Ei [16] for generalization to systems of PDEs. For results using the energy approach, see Bronsard-Kohn [6], Kalies, Van der Vorst, Wanner [27]. Remark that our approach by sub and super solutions seems new (even in this context). We also refer to the paper of Chen [11], for an interesting tour of the different regimes of the solution to the non linear heat equation that appear for general initial data.
Let us mention that similar results have also been obtained for Cahn-Hilliard equations or their generalization to systems, called Cahn-Morral systems (see Grant [24]).
We expect (even if we have no proof) Theorem 1.1 to be true for any power of the Laplacian (−∆) α , α ∈ (0, 1), once the layer solutions of (1.1) are found. The existence of such layer solutions is the content of the work in preparation [7], that tries to generalize the work of Cabré and Solà-Morales [8] for layer solutions of the half Laplacian. The extension problem (see [9]) that would be needed in this case is a degenerate elliptic equation with A 2 weight, that has been well understood in the classical reference [18].
We should mention the result by Forcadel-Imbert-Monneau [20], where they homogenize in particular system (1.4) and more generally non-local first order Hamilton-Jacobi equations describing the dynamics of dislocation lines. Similarly, homogenization results have been obtained in [31] for equation (1.2) for periodic stress σ.

Organization of the paper
In Section 2 we review of the notion and properties of viscosity solutions for this type of non-local equations. Section 3 contains all the preliminary results that will be required later (layer solutions, corrector) together with some motivation of our result and a study of the ODE system of particles (1.4). Section 4 is the proof of the main result (convergence is proven by finding suitable sub and supersolutions of the PDE). Section 5 and Section 6 are almost independent of the rest of the paper: Section 5 is a necessary review of known and further results (used in Section 6) for the half-Laplacian operator, while Section 6 contains the proof of the properties of the layer solution (Theorem 3.1) and of the existence of the corrector (Theorem 3.2).

Viscosity solutions
We recall the definition of a viscosity solution v for equation (1.3) with ε = 1 to simplify, namely for T ∈ (0, +∞): Definition 2.1. (Viscosity sub and supersolutions) An upper (respectively lower) semicontinuous function v ∈ L ∞ ([0, T ) × R) is said to be a subsolution (resp. a supersolution) of (2.10) if and only if and for any C 2 test function ϕ satisfying where for any functions Φ(x), w(x) we set is said to be a viscosity solution of (2.10), if and only if v * is a subsolution and v * is a supersolution. Finally, a function v ∈ L ∞ loc ([0, +∞) × R) is said to be a viscosity solution on [0, +∞) × R, if and only if its is a viscosity solution on [0, T ) × R for any T > 0.
is indeed a viscosity solution if and only if it is a classical solution of (2.10). This is mainly due to the fact that v t and Lv(t, ·) are then defined pointwise.
Without entering in the details of the viscosity solutions (for which we refer the interested reader the paper by Barles-Imbert [4]), let us recall the known results that we will use.
Then there exists a viscosity solution v of (2.10) satisfying Let us also recall the comparison principle: Then In particular, when the initial data v 0 is continuous, the comparison principle implies the uniqueness and the continuity of the solution.

Preliminary results
In this section, we collect several results that will be used in the section 4 for the proof of our main result.

Formal ansatz
In the following, we try to explain formally why we have to expect an ansatz as in (1.9) for the solutions of (1.3).
Namely, we are looking for an ansatz for the solutions of Assume to simplify that σ is a constant here, and that there is only one transition layer.
First try for the ansatz As a first try, in the case whereσ is constant, we could consider the Ansatz where φ is the transition layer solution of When we plug this expression in (3.11), we geṫ where we have used (3.12) for the second line, and a Taylor expansion to get the third line. From this computation, we learn two things. First for φ ≃ 0 we have to choose The second information, is that it is impossible to satisfy the equation with this ansatz, and we need to introduce a corrector.
Second try for the ansatz Let us takeṽ forσ satisfying (3.13) and for some constant c to fix later. When we plug this expression in (3.11), we getẋ where we have used a Taylor expansion to get the last line. Finally using (3.13), we see that ψ has to solve (3.14) We see that it is convenient to choose (in order to get later ψ independent onẋ 1 ) Moreover, multiplying (3.14) by φ ′ we get formally by integration: On the other hand using the fact that L is self-adjoint, we get formally where the last equality comes from the fact that φ solves (3.12) so then φ ′ solves Therefore, we get

The layer solution and the corrector
In this subsection, we state without proofs some results on the layer solution and the corrector. These results will be proven in Section 6. We recall that the layer solution satisfies (1.1), namely Then we have Under assumption Ai), there exists a unique solution φ of (3.15). Moreover φ ∈ C 1,β loc (R) for some 0 < β < 1, and there exists a constant C > 0 such that where H is the Heaviside function and α = W ′′ (0).
We are also looking for a corrector ψ, solution of the following equation

The ODE system
Recall that we consider a solution (x i (t)) i=1,...,N of (1.4), namely , on (0, +∞), We recall the following result Lemma 3.3. (Lower bound on the distance between particles, [20]) be the minimal distance between particles. Then under assumptions Ai) and Aii), we have This lemma prevents the crossing of particles in finite time. As a consequence, we easily get the following long time existence result:  We want to build a supersolution with a parameter δ > 0 to fix later. We define ( and φ is given in Theorem 3.1 and ψ in Theorem 3.2. Then we have Under assumption (A), there exists δ 0 > 0 such that for all 0 < δ ≤ δ 0 , we have Proof of Proposition 4.1 First we choose δ > 0 small enough such that Let A > 0 be large enough such that |ψ|.
Therefore for ε small enough, we have Using again the monotonicity of φ, we conclude in case 2 that Finally case 1 and 2 prove that (4.21) holds for any x ∈ R. This ends the proof of the Proposition.

Let us define for
where H is the Heaviside function.

Lemma 4.2. (Computation using the ansatz)
Let Then for any i 0 ∈ {1, ..., N}, we have where η is defined in (3.19) and the error e ε i 0 is given by Proof of Lemma 4.2 We have, using the equation for the corrector, that We first remark that all the terms in φ ′ vanish (because c i =ẋ i ). Using assumption (A), we can boundσ t , Lσ andċ i . Moreover using Theorem 3.2, we can also bound ψ ′ . Finally, collecting all the terms of order ε together, we get simply Now let us choose any index i 0 ∈ {1, ..., N} and let us make a Taylor expansion of W ′ (·) when the argument of this function is close toφ i 0 . We get We remark that the terms in W ′ (φ i 0 ) vanish and the terms in ψ i 0 vanish. Now taking into account the definition of e ε i 0 , we see that we get and then This ends the proof of the lemma.
We are now ready to prove the following result: Under assumption (A), there exists δ 0 > 0 such that for all 0 < δ ≤ δ 0 and given T > 0, we have Recall also that by (3.17), we have We deduce that Therefore using (4.23) where in the first inequality we have used the fact that αγη = 1. To get the last line, we have used the fact that e ε i 0 → 0 as ε → 0.
Finally case 1 and 2 prove that for ε small enough we have which ends the proof of the Proposition.
Proof of Theorem 1.1 Step 1: existence and uniqueness of the solution First remark that for ε small enough the initial condition satisfies are respectively sub and supersolutions of (1.3) on [0, +∞) × R. From the Perron's method, it follows the existence of a solution v ε of (1.3) on [0, +∞) × R which is moreover unique thanks to the comparison principle (Proposition 2.4).
Step 2: sub and supersolutions as ε → 0 Now, given any fixed T > 0, and thanks to Propositions 4.1 and 4.3, we can find some δ > 0 small enough such that v ε is a supersolution of (1 Similarly we can build a subsolution v ε (defined simply as v ε but with δ < 0). At the initial and then from the comparison principle, we get that Now for fixed T > 0, we get the result (namely (1.7) and (1.8)) on [0, T ) × R, passing to the limit as ε → 0. Finally, because T > 0 was arbitrary, we recover the result for all time on [0, +∞) × R. This ends the proof of the Theorem.

Known and further results on the half-Laplacian
In this section, we recall several results that will be used in Section 6 for the proof of Theorems 3.1 and 3.2. We first recall four definitions (that are equivalent for smooth enough functions) of the half-Laplacian operator (Fourier, Lévy-Khintchine, classical harmonic extension, notion of weak solutions). We will essentially use the notion of weak solution (via the harmonic extension) for which we will show that it coincides with the notion of viscosity solutions (via the Lévy-Khintchine formula), when the function is in the space Then we recall results from [8] on the layer solutions and the comparison principle (for the harmonic extension). We also recall regularity results for harmonic extensions and give new useful results for the L ∞ regularity of solutions to half-Laplacian type equations.
There are many points of view to handle half-Laplacian operators. Here we have chosen to consider it through a harmonic extension problem, although another more direct approach could have been taken. However, our approach has two advantages: first, many of the results reduce to its analogous for harmonic functions, and second, we use the setting already established by Cabré and Solà Morales in [8]. Another advantage is that this is precisely the setting of Cabré-Sire [7] for generalization to any power α ∈ (0, 1).

The Fourier approach
We recall our definition of the Fourier transform for w ∈ S(R) (where S(R) is the Schwartz space): and recall the definition of the following Hilbert space where v * denotes here the complex conjugate of v. We also set the following Hilbert space with the scalar product It is known that H − 1 2 (R) is the dual of H In particular, L is a continuous linear mapping from H . Moreover L is self-adjoint as follows: for any v, w ∈ H 1 2 (R), we have < Lv, w > H − 1 Finally let us remark that for any w ∈ H 1 2 (R) we have

The approach in real space
We have the following result (see [28], or the more recent paper [13], Theorem 1): Moreover this formula allows also to define Lw for instance for w ∈ L ∞ (R) ∩ C 1,β loc (R). Remark 5.2. In the previous formula the characteristic function 1 {|z|≤1} can be replaced by 1 {|z|≤r} for any r > 0, which does not change the value of the integral.
The operator L is called a Lévy operator associated to the Lévy measure dz/z 2 . In particular, the Lévy-Khintchine formula allows to see that for any w ∈ H 1 2 (R), we have This last formula is a simple exercise and can be found in the book by Lieb-Loss [29], theorem 7.12, part ii.

The approaches by harmonic extension on R 2 + and notion of weak solutions
In the following, we give a characterization of the half-Laplacian operator in R through a harmonic extension to the upper half-plane as a Dirichlet-to-Neumann operator. This relation was well known but it has been recently rediscovered by Caffarelli-Silvestre in [9], where they also give the generalization to any fractional power of the Laplacian between zero and one.
The main advantage of this interpretation is that the regularity results and maximum principles for the half-Laplacian can be shown through the analogous results for harmonic functions in the extension, as it has been considered by Cabré and Solà Morales in [8].

Classical harmonic extension for smooth functions with compact support
Assume that w ∈ C ∞ c (R) (the space of smooth functions with compact support) and consider its harmonic extension u(x, y) on on Ω, for all x ∈ ∂Ω.
Then it is known that Here, and for the rest of the paper, ν denotes the exterior normal along ∂Ω.
The half-Laplacian appears to be a Dirichlet-to-Neumann operator. This is also called a Steklov-Poincaré operator. Moreover, we can use the Poisson kernel P for the upper half-plane to write u = P * x w, i.e., (5.30) u where c 2 is a dimensional constant.

Harmonic extension of functions in H
Then we have the following result: Proof. For u smooth and with compact support, the classical embedding trace inequality reads A proof of the previous inequality can be found in corollary 6.3 in [2]. Then , there exists a unique u ∈ H which is harmonic on Ω, such that T u = w, and it is written as u = P * x w. Moreover for any v ∈ H, we have In particular, .
The previous theorem allows to define weak solutions (in the sense of distributions) for half-Laplacian equations, that may not be necessarily equivalent to the notion of viscosity solutions introduced in Definition 2.1. Although the introduction of a new concept may seem confusing at first, it will allow to use the available functional analysis tools. Moreover, this definition is a particular case of the one introduced in [8]. Thus we set:

Definition 5.5. (Weak sub/super/solution)
Let d be a measurable bounded function on R, and let g ∈ L 2 (R). We say that w ∈ H 1 2 (R) is a weak supersolution for the equation we have that Respectively, w is a weak subsolution if Accordingly, we say that w is a weak solution if it is both a weak sub and supersolution.
We summarize the different concepts in the following lemma: Proof. We refer the reader to the paper [13].

Miscellanea
Here we remind the reader some basic facts.
For harmonic extensions, we have (cf. [35], chapter II.2): Theorem 5.7 (L p bound). If w ∈ L p (R), 1 ≤ p ≤ ∞, and u = P * x w is the Poisson integral of w, then u is harmonic in R 2 + , lim y→0 u(x, y) = w(x) for almost every x ∈ R and for all y > 0. If w is bounded on R, then also u is uniformly bounded in R 2 + .

Layer solutions for half-Laplacian problems
The paper [8] is concerned with the boundary reaction problem for some non-linearity f . We say that u is a layer solution of (5.38) if it satisfies (5.38), u x > 0 on ∂Ω and lim Since we are only concerned with smooth solutions of (5.38), then we do not need to worry here about the different concepts of weak or viscosity solutions for half-Laplacian problems previously introduced.
In the next theorem we summarize the basic facts about existence and behavior of layer solutions for (5.38) (see theorems 1.2, 1.4, 1.6, and equation (2.27) from [8]).

We have the estimate
3. Set φ(x) := u W (x, 0). We have the bounds for all x ∈ R, for some positive constants C, C 1 , C 2 .

A layer solution is always a stable solution. This means that
for every function v ∈ C 1 (Ω) with compact support in Ω.
One of the crucial ingredients in the present work is the precise knowledge of the asymptotics for φ as given in (3.17). In order to improve from the known (5.41), we will use a comparison argument with the explicit layer solution of the Peierls-Nabarro problem (see [8], Lemma 2.1, for the proofs, or the original paper [36]): for each a > 0, consider the equation with potential given by It is well known that the function is a layer solution for the problem (5.44) and, in fact, it is the only one.
A simple computation gives that 0). We know explicitly its asymptotic behavior an improvement from (5.41).
We can see that v is a (smooth) solution of in Ω, v(x, 0) = 0 on ∂Ω.
We know (5.40) which implies that v ∈ L 2 (R × [0, R]). Because the right hand side of the equation is bounded in L 2 (R × [0, R]), and here we are using again the precise decay of φ ′ , then we get also that v ∈ W 2,2 loc using the boundary gradient estimates for harmonic functions in [23], Lemma 9.12. As a consequence,ū ∈ W 1,2 (R × [0, R]).
Step 2: {y>R} |∇ū| 2 < +∞. Far away from the boundary of Ω, we can use standard interior gradient estimates (Lemma 5.8) for the harmonic functionū, say, at a ball B 1 (x, y) ⊂ Ω so that where we have used (5.40) in the last inequality.

Comparison principles for smooth enough solutions
Next, we consider some comparison results for half-Laplacian equations: Lemma 5.12 (Strong comparison principle, Lemma 2.8 in [8]). Let v be a bounded function in Ω, v ∈ C 2 (Ω) and C 1 up to the boundary ∂Ω, satisfying for some bounded function d satisfying d(x) ≥ τ for all x ∈ R and some τ > 0. Then v > 0 in Ω unless v ≡ 0.
We will also need the following variation: Corollary 5.13. Let v be a bounded function in Ω, v ∈ C 2 (Ω) and C 1 up to the boundary ∂Ω, satisfying where d is a bounded function satisfying d(x) ≥ τ for all x ∈ R and some τ > 0. Assume, in addition, that the right hand side M can be estimated by for some constant C ≥ 0. Then there exists a non-negative constantC depending on C, τ such that Proof. Fix a := 2/τ , and consider the model layer solution u a given in (5.45), with trace φ a . Because of hypothesis on M and the decay of (φ a ) x from (5.42), we can find a positive constant K such that |M(x)| ≤ Kφ a x (x) for all x ∈ R. Next, compute where we have used (5.46) in the second inequality. Now we apply Lemma 5.12 to the difference 2K τ u a x − v to get v ≤ 2K τ u a x inΩ. The same argument, applied to −v gives that |v| ≤ 2K τ u a x . Taking into account the estimate (5.42), we achieve the conclusion of the corollary.

Further properties of weak solutions
The aim of this subsection is to extend the classical results for harmonic functions to the boundary ∂Ω, in order to obtain regularity and maximum principles for weak solutions (in the sense of definition 5.5). We set u − := − inf{u, 0}, u + := sup{u, 0}.
Our first result is a weak maximum principle for non-smooth solutions: Proposition 5.14. (Weak maximum principle). Let u ∈ H be a weak supersolution of (5.35) with g ≡ 0, and let d(x) be a measurable bounded function. Set u = P * x w. Assume that u ≥ 0 in B + R , and that d > 0 on R\[−R, R]. Then and also, T u ≥ 0 on R.
Proof. This is a small variation of the classical proof for the Laplacian, that can be found in Gilbarg-Trudinger [23], Theorem 8.1. Note that (T u) − = T (u − ) a.e. in R. Set We know that Ω − is contained in R 2 + \B + R and Θ ⊂ R\(−R, R). Use u − ∈ H as an admissible test function in (5.36); then The proof is easily completed.
Note that a function u ∈ H must have trace T u belonging to L p (R) for all p > 1. However, it may not belong to L ∞ (R). In the next result, we use a Moser iteration scheme to show that weak solutions are actually bounded.
Proposition 5.15. (L ∞ bound for g = 0). Let w ∈ H 1 2 (R) be a weak subsolution of (5.35) with g ≡ 0 for some uniformly bounded function d defined on R. Then If w is a supersolution, the conclusion reads The constant does not depend on w. As a corollary, u := P * x w is also uniformly bounded on Ω.
Proof. This is the classical proof by Moser for the Laplacian case. We follow closely the arguments from Theorems 8.15 and 8.18 in Gilbarg-Trudinger [23], so we will not give every single detail here. Assume that w is a subsolution, and set u := P * x w. Given σ ≥ 1, we insert into (5.37), the test function v = (u + ) σ .
Note that T v = (T u) σ a.e.. Here we point out that this v might not belong to H; in order to make the argument rigorous, a truncated version must be used instead in the definition of v, say replacing u + by (5.50) min{u + , n}, for n ∈ N.
We obtain σ Next, we set u 1 := (u + ) γ and w 1 := T u 1 for γ = σ+1 2 , and use that |d(x)| ≤ C 0 . Thus we get For the left hand side, we proceed as follows. We use the trace inequality (5.32) to estimate But, on the other hand, by Theorem 5.9, for every p > 1, there exists a constant C p > 0 such that where we have used (5.51) in the last line. If we denote , then we have shown that Fixed p > 1 and choose γ m = p m for m ∈ N; by iteration we know It is then easy to check that C m ≤ C ∞ uniformly bounded in m ∈ N (which is also shown, for instance, in page 190 of [23]). Remark that Letting m to infinity in (5.52), we have for every R > 0. We get the first part of the conclusion, taking R → +∞ and then n → +∞ in (5.50). The second part is analogous, using u − instead. The final assertion is an immediate consequence of Theorem 5.7.
We will use also the following particular case (that is not the optimal one can prove, but it suffices for our purposes): Corollary 5.16. (L ∞ bound when g is bounded). Let w ∈ H 1 2 (R) be a weak solution of (5.35) with non-vanishing right hand side g. If there exists p 0 > 1 such that (with c independent on p), then w is bounded.
We now quote the regularity theory for bounded weak solutions of [8].
Proposition 5.17. (Regularity of weak solutions). Let R > 0, β ∈ (0, 1), and let w ∈ H 1 2 (R) be a weak solution of (5 for some constant C R that depends only on β, R and on the upper bounds for

Proof.
This is a consequence of Lemma 2.3 from [8], using definition 5.5 for weak solutions.
In particular, we obtain C 1,β loc regularity up to the boundary ∂Ω: Corollary 5.18. (Uniform regularity of weak solutions). Given R, β, w, u, d, g as above, assume, in addition, that d C β (R) ≤ C, g C β (R) ≤ C. Then u ∈ C 1,β (R × [0, R]) and for some constant C R that depends only on β, R and on the upper bounds for u H , d C β (R) and g C β (R) .
Proof. We apply the previous proposition to every ball B R (x 0 , 0), x 0 ∈ R, as in Lemma 2.3, part (b) of [8]. It uses the condition that u is uniformly bounded on Ω, which in our case follows from Corollary 5.16.

Results on the layer solution and the corrector
In this last section, we give the proofs of Theorems 3.1 and 3.2.

Proof of Theorem 3.1
Let W be a potential satisfying hypothesis Ai). Theorem 5.10 shows existence, regularity, uniqueness and asymptotic behavior of a layer solution 0 < u W < 1 satisfying u W (0, 0) = 1/2, u ∈ C 2,β loc (Ω), for the problem (6.54) We denote its trace value by φ := u W (·, 0). The aim of the next paragraphs is to show the more precise estimate (3.17) for the asymptotic behavior near infinity.
Set a = 1/α, where α = W ′′ (0) as defined in (1.6), and choose the model layer solution u a as given in (5.45). We know explicitly because of (5.47) that φ a := u a (·, 0) has the right asymptotic behavior (3.17). We would like to compare u W to u a , in order to obtain the analogous estimate for u W . Therefore, we set v : thanks to the growth estimate (5.41). And also, Because of the choice of a, we have that W ′′ (0) = V ′′ a (0). Thus adding the previous two lines we conclude that v is a solution of for some function M satisfying Therefore, (3.17) follows from Corollary 5.13 (comparison principle for smooth solutions) and the fact that W ′′ (0) > 0.
The proof of Theorem 3.1 is completed due to the fact that a smooth, bounded solution u of (6.54), with trace φ, is a viscosity solution in the sense of Definition 2.1 of with L defined through the Lévy-Khintchine formula, as it is noted in Lemma 5.6.

Proof of Theorem 3.2 -coercivity
Here we remind the reader of the notation introduced in Section 5: let φ be the layer solution constructed Theorem 3.1, with harmonic extension u W (some properties are given also in Theorem 5.10). Its asymptotic behavior is given in Theorem 3.1: for all x ∈ R. We also fixū := (u W ) x ∈ C 1,β loc (Ω), and note thatū = P * x φ ′ ∈ H. For some of its properties, see also Lemma 5.11. In the following, we set up the necessary functional analysis ingredients for Theorem 3.2, while the proof will be completed in the last subsection.
We set H 0 to be the subspace in H consisting of functions whose trace is orthogonal to φ ′ in L 2 (∂Ω), i.e, using the bounds for φ ′ given in (6.55). We conclude that u 0 ∈ H 0 , and the lemma is shown. Now, given u, v ∈ H, we define the bilinear functional It is easy to check that B is a continuous and symmetric bilinear functional in H.
The key lemma in this section is the following: Lemma 6.2. (Lower bound for Q). There is some constant C 0 > 0 such that for all u ∈ H 0 , where H 0 is defined in (6.56), we have that So we immediately have that Corollary 6.3. (Coercivity). There exists some C > 0 such that for all u ∈ H 0 , Proof. It easily follows from Lemma 6.2. Indeed, Because W ′′ is bounded, there exists µ > 0 small enough such that (1−µ)C 0 +µW ′′ (φ) > C 0 2 . Therefore, as claimed.
Proof of Lemma 6.2: It is divided into several steps: Step 1: Stability. Because of the construction of u W and φ, we know from Theorem 5.10 that Q(u) ≥ 0 for all u ∈ C ∞ c (Ω) (i.e. layer solutions are stable). Thus the functional Q is bounded from below also in H.
Step 2.1: Preliminaries. We seek a minimizer for the functional subject to the constraints u ∈ H 0 and ∂Ω u 2 = 1.
Note that W ′′ (φ) is not bounded from below by any positive constant. However, due to our assumptions Ai) on W and the growth condition (6.55) for φ, we still can find some τ > 0 and h a non-negative smooth function supported on a compact set K : Let {u k } ∞ k=1 ⊂ H 0 be a minimizing sequence for Q with ∂Ω u 2 k = 1. This L 2 bound for u k tells us that, in particular, |Q 2 (u k )| ≤ sup h ≤ C for all k, for some constant C > 0. As a consequence, we also know that there exists a constant C ′ such that Q 1 (u k ) ≤ C ′ . From here we get a uniform bound for the H-norm of the sequence {u k }, depending on τ and C ′ .
Next, we concentrate on the space H 1 2 (R). We define an equivalent Hilbert space norm · H on R as follows: for any w ∈ H 1 2 (R), we consider its harmonic extension u := P * x w, and define w H := Q 1 (u).
We have shown that u k H ≤ C for all k. This implies that for w k := T u k , the norm w k H ≤ C ′ for another constant C ′ . BecauseH is a Hilbert space, then there is a subsequence (still denoted by w k ), such that w k ⇁ w 0 , weakly inH, for some w 0 ∈ H 1 2 (R). Weak convergence implies, for instance ( [5], proposition III.12), that Setũ k := P * x w k , that has the same trace as u k , and u 0 := P * x w 0 . It is clear that and that equality holds when u k is already harmonic. Then we have shown that (6.60) On the other hand, the embedding H 1 2 (R) ֒→ L p (K) is compact for any compact subset K ⊂ R and p > 2, as stated in Theorem 5.9. Thus, we can find some subsequence (still denoted by w k ) such that w k →w 0 strongly in L p (K). As a consequence, also w k →w 0 in L 2 (K). The uniqueness of the limit will give thatw 0 = w 0 , at least a.e. in K. In addition, since Q 2 lives only on the compact subset K ⊂ R, we also have that From (6.60) and (6.61) we deduce that u 0 ∈ H satisfies Remark that if lim inf k→∞ Q(u k ) > 0, then we have already finished the proof, because this implies (6.58). Therefore we can assume that (6.62) lim inf Q(u k ) = 0 Step 2.2: Concluding. Case A: u 0 ≡ 0 Remark first that On the other hand, we have Q 2 (u k ) → Q 2 (u 0 ) = 0. Therefore we get which is in contradiction with (6.62). Therefore only the next case occurs. Case B: u 0 ≡ 0 Then we have 0 ≤ Q 1 (u 0 ) = 0 and we will show that T u k → T u 0 in H 1 2 (R). Since {u k } was a minimizing sequence, then Q(u k ) → 0 = Q(u 0 ) when k → ∞. In addition, we have shown that T u k → T u 0 strongly in L 2 (K), so that Q 2 (u k ) → Q 2 (u 0 ). Because Q(u 0 ) = 0 ≤ Q(ũ k ) ≤ Q(u k ) → 0, we deduce that Q(ũ k ) → Q(u 0 ). Moreover we have Q 2 (ũ k ) = Q 2 (u k ) → Q 2 (u 0 ) and Q = Q 1 + Q 2 which implies that By definition (6.63) means that Proposition III.30 in [5] assures that if we have a weakly convergent sequence w k ⇁ w 0 in the spaceH and such that we have convergence of the norms (6.64), then the convergence w k → w 0 is strong inH, and then in H 1 2 (R). As a consequence, w k = T u k → T u 0 = w 0 in L 2 (R), which, in particular, implies the non-degeneracy of the L 2 (∂Ω) norm of w 0 : To finish, just remark that strong convergence in L 2 (R) implies that ∂Ω u 0 φ ′ = 0. Thus, u 0 ∈ H 0 .
Step 3: Euler-Lagrange equation. Once the minimizer u 0 is found, the theory of Lagrange multipliers for minimization problems (see, for instance, [17], chapter 8.4) gives that u 0 ∈ H must be a weak solution of (6.65) where λ, µ ∈ R. The value of the multipliers can be found by choosing the right test function v. Thus if we test the equality (6.65) with v = P * x φ ′ , we obtain µ = 0 (see Lemma 5.11), while testing against v = u 0 gives that Step 4: Regularity. First, Proposition 5.15 with d(x) = W ′′ (φ(x)) − λ implies a uniform L ∞ bound for both w 0 and u 0 = P * x w 0 . Then, Proposition 5.17 assures that u 0 ∈ C 1,β loc (Ω), with the following boundary estimate . Note that u 0 is as smooth as we want in the interior.
Step 5: Positivity. In the following we show that if λ = 0, then w 0 = T u 0 must necessarily be a multiple of φ ′ , which is a contradiction with the fact that u 0 ∈ H 0 and ∂Ω u 2 0 = 1. Thus we take λ = 0 in (6.65). Then we have a solution u 0 satisfying B(u 0 , v) = 0 for all v ∈ H. We remind the reader that have defined u W to be the layer solution for our potential W , with trace φ := u W (·, 0), andū := ∂ x u W , which is also uniformly bounded and has tracē u(·, 0) = φ ′ . Because of the hypothesis on the potential W and the asymptotic behavior of the layer solution φ, there exists some τ > 0 such that W ′′ (φ) ≥ τ on R\[−R, R] for some R > 0.
We have shown, in particular, that u 0 is uniformly bounded in Ω. Sinceū is positive from (5.39), then there exists κ ≥ 0 such that |u 0 | < κū in B + R . We set v := κū − u 0 , and note that v satisfies the conditions of Proposition 5.14. Thus we can conclude that v ≥ 0 in R 2 + so that u 0 ≤ κū. An analogous argument gives that −u 0 < κū. We have proved that |u 0 | < κū on R 2 + and, in particular, |w(x)| < κφ ′ (x) for all x ∈ R.

Proof of Theorem 3.2 -regularity
Here we seek a solution for It is easy to check that g ∈ H We define also the continuous linear functional on H given by The previous Corollary 6.3 asserts that the bilinear functional B from (6.57) is coercive on the space H 0 . Then, Lax-Milgram theorem can be applied to obtain a unique weak solution u 0 ∈ H such that B(u 0 , v) = Ev for all v ∈ H. Moreover, the solution depends continuously on the initial datum: u 0 H ≤ C g H 1 2 (R) . We set ψ := T u 0 ∈ H 1 2 (R).
Proof. Note that u 0 ∈ H so, in particular, ψ ∈ H 1 2 (R) (and is also in all L p (R) for p > 1).
Claim 1: ψ is bounded; as a consequence, also u 0 . This follows from Corollary 5.16, after estimating the norm of g. We use the asymptotic behavior and boundedness of φ, φ ′ from Theorem 3.1, and the Hölder regularity of W ′′ . Then In particular, for R = 1, and p > 1 β we have, g L p (R) ≤ Kc.
Claim 2: C 1,β loc (R) regularity for ψ follows from Proposition 5.17, while the uniform bound for ψ ′ is a consequence of Corollary 5.18. For the values of ψ at infinity, just note that ψ is a smooth function in L 2 (R).
Finally, a weak solution ψ ∈ H 1 2 (R) ∩ C 1,β loc (R) ∩ L ∞ (R) of (6.69) is actually a viscosity solution of the equation (3.18), thus completing the proof of Theorem 3.2. This is so because of Lemma 5.6.