MINIMIZATION

. The aim of this paper is to investigate the minimization problem related to a Ginzburg–Landau energy functional, where in particular a nonlinear diffusion of mean curvature–type is considered, together with a classical double well potential. A careful analysis of the corresponding Euler–Lagrange equation, equipped with natural boundary conditions and mass constraint, leads to the existence of an unique Maxwell solution , namely a monotone increasing solution obtained for small diffusion and close to the so–called Maxwell point . Then, it is shown that this particular solution (and its reversal) has least energy among all the stationary points satisfying the given mass constraint. Moreover, as the viscosity parameter tends to zero, it converges to the increasing (decreasing for the reversal) single interface solution , namely the constrained minimizer of the corresponding energy without diffusion. Connections with Cahn–Hilliard models, obtained in terms of variational derivatives of the total free energy considered here, are also presented.


Introduction
Following the original Van der Walls' theory, the energy of a fluid in a one dimensional vessel is given by the Ginzburg-Landau functional where F (u) stands for the free energy per unit volume, and ε is a positive (small) viscosity coefficient.The study of the minimization of (1.1) with the constant mass constraint, and the complete description of the corresponding minimizer(s), has been carried out in the seminal paper [7], together with the connection of the latter to the minimizer of the purely stationary case ε = 0, when interfaces (jump in the density u) are allowed without any increase of energy.We also recall that the same problem in the multidimensional case was addressed in [17].Following the blueprint of [7], the main goal of our paper is then to minimize the functional over all u ∈ H 1 (−1, 1) satisfying the mass constraint In (1.2), the function Q is explicitly given by while for the free energy we assume that there exist 0 < α < β ∈ R such that the function F ∈ C 5 (0, ∞) satisfies F ′′ > 0 on (0, α) ∪ (β, ∞), F ′′ < 0 on (α, β); The above conditions are summarized in Figure 1, from which the connection with the aforementioned 1-d Van der Walls' theory is manifest.Moreover, the specific form of the function Q in the energy functional (1.2) considered in the present paper is motivated by the discussion made by Rosenau in [19,20].In the theory of phase transitions, in order to include interaction due to high gradients, the author extends the Ginzburg-Landau free-energy functional (1.1) and considers a free-energy functional with Q as in (1.4), and so, with a linear growth rate with respect to the gradient.We also mention here the papers [9,12,13,15,16], where the same nonlinear diffusion of mean curvature-type is also considered in different contexts.Finally, it is worth observing that 0 ≤ Q(s) ≤ s 2 for any s ∈ R and therefore (1.2) is well-defined for u ∈ H 1 (−1, 1).Nevertheless, we underline that the function Q is convex, but fails to be coercive, and therefore, to the best of our knowledge, there is no straightforward evidence of existence of minimizers for the functional (1.2).
Remark 1.1.We point out that the explicit choice (1.4) for the function Q, which is the paradigmatic example given in [19,20], is made here for simplicity and readability of the paper.However, we claim that the main results contained in this article can be proved for a generic even function Q ∈ C 2 (R) satisfying (1.6) Notice that, under these assumptions, we readily obtain 0 ≤ Q(s) ≤ Cs 2 for any s ∈ R, so that the minimization problem under discussion here is still well defined.Actually, we emphasize that the specific form of the nonlinear function Q is used only in Sections 3-4, and in particular in the explicit formulas (3.4), (3.19) and (4.1)-(4.2).We sketch the argument to handle the more general case in Remarks 3.3 and 4.1.
In the case ε = 0, the problem we are dealing with reduces to the minimization of with the constraint (1.3), which leads to the study of the auxiliary functional where σ stands for a (constant) Lagrange multiplier.The function inside the above integral is referred to as Gibbs function [7] Φ σ (z) := F (z) − σz, (1.9) and its properties, depending on the assumptions (1.5) on F , are crucial in the minimization problems listed above.We depict in Figure 2 the graph of the Gibbs function for different values of σ ∈ (σ, σ).The value σ 0 refers to the so-called Maxwell point, which can be obtained from the celebrated Maxwell construction, also known as equal area rule [18]; see Figure 1.This construction clearly leads to the following relations: or, in terms of the Gibbs function, Φ σ 0 (α 0 ) = Φ σ 0 (β 0 ).Hence, the Maxwell point is defined by the following pair of parameters: ∆ 0 := (σ 0 , b 0 ), with b 0 := Φ σ 0 (α 0 ) = Φ σ 0 (β 0 ).(1.11) Conditions (1.10) are related to the minimization problem for (1.8), for which the associated Euler-Lagrange equation F ′ (u) = σ should hold at points of continuity of u, while the function F (u) − σu should be continuous across jumps of u (the Weierstrass-Erdmann corner condition).In addition, as pinpointed in Figure 2, the Gibbs function for σ = σ 0 becomes a double well potential with wells α 0 , β 0 of equal depths.
At this stage, we can define the single-interface solutions where As proven in [7], this particular minimizer of (1.7) with the constraint (1.3) is the physically most relevant, because, for ε > 0 sufficiently small, there exists a unique (modulo reversal) global minimizer of (1.1) with the same constraint; this minimizer is strictly monotone and it converges, as ε → 0, to (1.12).The present investigation will lead to the same conclusions concerning stationary points of the generalized energy functional (1.2), again with the mass constraint (1.3).In particular, we shall first characterize all possible smooth minimizers of this problem by solving the corresponding Euler-Lagrange equation with natural boundary conditions.Then, for ε sufficiently small, we shall prove that, among them, there exists a unique (modulo reversal) solution with least energy, which turns out to be monotone and to converge to the same single-interface solutions (1.12) as ε → 0 + .
1.1.Plan of the paper.This paper is organized as follows.In Section 2 we connect our minimization problem with a Cahn-Hilliard-type equation, which is derived by using (1.2) instead of the classical Ginzburg-Landau functional (1.1).Section 3 is devoted to the characterization of all possible smooth minimizers of the functional (1.2) satisfying the mass constraint (1.3).Moreover, we shall state in this section the theorem which identifies, for ε small, the Maxwell solution, that is a monotone increasing solution of the Euler-Lagrange equation satisfying both the natural boundary conditions and the mass constraint (1.3), see Theorem 3.4.The existence of the Maxwell solution is then proved in the subsequent Section 4. Finally, in Section 5 we prove that the Maxwell solution (and its reversal) is the stationary point of (1.2)-(1.3)with least energy, and it coincides with the unique minimizer of the problem, provided the latter exists.
1.2.Outline of the main results.Since our analysis is quite technical, we conclude this Introduction by outlining our main results, the strategy used to prove them, and the main differences with respect to [7].
It is well known that any smooth stationary point of (1.2) must satisfy the corresponding Euler-Lagrange equation together with natural boundary conditions.In addition, we need to consider the mass constraint (1.3), which leads to the problem (3.2) in Section 3.Then, in order to characterize all stationary points, we first study the Euler-Lagrange equation in the whole real line (hence, without the boundary conditions and the mass constraint).It turns out that there exist periodic solutions for any admissible pair ∆ = (σ, b); see Section 3 and specifically Proposition 3.2.Let us underline that the proof of Proposition 3.2 is an important preliminary step of our analysis, which is not needed in [7].Next, in Section 4 we prove the main result of Section 3, Theorem 3.4, which establishes the existence of a special monotone solution of (3.2) (called Maxwell solution) by showing that it is possible to chose the parameters σ, b (near the Maxwell point) such that the restriction to the interval [−1, 1] of the periodic solution also satisfies the boundary conditions and the mass constraint.Hence, the Maxwell solution is a monotone stationary point of (1.2)- (1.3).The existence and uniqueness of the Maxwell solution is obtained by rewriting the problem in an appropriate way (see (3.18)) and then by using the Implicit Function Theorem.To obtain our result, the main extra difficulty with respect to [7] is to prove Lemma 4.4, which is needed here in view of to the presence of the nonlinear function Q.
We claim that, using the same strategy, it is possible to show the existence of other solutions of (3.2), that is stationary points with an arbitrary number of transitions, as well as other monotone solutions.However, we will not investigate the existence of these solutions in detail, because we shall see in Section 5 that they can not minimize the energy (1.2) with constraint (1.3).Indeed, we evaluate (1.2) at the Maxwell solution (Proposition 5.1) and show that it minimizes (1.2)-(1.3)along all possible stationary points, see Propositions 5.3, 5.5 and 5.6.In particular, we shall prove that the energy (1.2) of the Maxwell solution has the asymptotic expansion E r + o(ε), when ε is sufficiently small, where E r is the minimum of (1.7)- (1.3).Finally, we show that the Maxwell solution and its reversal converge as ε → 0 + to the single-interface solutions defined in (1.12), see Theorem 5.7.
Summarizing, the main result of this paper is that, if r ∈ (α 0 , β 0 ), then (i) when ε > 0 is small enough, there exists a smooth strictly increasing stationary point ε ] converges as ε → 0 + to the minimum of (1.7), that is the minimum energy in the case without diffusion; (iii) u ε and u R ε converge as ε → 0 + to the single-interface solutions defined in (1.12), namely the two-phase solutions with least energy in the case without diffusion.

Connections with Cahn-Hilliard models
As a possible motivation for our studies, in this section we derive the evolutionary equation related to the minimization problem under investigation in the paper.In particular, we show that a Cahn-Hilliard type equation with nonlinear diffusion can be derived from the energy functional (1.2).
First, let us recall that the celebrated Cahn-Hilliard equation, which in the onedimensional case reads as where ε > 0 and F : R → R is a double well potential with wells of equal depth, has been originally proposed in [4,6] to model phase separation in a binary system at a fixed temperature, with constant total density and where u stands for the concentration of one of the two components.Generally, (2.1) is considered with homogeneous Neumann boundary conditions which are physically relevant since they guarantee that the total mass is conserved.Moreover, it is well-known [10] that equation (2.1) is the gradient flow in the zero-mean subspace of the dual of H 1 (a, b) of the Ginzburg-Landau functional (1.1) and that the only stable stationary solutions to (2.1)-(2.2) are minimizers of the energy (1.1) [21].Therefore, thanks to the aforementioned work [7], we can state that solutions to (2.1)-(2.2) converge, as t → ∞, to a limit which has at most a single transition inside the interval [a, b].It is also worth mentioning that the convergence to the monotone steady states is incredibly slow and if the initial profile has an N -transition layer structure, oscillating between the two minimal points of F , then the solution maintains the unstable structure for a time T ε = O (exp(c/ε)), as ε → 0 + , see [1,2,3].This phenomenon is known in literature as metastable dynamics, and it is studied in various articles and with different techniques for reaction-diffusion models, among others see [12,13] and references therein.The analysis of this property can be carried out also for equation (2.1) with nonlinear diffusions, as in the case under investigations here, but it is not in the main aims of the present paper and it is left for future investigations; the interested reader can refer to [11] for the case of the p-Laplace operator.
Equation (2.1) can be also seen as the simplest 1-d case of a very general Cahn-Hilliard model introduced by Gurtin in [14], which reads as where D is a non constant mobility (which may depends on u and its derivatives), Ψ is the so-called free energy, γ is an external microforce and m is an external mass supply, for further details see [14].In particular, the standard Cahn-Hilliard equation (2.1) corresponds to the one-dimensional version of (2.3), with the choices D ≡ 1, γ ≡ m ≡ 0 and the free energy If we consider a concentration-dependent mobility (cfr.[5] and references therein) D(u) : R → R + , γ = m = 0 as in the standard case and the free energy we end up with the following Cahn-Hilliard model with mean curvature type diffusion For the sake of completeness, we recall here how one can derive the model (2.6) from the (multi-d version of the) functional (1.2).Let us consider the total (integrated) free energy [14] The formal variation with respect to fields ϕ that vanish on the boundary ∂Ω is given by where ⟨•, •⟩ denotes the H −1 , H 1 0 pairing.As a consequence, the variational derivative is given by δE δu = Ψ u (u, ∇u) − divΨ v (u, ∇u). (2.7) The standard Cahn-Hilliard equation is derived from where D ≡ 1 and the free energy is given by (2.4); indeed, in this case one has On the other hand, in the case of free energy defined as in (2.5), one obtains the total free energy that is, the multi-dimensional version of (1.2), and since substituting in (2.7)-(2.8),one deduces the model that is (2.6) with the explicit formula As an alternative viewpoint for the derivation of (2.10), we recall the continuity equation for the concentration u u t + divJ = 0, (2.11) where J is its flux.The standard Cahn-Hilliard equation follows from (2.11) and the constitutive equation J = −∇µ, which relates the flux J to the chemical potential [14] (2.12) On the other hand, by considering (again) the variational derivative (2.7) with free energy (2.5) and a concentration dependent mobility, we obtain the flux By substituting (2.13) in the continuity equation (2.11), we end up with (2.10).We emphasize that there are two novelties in (2.13): the concentration-dependent mobility and the mean curvature operator replace the constant mobility and the Laplacian, which are a peculiarity of the classical choice (2.12).
Let us now show that the physically relevant (no-flux) boundary conditions for (2.10) where ν is the unit normal vector to Ω. Indeed, the boundary conditions (2.14) guarantee that the energy (2.9) is a non-increasing function of time along the solutions to (2.10) and that these solutions preserve the mass.Precisely, differentiating (2.9) with respect to time and using the first condition in (2.14), we get Moreover, (2.11)-(2.13)and the second condition in (2.14) imply where we used the positivity of the function D. Finally, by using (2.11) and (2.14) we infer for any t > 0.
In the one-dimensional case, we expect that the aforementioned results of [21], valid for (2.1)-(2.2),can be extended to the boundary value problem (2.10)-(2.14), that is its solutions converge as t → +∞ to the minimizers of (1.2)-(1.3).In the following sections, we rigorously prove that, if ε > 0 is sufficiently small, the stationary points of (1.2)-(1.3)with least energy are monotone and, as a consequence, we expect that any solution of (2.10)-(2.14)will eventually have at most one transition.However, inspired by the classical results [1, 2, 3] and using the same strategy of [11,12,13], we are confident that, in the one-dimensional-case, there exist metastable solutions of (2.10)-(2.14),which maintain an unstable structure with an arbitrary number of transitions for an extremely long time.Up to our knowledge, these results concerning the asymptotic behavior of the solutions are an open problem, which needs further investigation.

Maxwell solution
As it was already mentioned, the main goal of this paper is to minimize the functional A classical result of calculus of variations asserts that all smooth stationary points u = u(x) satisfy the Euler-Lagrange equation with homogeneous Neumann boundary conditions u ′ (±1) = 0. Expanding the derivative, imposing the constraint (3.1) and introducing the function where the constant σ is a Lagrange multiplier.In order to determine all solutions of (3.2), we multiply by z ′ the ordinary differential equation in (3.2) and study the equation in the whole real line, where b is a constant and we introduced the function and the Gibbs function Φ σ , depicted in Figure 2, is defined in (1.9).As pointed out in [7] (see Propositions 2.1 and 4.1), we briefly recall here that for each σ ∈ (σ, σ): (1) Φ σ has exactly three critical points, namely , where α 0 := α σ 0 , β 0 := β σ 0 , ζ 0 := ζ σ 0 ; see also Figure 1, where in particular the behavior of the critical points α σ , ζ σ , and β σ , summarized in points ( 1) and ( 2), is manifest.These properties are instrumental in the study of the behavior of the solutions to the ODE that is the ODE satisfied by the minimizers in the case of the classical Ginzburg-Landau functional (1.1) (the ODE can be obtained by In particular, it is possible to choose the constants σ, b such that there exist periodic solutions of (3.5), see the phase portraits in Figure 3.To be more precise, σ has to belong to the interval (σ, σ) and b must satisfy ) for any σ ∈ [σ, σ], in view of point (4) above.We refer to such a pair ∆ = (σ, b) as admissible.It is worth observing here that the pair ∆ 0 , corresponding to the Maxwell point defined in (1.11), is not an admissible pair.Indeed, notice that Φ σ 0 is a double well potential with wells α 0 , β 0 of equal depths, and, as a consequence, the choice b 0 = Φ σ 0 (α 0 ) = Φ σ 0 (β 0 ) corresponds to the heteroclinic orbit connecting the two wells α 0 and β 0 , see Figure 3. Hence, the corresponding solution never satisfies the Neumann boundary condition in (3.2).
On the other hand, to any admissible pair ∆ corresponds a periodic solution to (3.5); our goal is to extend the latter result to equation (3.3).To this aim, we rewrite this equation as The crucial properties of the function P ε defined in (3.4), in particular its invertibility, needed to investigate equation (3.3) (or (3.7)) are consequences of the bounds for f ∆ (z) collected in the next lemma.
We are left with the proof of (3.15), namely, of the boundedness of the improper integral defining T (∆).Since H + ε (s) = √ 2s+o(s), as s → 0 + , using the definition (3.7), we deduce Finally, F ′ (z i ) − σ ̸ = 0 for i = 1, 2 because z i are not critical points of the Gibbs function Φ σ , and in particular Φ ′ σ (z 1 ) > 0 and Φ ′ σ (z 2 ) < 0. As a consequence, T (∆) is bounded, being and the proof is complete.□ Remark 3.3.Proposition 3.2 guarantees the existence of periodic solutions in the whole real line, which is instrumental in constructing particular solutions of (3.2), as we shall see below.In addition, we emphasize that, being P ε (s) = 0 if and only if s = 0, the phase portrait of equation (3.7) is (qualitatively) the same of the one depicted in Figure 3, and therefore all possible solutions of (3.2) are constants or restriction of periodic solutions in the real line, with half period T (∆) defined in (3.16) appropriately chosen to fulfill the boundary conditions, that is nT (∆) = 2ε −1 , n ∈ N. In particular, if T (∆) = 2ε −1 , then we obtain a monotone solution of (3.2), while for n ≥ 2 we obtain all other possible (non constant and non monotone) solutions.Finally, we emphasize that the crucial property needed in the proof of Proposition 3.2 is the bound (3.15).In the present case, we prove it taking advantage of the given formula (1.4) for Q to compute P ε and its inverses H ± ε given by (3.11) explicitly.Hence, to extend the existence result to the case of a generic Q satisfying (1.6), it is sufficient to prove that P ε has inverses which satisfy H ± ε (ξ) ≈ ε −1 √ ξ, when ξ ∼ 0, which clearly implies (3.15).The aforementioned behavior close to zero of H ± ε (ξ) can be proved proceeding as in [8, Lemma 2.1].
The next goal is to prove that we can choose an admissible ∆ such that T (∆) = 2ε −1 , namely the existence of monotone solutions to (3.2).For definiteness, we focus our attentions to (non-constant) increasing solutions of (3.2), referred to as simple solutions, following [7].Specifically, in order to select such ∆, we need to find conditions on σ and b such that the restriction in (−ε −1 , ε −1 ) of the corresponding periodic solution satisfies the boundary conditions and the mass constraint in (3.2).In particular, the boundary conditions imply Φ σ (z(±ε −1 )) = b or, equivalently, where z 1 = z(−ε −1 ), and z 2 = z(ε −1 ).
(3.17)Both conditions can be recast as where we use the notation In other words, the existence of a simple solution to (3.2) is equivalent to find ∆ admissible which solves system (3.18).To this aim, since a simple solution satisfies z ′ (x) > 0 for any x ∈ (−ε −1 , ε −1 ), we take advantage of (3.17) and (3.7) to convert the above integral into one of the form where in the last passage we substitute the formula for H + ε in (3.11).The above rewriting of the integrals I n (∆) is instrumental in solving system (3.18), which then leads to the main result of this section.Theorem 3.4.For any δ ∈ (0, (β 0 − α 0 )/2), there exist ε δ > 0 and a neighborhood N δ of the Maxwell point ∆ 0 , defined in (1.11), such that for ε ∈ (0, ε δ ) and problem (3.2) has exactly one simple solution with corresponding admissible pair ∆(ε, r) ∈ N δ .Moreover, there is a constant as ε → 0 + , uniformly for r ∈ R δ .
Hence, the Maxwell solution, mentioned in Section 1, is the one whose existence and uniqueness is established by Theorem 3.4.The Maxwell solution plays a crucial role, because it minimizes the energy functional (1.2) with mass constraint (3.1), among all its stationary points, as we shall see in Section 5. We prove Theorem 3.4 in the next section.
Remark 3.5.Condition (3.20) is necessary to prove the existence of simple solutions to (3.2).Indeed, as we mentioned in Remark 3.3, all possible solutions of (3.2) are constants or restriction of periodic solutions in the real line, and the latter, oscillating between z 1 and z 2 in (3.17), have mass in [2z 2) has the unique solution u ≡ r in view of (4.3) below.Actually, it is possible to prove a stronger result: given ε > 0, there exists δ ε > 0 such that problem (3.2) does not admit simple solutions if r < α 0 + δ ε or r > β 0 − δ ε .We omit the proof of this fact, which can be easily done by proceeding as in [7,Theorem 7.1] and by taking advantage of the analysis we present in the subsequent Section 4.

Proof of Theorem 3.4
The strategy is to solve system (3.18) for small ε using the Implicit Function Theorem and thus we shall analyze carefully the behavior of I 0 and I 1 with respect to this parameter.To do this, notice that, by rewriting 1/H + ε as 1 and substituting in (3.19), we deduce where In particular, we shall single out the terms in (4.1) which are singular in ε, while the remainder should be treated implicitly.For this, as the involved functions are not globally C 1 close to ∆ 0 , we need to introduce the following notions as stated in [7]. where The functions z i (∆), i = 1, 2, are continuous in Σ and z i ∈ C 4 (Σ); moreover, while in the boundary one has The proofs of (4.3), (4.4), (4.5) can be found in [7,Proposition 4.2].Notice in particular that, in view of (4.3), the assumption ε ∈ (0, F ) guarantees the validity of (3.12), with f ∆ (z) defined in (3.7), for any z ∈ (z 1 (∆), z 2 (∆)), i = 1, 2, and ∆ ∈ Σ.
In order to solve system (3.18),we introduce the transformation h = π(∆), defined by From the definition of the Gibbs function (1.9) and the fact that α σ , β σ are critical points of Φ σ , it follows that the Jacobian determinant of the mapping π is Hence, π is a C 4 diffeomorphism of Σ onto H, where H = π(Σ).In particular, the definitions of ∆ 0 , ∆, ∆ give Finally, for later use, it is worth mentioning that, in view of (3.6), π i ≥ 0 in Σ and π i > 0 in Σ, i = 1, 2.

Definition 4.2 ([7]
).We say that a function ψ := ψ(h) is nearly regular at h = 0 if there exists a neighborhood U ⊂ H of 0 and s < 1 such that ψ ∈ C(U), ψ ∈ C 1 (U\{0}), and In the sequel, we will use the following notations: given a function φ : Σ → R, we write Moreover, if Ω ⊂ R 2 , then we denote and let Ω 0 , Ω 1 and Ω 2 be arbitrary closed subregions of Σ with Notice that in the definition (4.1) of I n the first integral has an integrand which is singular at z = z i (∆), because f ∆ (z i (∆)) = 0, while R n is continuous as a function of ∆ in Σ, thanks to (4.2) and Lemma 3.1.Therefore, the most relevant case in our analysis is when ∆ 0 ∈ Ω 0 , that is where the integrals I 0 , I 1 defined in (3.19) blow up, and we shall use the notion introduced in Definition 4.2.Our fundamental result which is instrumental to solve system (3.18)close to ∆ = ∆ 0 , or, equivalently, close to h = 0, is contained in the next proposition and concerns the precise nature of the singularities of I 0 , I 1 at the Maxwell point ∆ 0 .With this aim in mind, let us first recall that z i (∆) belong to N R( Σ) and for h = π(∆), one has It is worth mentioning that (4.7) and [7, Lemma 4.2] imply Proposition 4.3.Assume F satisfies (1.5) and ε ∈ (0, F ), where F is defined in (3.10).
Then, I n defined in (3.19) verifies , n = 0, 1, (4.11) In view of [7, Proposition 5.1], (4.1), (4.2), and the continuity of R 0 (∆; ε) and R 1 (∆; ε) in Ω 0 ∪ Ω 1 ∪ Ω 2 , the above result is a direct consequence of the following lemma.Proof.From (4.2), it follows that We need to control the above integrals only close to the boundary points z 1 (∆) and z 2 (∆), namely where f ∆ vanishes; see (3.7).It is worth observing that, contrary to I 0 and I 1 , these integrals will not blow up, and actually they are continuous in the whole region Σ, at least for ε smaller than F defined in (3.10).However, they are not smooth close to the boundary points, yet we shall be able to prove they are nearly regular.For this, we focus our attention in the region (z 1 (∆), z 1 (∆) + η) for a fixed η > 0 sufficiently small; the analysis close to z 2 (∆) being analogous.
Let us start with the study of R 0 , and define the following integral where Notice that the function G ε ∈ C ∞ (0, 2ε −2 ) and It follows that and since the function In order to study the latter integral, we expand f ∆ (z 1 (∆) + t) about t = 0 using Taylor's formula: first, we have where Θ F (y, t) is a C 1 function for y > 0 and we used f (z 1 (∆)) = 0 and the formulas (3.7)-(1.9).As a consequence, we can rewrite with Q 1 defined in (4.9) and In particular, λ, c ∈ C(Ω 0 ), ρ ∈ C([0, ∞) × Ω 0 ) and λ > 0 on Ω 0 \∂ 1 Σ because of (4.7).Moreover, one has and substituting in (4.13), we deduce By using the binomial formula with a m appropriate binomial coefficients, we infer Let us analyze all the contributions L i (∆), i = 1, 2, 3, 4 and, being explicitly computed, we carried out in full details the first two terms.For this, routine integrations show that Then, since λ(∆) ∈ N R(Ω 0 ) and λ(∆) ln λ(∆) ∈ N R(Ω 0 ) (see [7,Lemma 5.1]), we can state that L i (∆) ∈ N R(Ω 0 ), for i = 1, 2, that is the "leading terms" in Rη 0 (∆).The analysis of L i (∆), for i = 3, 4 is straightforward, in view of the arguments carried out in [7].Indeed, the integrand in these terms can be recast as the corresponding ones of [7] multiplied by 2λ(∆)t + t 2 and thus they are more regular at t = 0 than the ones already discussed there.Hence, we can conclude that Rη 0 (∆) ∈ N R(Ω 0 ).Concerning R 1 (∆), we can proceed in the same way and we arrive to study Therefore, we can say that Rη )) and the last integral can be treated exactly as in the proof of Rη 0 (∆) ∈ N R(Ω 0 ).In conclusion, R 0 and R 1 belong to N R(Ω 0 ) and the proof is complete.□ Proof of Proposition 4.3.In view of [7, Proposition 5.1], we readily obtain for n = 0, 1 and 1 √ 2 Then, the continuity of R n implies (4.11), while Lemma 4.4 gives ) and the proof is complete.□ Remark 4.5.As already mentioned before, the regularity results (4.12) proved in in Proposition 4.3 refers solely to the set Ω 0 , as they are instrumental to prove the existence and uniqueness of Maxwell solution close to ∆ 0 ∈ Ω 0 .However, one could split the analysis of the integrals I n close to the two points z 1 (∆) and z 2 (∆) separately, and give more refined regularity results there, as done for instance in (4.10), see [7] for further details.
Analogously, a more refined version (again involving Ω 1 and Ω 2 ) of the results contained in Lemma 4.4 can also be proved.Finally, it is worth mentioning that the properties (4.11) are not needed at this stage, but they will be crucial afterwards, while studying the energy of other (non-Maxwell) simple solutions (if any), which are defined away from ∆ 0 ; see Proposition 5.4 and Proposition 5.5 below.
After studying the behavior of the functions I n (∆; ε), the next step of our proof of Theorem 3.4 will consist in solving system (3.18) in a neighborhood of the Maxwell point ∆ 0 , see (1.11).With this in mind and following [7], we introduce the scaling where The transformation (4.18) is defined for k , defined by (4.18), belongs to H = π(Σ), where the transformation π is given by (4.6), then we call (k, ε, r) compatible and ∆ = π −1 (h) is an admissible pair; let us denote the resulting map as Conversely, given any admissible pair ∆ and any ε > 0, we can define h = π(∆) and solve (4.18) for k; we denote this mapping by Given any function φ := φ(∆), we use the notation In order to study implicitly (3.18) close to the Maxwell point and for sufficiently small ε > 0, we need to extend smoothly functions φ * also for negative values of ε, namely to a neighborhood in R 4 of the set The needed extension is guaranteed by the following lemma, proved in [7], and where the regularity properties introduced in Definition 4.2 play a crucial role.We say that a neighborhood K of E 0 is compatible if every (k, ε, r) ∈ K, with ε > 0, is compatible.Finally, given any C 1 function Ψ = Ψ(k, ε, r), we write Lemma 4.6.Let Then, there exists a compatible neighborhood For the proof of Lemma 4.6 see [7, Lemma 6.1].Thanks to Lemma 4.6 we can then prove the following result.Proposition 4.7.There exist a compatible neighborhood K of E 0 and a such that, given (k, ε, r) ∈ K with ε > 0 and ∆ = D(k, ε, r), one has if and only if ∆ is a solution of (3.18).
Proof.By the definitions (4.19)-(4.20) of B i and (4.9), we have Hence, letting ξ 1 = α 0 , ξ 2 = β 0 , thanks to (4.10) and the fact that z i (∆) belong to N R( Σ) (because of (4.8)), we end up with for i = 1, 2, where p i , q i ∈ N R(Ω 0 ) and p i (∆ 0 ) = q i (∆ 0 ) = 0. Thanks to Proposition 4.3, in particular (4.15)-(4.16),and using (4.24), we obtain Then, we can apply Lemma 4.6 to the right hand side of the above equalities and deduce that there exist a compatible neighborhood K of E 0 and functions S 0 , S 1 ∈ C 1 (K), with ∇S 0 = ∇S 1 = 0 on E 0 such that for all (k, ε, r) ∈ K, with ε > 0. Thus, system (3.18) is equivalent to which readily gives Therefore, (4.22) and (4.23) hold true with and the proof is complete.□ Thanks to (4.22) and the Implicit Function Theorem we can state that there exists a unique solution k = k(ε, r) of (4.23) in a neighborhood of a point (a, 0, r), where a ∈ R 2 is fixed and for any r satisfying (3.20).Indeed, (4.22) implies for some a = (a 1 , a 2 ) ∈ R 2 .Hence, (a, 0, r) is a solution of (4.23) for any r ∈ (α 0 , β 0 ).Moreover, (4.22) also gives where W (k, ε, r) = k − W (k, ε, r) and I 2 is the identity matrix in R 2 , and a simple application of the Implicit Function Theorem gives the following result.
Proposition 4.8.Choose δ ∈ (0, (β 0 − α 0 )/2) and define R δ as in (3.20).Moreover let a be given by (4.25) and let K be the neighborhood of E 0 given by Proposition 4.7.Then, there exist a neighborhood Now, we have all the tools to prove Theorem 3.4.

Properties of the Maxwell solution and minimization of the energy
The goal of this section is to prove that the Maxwell solution given by Theorem 3.4 minimizes the energy functional (1.2) with constraint (1.3) among all its stationary points.In view of the change variable x = εt, (1.2) becomes so that, if we define z(t) = u(εt), we obtain the functional Hence, we look for a minimum of (5.1) over the class Let us start by computing the energy of simple solutions, which is a function depending on the admissible pair ∆ ∈ Σ denoted by E εr (∆): where z ∆ stands for the simple solution of (3.2) corresponding to ∆.As we have seen before, z ∆ satisfies (3.7) and the definition (3.4) of P ε gives .
By using the explicit formula for Q (1.4), we deduce and, as a trivial consequence, using the definitions (1.9) and (3.7), one has Substituting in the definition of E εr (∆), we obtain where in the last passage we used the change of variable s = z ∆ (t) and (3.12).Hence, from (3.11) it follows that for any ∆ ∈ Sol(ε, r).Thanks to the formula (5.3), we can prove the following result about the energy E 0 εr of the Maxwell solution denoted by z εr : E 0 εr := E ε [z εr ].We recall that the Maxwell solution is the unique simple solution close to the Maxwell point ∆ 0 given by Theorem 3.4 for ε ≪ 1.
Proposition 5.1.The energy (5.1) of the Maxwell solution has the asymptotic form where Proof.By proceeding as in the proof of Lemma 4.4, we obtain that the integral function in (5.3) belongs to N R(Ω 0 ), that is, changing variables, Êεr (h) is nearly regular: there exists s < 1 such that Êεr (h) = Êεr (0) + O(|h| 1−s ).
Since h = 0 correspond to the Maxwell point ∆ 0 , (1.11) gives where c ε is defined in (5.5) and we used (1.9)-(1.11)-(3.7).Therefore, we can conclude that (5.4) holds true thanks to exponentially smallness of h given by (4.26).□ Remark 5.2.Notice that the quantity 2(σ 0 r + b 0 ) represents the minimum energy of (1.2) in the case ε = 0, that is with the mass constraint (1.3); see [7].Moreover, the constant c ε , defined in (5.5) and describing the first order correction in ε, satisfies lim where c 0 is a strictly positive constant depending only on the potential F .In addition, the leading term in the difference Finally, if we assume that F is a double well potential with wells of equal depth and F (α 0 ) = F (β 0 ) = 0, then σ 0 = b 0 = 0 and The latter constant represents the minimum energy of a transition between α 0 and β 0 in the case of the renormalized energy studied in [12, Proposition 2.5] and [13,Proposition 3.3].Now, we shall compare E 0 εr given in (5.4) with the energy of all other possible stationary points of (5.1)-(5.2),that is, solutions of (3.2).Among them, let us start by observing that, given any simple solution z := z(t) of (3.2), its reversal z that is the energy of a simple solution and its reversal are the same.Therefore, we are left to compare (5.4) with the energy of remaining solutions of (3.2), namely constants, (other) simple solutions, and non-monotone solutions.We start by analyzing the first case.Proposition 5.3.For any δ ∈ (0, (β 0 − α 0 )/2) and r ∈ R δ , we can choose ε δ > 0 such that if ε ∈ (0, ε δ ), then E 0 εr is strictly less than the energy of constant solutions of (3.2).Proof.First of all, notice that the unique constant solution of (3.2) is exactly z ≡ r and its energy (5.1) is given by E ε [r] = 2F (r).On the other hand, from (5.4), by using (1.11) and (1.9), it follows that The second step is to compare E 0 εr with the energy of other simple solutions, namely of simple solutions with admissible pair ∆ that is not close to the Maxwell point ∆ 0 .We are not interested in the study of the existence of such solutions, but we can prove that if there exists a simple solution different from the Maxwell one, then σ is bounded away from σ 0 , ∆ is close to ∂ 1 Σ or ∂ 2 Σ, and the solution is close to a constant one.To be more precise, we have the following result.
Proposition 5.4.Choose ρ, δ > 0.Then, there exist ε = ε(ρ, δ) > 0 and ρ 0 = ρ 0 (δ) > 0 such that for any ε ∈ (0, ε), r ∈ R δ and ∆ ∈ Sol(ε, r), with ∆ that is not the pair corresponding to the Maxwell solution, it follows that The proof of Proposition 5.4 is very similar to the corresponding one of [7, Theorem 7.2] and we omit it.It is worth observing that, as already mentioned in Remark 4.5, the continuity properties in the subregions Ω 1 and Ω 2 stated in (4.11) of Proposition 4.3 play here a crucial role.
The desired property for E 0 εr is clearly implied by the following result.Proposition 5.6.Non monotone solutions of (3.2) can not even be local minimizers of E ε .

Declaration of competing interests
Authors have no competing interests to declare.

Figure 1 .
Figure 1.Graph of F ′ and possible choices of σ.

Figure 2 .
Figure 2. Graph of the Gibbs function Φ σ for different values of σ.

Remark 4 . 1 .
The specific form (1.4) for Q again plays a crucial role in (4.1)-(4.2) because it leads to the explicit formula (3.11).In the case of a generic function Q satisfying (1.6), one can use solely the expansion of the inverse H + ε close to zero (as already claimed in Remark 3.3, see [8, Lemma 2.1]) in the proof of the needed properties of the remainder R n , without having an explicit formula at our disposal.Let Σ be the set of all admissible pairs (σ, b) and let us write the boundary of Σ as the disjoint union

Lemma 4 . 4 .
Under the same assumptions of Proposition 4.3, R 0 and R 1 defined as in (4.2) belong to N R(Ω 0 ).