Cyril Imbert
email: cyril.imbert@ens.fr

Vinh Duc Nguyen
email: vinh.nguyen@math.cnrs.fr

) Cermics

EFFECTIVE JUNCTION CONDITIONS FOR DEGENERATE PARABOLIC EQUATIONS

Keywords: Mathematics Subject Classification. 49L25, 35K65, 35R02 Parabolic equations, viscosity solutions, networks, effective boundary conditions, vanishing viscosity limit, large deviation problems

.

Here there are 3 branches (N = 3) and the tangential dimension is 1 (d = 1). We did not illustrate the junction condition L = 0 on the junction hyperplane Γ (which is a line in this example).

1. Introduction 1.1. Degenerate parabolic equations on junctions. Multi-dimensional junctions [START_REF] Freidlin | Diffusion processes on an open book and the averaging principle[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF][START_REF] Oudet | Hamilton-Jacobi equations for optimal control on multidimensional junctions[END_REF] are union of half-spaces whose boundaries are identified -see Figure 1. Precisely:

J = N i=1 J i with J i = {x = (x , x i ) : x ∈ R d , x i ≥ 0} R d+1 + J i ∩ J j = Γ R d for i = j.
Given T ∈ [0, +∞], we consider a general degenerate parabolic equation posed on a junction, u t + F i (t, x, Du, D 2 u) = 0 (t, x) ∈ (0, T ) × J * i , i = 1, . . . , N, L(-u t , ∂ 1 u, . . . , ∂ N u, t, x , D u) = 0 (t, x) ∈ (0, T ) × Γ (1.1) where J * i denotes J i \ Γ, u t denotes the time derivative, Du and D 2 u respectively denote the gradient and the Hessian of u with respect to x, and for x ∈ Γ, ∂ i u(x ) denotes the derivative of u i (x) = u| J i (x) with respect to x i at x i = 0 (recall x = (x , x i )) and D u denotes the derivative with respect to x .

Example 1.1. The case N = 1 corresponds to the study of a degenerate parabolic equation posed on a half-space, subject to a non-linear boundary condition (dynamic or not). Example 1.11 illustrates how the main theorem can be applied in this special case. The case N = 2 corresponds to the two-domain case: a degenerate parabolic equation has coefficients which are continuous on either part of a hyperplane (or a smooth interface); the generalized junction condition can be thought as a transmission condition. Theorem 1.12 is an application of the main theorem with N = 2.

We make the following assumptions on each F i .

Assumption (F).

(F1). The function F i is continuous and degenerate elliptic.

(F2). For all R > 0, there exists C i,R > 0 such that for all y = (y , y i ), all p ∈ R d+1 , all B ∈ S d+1 (R) and all λ > 0 s ∈ (0, T ), (F4). There exists H i : (0, T ) × Γ × R d × R → R continuous such that for all (t, x , p , p i , B) ∈ (0, T ) × Γ × R d × R × S d+1 (R), F i (t, (x , 0), (p , p i ), B) = H i (t, x , p , p i ); for all t ∈ (0, T ), x ∈ Γ, for all λ ∈ R, the set {p = (p , p i ) ∈ R d+1 :

|y i | ≤ 1 |y | + |B| ≤ R ⇒ F i (s,
H i (t, x , p) ≤ λ} is convex.

In the assumption above, S d+1 (R) denotes the set of real-valued (d + 1) × (d + 1) symmetric matrices and e d+1 denotes the unit vector orthogonal to Γ and pointing inside J i . We recall that F i (t, x, p, A) is degenerate elliptic if it is non-increasing with respect to A (using the classical partial order on S d+1 (R)). The function H i appearing in (F) is referred to as the Hamiltonian from the branch J i .

Example 1.2 (First order case). The first example we give is the one coming from [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF]. It reduces to deal with F i (t, x, p, B) = H i (t, x, p) for any x ∈ J i (and not only x = (x , 0) ∈ Γ) and p ∈ R d with H i continuous, coercive in p uniformly in x, i.e. satisfying lim |p|→+∞ inf (t,x)∈(0,T )×J H i (t, x, p) = +∞ and quasi-convex in p, i.e. the sublevel sets {p ∈ R d+1 : H i (t, x, p) ≤ λ} are convex for all λ ∈ R and (t, x) ∈ (0, T ) × Γ.

Example 1.3 (The model case). Our results apply to the model case where F i (t, x, p, B) = H i (t, x, p) -Trace(σ i (x)σ T i (x)B) with H i is as in Example 1.2 and where the (d + 1) × m real matrix σ i is such that σ i ≡ 0 on Γ and the (d+1)-th line σ d+1 i of σ i satisfies |σ d+1 i (y)| ≤ c i |y d+1 |. Remark that this latter condition holds true if σ i ≡ 0 on Γ and σ i is Lipschitz continuous.

As far as the junction function L is concerned, we make the following assumption.

Assumption (L).

(L1). The function L is continuous. (L2). The function L(p 0 , . . . , p N , t, x , p ) is non-increasing in p i for i = 0, . . . , N . (L3). ∀i, p i < q i ⇒ L(p 0 , . . . , p N , t, x , p ) > L(q 0 , . . . , q N , t, x , p ). (L4). inf t,x ,p L(p 0 , . . . , p N , t, x , p ) → +∞ as min i=1,...,N p i → -∞.

(L5). sup t,x ,p L(p 0 , . . . , p N , t, x , p ) → -∞ as max i=0,...,N p i → +∞.

Example 1.4 (Kirchoff conditions). A model for L is

L(p 0 , . . . , p N ) = - N i=1 β i p i
with β i > 0 for all i. Such a condition is called a Kirchoff condition.

Example 1.5 (Flux-limited junction conditions). A second important example of junction functions L is the one related to flux-limited solutions [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF]. Given a flux limiter A,

A : (0, T ) × Γ × R d → R continuous for all (t, x ) ∈ (0, T ) × Γ, λ ∈ R, {p ∈ R d : A(t, x , p ) ≤ λ} convex
we consider the associated junction function L A defined by

(1.2) L A (p 0 , . . . , p N , t, x , p ) = -p 0 + max(A(t, x , p ), max i H - i (t, x , p , p i ))
where H - i (t, x, p , p i ) denotes the non-increasing part of p i → H i (t, x , p , p i ) [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF]: if p i → H i (t, x , p , p i ) reaches its minimum at π 0 i (t, x , p ), which is the minimal minimizer, then

H - i (t, x , p , p i ) = H i (t, x , p , p i ) if p i ≤ π 0 i (t, x , p ) H i (t, x , p , π 0 i (t, x , p )) if p i ≥ π 0 i (t,
x , p ). Remark 1.6. The flux-limited function F A defined in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] corresponds to

F A (p 1 , . . . , p N , t, x , p ) = max(A(t, x , p ), max i H - i (t, x , p , p i )) = L A (p 0 , p 1 , . . . , p N , t, x , p ) + p 0 .
The appropriate notion of weak solutions for Hamilton-Jacobi equations is the one of viscosity solutions, introduced by Crandall and Lions [START_REF] Michael | Viscosity solutions of Hamilton-Jacobi equations[END_REF] -see also [START_REF] Crandall | Some properties of viscosity solutions of Hamilton-Jacobi equations[END_REF]. It is explained in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] that two notions of viscosity solutions are needed in the study of Hamilton-Jacobi equations on networks, depending on the type of junction conditions we impose. We will see that it is also the case for the degenerate parabolic equations we consider in this work. For general junction functions L in (1.1), the junction condition has to be understood in the following weak sense: either the junction condition L = 0 or one of the equations u t + F i = 0 is satisfied. We refer to such viscosity solutions as relaxed solutions -see Definition 2.2 below. But for the special junction conditions L A given by (1.2), relaxed solutions satisfy the junction condition in a stronger sense: the junction condition L A = 0 is indeed satisfied (Proposition 2.10). Such viscosity solutions are referred to as flux-limited solutions -see Definition 2.8 below.

Main result.

The main result of this article is about equivalent classes of generalized junction conditions. Roughly speaking, we prove that imposing a general junction condition amounts to imposing an effective one. This effective junction condition corresponds to some L A given in (1.2) for some flux limiter A = A L . This flux limiter only depends on the junction function L and the Hamiltonians H i . Moreover, the effective junction condition L A L is satisfied in a strong sense: if the relaxed solution u is continuously differentiable in time and space up to the junction hyperplane Γ, then the boundary condition L = 0 on Γ can be lost (see the discussion above and Definition 2.2) but L A L = 0 on Γ is indeed satisfied in the classical sense.

Definition 1.7 (The effective flux limiter A L ). Let

(1.3) A 0 (t, x , p ) = max i=1,...,N min p i ∈R H i (t, x , p , p i )
and p 0 i ≥ π 0 i (t, x , p ) be the minimal p i such that H i (t, x , p , p i ) = A 0 (t, x , p ). For all (t, x , p ), the effective flux limiter A L (t, x , p ) is defined as follows: if Theorem 1.9 (Effective junction conditions). Assume (F), (L).

L(A 0 (t, x , p ), p 0 1 , . . . , p 0 N , t, x , p ) ≤ 0, then A L (t, x , p ) = A 0 (t, x , p ), else A L (t,
Then A L : (0, T ) × Γ × R d → R given in Definition 1.7 is well-defined, continuous, such that lim |p |→+∞ inf (t,x )∈(0,T )×Γ A L (t, x , p ) = +∞
and such that any L-relaxed sub-solution (resp. super-solution) of (1.1) is an A L -fluxlimited sub-solution (resp. super-solution) of (1.1). Moreover, if {p ∈ R d : L(p 0 , p 1 , . . . , p N , t, x , p ) ≤ λ} is convex for all p 0 , . . . , p N , λ ∈ R and all (t, x ) ∈ (0, T ) × Γ, then

{p ∈ R d : A L (t, x , p ) ≤ λ} is convex for all λ ∈ R and (t, x ) ∈ (0, T ) × Γ.
Remark 1.10. Applying Theorem 1.9 in the case N = 1, effective boundary conditions for degenerate parabolic equations posed on a domain are obtained; see Example 1.11 for instance. In the case N = 2, we get effective transmission conditions; see Theorem 1.12 for instance.

Example 1.11 (The 1D Neumann problem on a half-line). We illustrate our result on the simplest example:

u t + H(u x ) -x 2 u xx = 0, x > 0, -u x = 0, x = 0
where H is a quasi-convex function (i.e. {p ∈ R : H(p) ≤ λ} convex for all λ ∈ R) as illustrated in Figure 2. This example corresponds to the case N = 1 (number of branches) d = 0 (dimension of the tangential space) and On the left at the top, the line {p = 0} intersects the graph of H in its increasing part. On the right at the top, the vertical line {p = 0} intersects the graph of H in the non-decreasing part, but on a constant part. This second case illustrates that we exhibit equivalent classes of boundary conditions; indeed, different junction conditions can be equivalent to the same effective one: other vertical lines (corresponding to u x = const at x = 0) have the same effective boundary condition (because they have the same effective flux limiter). This is also illustrated in the last case: the vertical line {p = 0} (and others) intersects the graph of H in its non-increasing part, which implies that the flux limiter coincides with A 0 = min H = max i (min H i ) for all the vertical lines appearing in this third picture.

H 1 = H and L Neu (-u t , ∂ 1 u) = -∂ 1 u. In
1.3. Comments on the main result. Our main result, Theorem 1.9, extends the results contained in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] in two directions: first, we can deal with Kirchoff conditions (see Example 1.4), second we can deal with second order terms (but degenerating along the junction).

As in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF], the effective junction condition result is quite a straightforward consequence of the following important fact (Theorem 3.2): in order to check that a function is a flux-limited sub-and super-solution, it is enough to use a reduced set of test functions ϕ whose normal derivatives ∂ i ϕ have specific values along Γ. For instance, these normal derivatives are equal to π + i (p , A(p )) if the Hamiltonian has no constant parts and does not depend on x . We recall that, roughly speaking, π + i is the inverse function of the non-decreasing part of H i , see (1.7) below.

The first version of this paper contained a comparison principle for (1.1), under stronger assumptions on F . On the one hand, the proof was quite difficult, relying on the vertex test function introduced in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF], for which C 2 regularity was to be proved in the multidimensional setting. On the other hand, new and simpler techniques now emerge to attack this problem, see for instance [START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF][START_REF] Guerand | Effective nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations[END_REF][START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Monneau | [END_REF]. In particular, it is explained in [START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF] that the equations considered in the present work can be handled in the two-domain case. For these two reasons, we decided to restrict ourselves to the core of the work, that is to say the study of effective junction conditions.

1.4. Comments on assumptions. Assumptions (F1), (L1), (L2) are natural (if not necessary) when dealing with viscosity solutions of continuous Hamilton-Jacobi equations. In particular, (L2) ensures that the junction condition is compatible with the maximum principle. We recall that our goal is to exhibit effective junction conditions for degenerate parabolic equations. In particular, we want to understand what are the effective junction conditions that are imposed at the junction. From this point of view, it is necessary to consider degenerate parabolic equations which actually degenerate along Γ. This is exactly (F4). We also assume that the Hamiltonians have convex sublevel sets, see (F4). This condition can probably be relaxed but until very recent contributions [START_REF] Guerand | Effective nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations[END_REF][START_REF] Monneau | [END_REF][START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] (none of these contributions were not available when the first version of this work appeared), the non-convex case was out of reach. As far as (F3) is concerned, it ensures that the Hamiltonians are coercive, a property which is used repeatedly and is at the core of most proofs. It is used together with (L4) for instance to derive the "weak continuity" of sub-solutions (see Lemma 2.3 below). Condition (F2) is used in an essential way when proving that the set of test functions can be reduced (see the proof of Lemma 3.5 about critical slopes below). Remark that this condition is weaker than the one which is needed in order to prove uniqueness, see [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]Condition (3.14)]. To finish with, (L3) and (L5) are used when proving the main result.

1.5. An application: the vanishing viscosity limit. Because we are able to deal with Kirchoff conditions, we are in position to adress an open problem about Hamilton-Jacobi equations from "regional control" problem: the identification of the vanishing viscosity limit.

We study the limit as ε → 0 of the equation posed in (0, +∞) × R d+1

(1.4)

     v ε t + H1 (t, x, Dv ε ) = ε∆v ε , x d+1 < 0, t > 0 v ε t + H2 (t, x, Dv ε ) = ε∆v ε , x d+1 > 0, t > 0 v ε (0, x) = v 0 (x), x ∈ R d+1
where x = (x , x d+1 ) ∈ R d+1 . In the previous equation, we do not need to impose any condition since the Laplacian is strong enough to ensure the existence of solutions that are continuously differentiable in the space variable x ∈ R d despite the discontinuity of the first order term. In particular, the following condition holds at x d+1 = 0,

(1.5) ∂ x d+1 v ε (t, x , 0+) = ∂ x d+1 v ε (t, x , 0-).
In this specific singular perturbation problem, the limit is identified by remarking that (1.5) is a Kirchoff condition and that consequently we can pass to the limit using relaxed solutions; more precisely, the limit of v ε corresponds to a relaxed solution associated with this specific generalized junction condition. But the main theorem tells us that the limit thus corresponds to a flux-limited solution associated with a flux limiter A that is explicitly given by a formula (see Definition 1.7). Looking closely at this formula, we can prove that it corresponds to the maximal Ishii solution of the limit equation recently identified by Barles, Briani and Chasseigne [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF].

Theorem 1.12 (The vanishing viscosity limit selects the maximal Ishii solution). Assume

       Hi continuous ∀(t, x ) ∈ (0, T ) × R d , ∀λ ∈ R, {p = (p , p i ) ∈ R d+1 : Hi (t, x , p) ≤ λ} convex lim |p|→+∞ inf (t,x )∈(0,T )×R d Hi (t, x , p) = +∞ and v 0 is uniformly continuous in R d+1 . Let v ε be solution of (1.4) such that there exists C > 0 (independent of ε) such that |v ε (t, x) -v 0 (x)| ≤ Ct for all (t, x) ∈ (0, T ) × J. Then v ε converges towards the maximal Ishii solution v of (1.6) v t + H1 (t, x, Dv) = 0, x d+1 < 0, t > 0 v t + H2 (t, x, Dv) = 0, x d+1 > 0, t > 0 subject to the initial condition v(0, x) = v 0 (x), x ∈ R d+1 .
Remark 1.13. The function v is associated with the unique flux-limited solution u of the previous Hamilton-Jacobi equation for some flux limiter A - I (t, x , p ) that was identified in a previous work (see (5.5) in Proposition 5.6 below, corresponding to [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF]Proposition 4.1]). The functions v and u satisfy the following equality: v(t, x , x d+1 ) = u(t, x , |x d+1 |), see Theorem 5.8 in Section 5.

1.6. Review of literature. Semi-linear uniformly parabolic equations on compact networks were studied in [START_REF] Von | Classical solvability of linear parabolic equations on networks[END_REF][START_REF] Von | Dynamical interface transition in ramified media with diffusion[END_REF][START_REF] Kramar Fijavž | Variational and semigroup methods for waves and diffusion in networks[END_REF][START_REF] Yu | Differential equations on networks (geometric graphs)[END_REF] where uniqueness, existence, strong maximum principle among other results were proved to be true.

The first results for Hamilton-Jacobi equations on networks were obtained in [START_REF] Schieborn | Viscosity solutions of Hamilton-Jacobi equations of eikonal type on ramified spaces[END_REF] for eikonal equations. Some years later, the results were extended in [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF][START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]. Many new results were obtained since then, see for instance [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] and references therein.

In [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF], the authors study regional control, i.e. control with dynamics and costs which are regular on either side of a hyperplane but with no compatibility or continuity assumption along the hyperplane. They identify the maximal and minimal Ishii solutions as value functions of two different optimal control problems. They also use the vanishing viscosity limit on a 1D example in order to prove that the two Ishii solutions can be different. Moreover, the authors ask about the vanishing viscosity limit in the general case.

In [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF], the authors study the vanishing viscosity limit associated with Hamilton-Jacobi equations posed on a junction (the simplest network, see above). The main difference with our results is that the authors impose some compatibility conditions on Hamiltonians. In particular, this allows them to construct viscosity solutions which satisfy Kirchoff conditions in a strong sense. We proceed in a different setting and in a different way: no compatibility conditions on Hamiltonians are imposed, and Kirchoff conditions are understood in a relaxed sense, which is stable under local uniform convergence (and even relaxed semi-limits). We then use Theorem 1.9 to prove that imposing Kirchoff conditions reduce to the study of a flux-limited problem (for which uniqueness holds true).

In his lectures at Collège de France [START_REF] Lions | Lectures at Collège de France[END_REF], Lions also treats problems related to Hamilton-Jacobi equations with discontinuities. After posting a first version of this paper, Lions and Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] wrote a note about a new approach for Hamilton-Jacobi equations posed on junctions with coercive Hamiltonians that are possibly not convex.

We previously mentioned that, since the first version of this paper were posted, Guerand and Monneau studied independently effective non-linear boundary conditions in the nonconvex case. On the one hand Guerand [START_REF] Guerand | Effective nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations[END_REF] studied the case N = 1 in the 1D setting, which amounts to studying first order non-convex Hamilton-Jacobi equations with nonlinear boundary conditions of Neumann type. On the other hand Monneau [START_REF] Monneau | [END_REF] mentioned to us that he studies effective junction conditions for non-convex Hamilton-Jacobi equations posed on multi-dimensional junctions.

As far as effective boundary conditions are concerned, we would like to mention that there are some results for motion of interfaces by Elliott, Giga and Goto [START_REF] Elliott | Dynamic boundary conditions for Hamilton-Jacobi equations[END_REF] and for conservation laws by Andreianov and Sbihi [START_REF] Andreianov | Scalar conservation laws with nonlinear boundary conditions[END_REF][START_REF] Andreianov | Strong boundary traces and well-posedness for scalar conservation laws with dissipative boundary conditions[END_REF][START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF].

To finish with, the link between the theory developed in [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF] and flux-limited solutions from [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] is explored in [START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF]. In particular, [START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF] contains alternative proofs in the twodomain case of the comparison principle from [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] and of the vanishing viscosity limit obtained in the present work. 1.7. Organization of the paper. In Section 2, the notions of relaxed and flux-limited solutions are presented and their properties studied. In Section 3, it is proved that in order to check that a function is a flux-limited solution, the set of test functions can be reduced. In Section 4, we prove the main result of this paper, Theorem 1.9. Section 5 is devoted to the study of the vanishing viscosity limit. The last section (Section 6) is devoted to the proof of a known result about large deviations using the main result of this work. 1.8. Notation. A distance is naturally associated with the junction J: for x ∈ J i and y ∈ J j ,

d(x, y) = |x -y | + |x i -y i | if i = j, |x -y | + x i + y j if i = j. The open ball B r (t 0 , x 0 ) centered at (t 0 , x 0 ) ∈ R × J is defined as (t 0 -r, t 0 + r) × {y ∈ J : d(y, x 0 ) < r}.
The junction hyperplane Γ is the common boundary of J i : we have Γ = ∂J i . We identify Γ with R d and we do not write the injection of R d into J i : x → (x , 0). For this reason, we write indisctinctively x = (x , 0) ∈ Γ and x ∈ Γ.

The Hamiltonian H i (t, x , p , p i ) is defined for x ∈ Γ and p ∈ R d+1 . The minimal minimizer of p i → H i (t, x , p , p i ) is denoted by π 0 i (t, x , p ). The functions H - i and H + i are defined as follows

H - i (t, x , p , p i ) = H i (t, x , p , p i ) if p i ≤ π 0 i (t, x , p ) H i (t, x , p , π 0 i (t, x , p )) if p i ≥ π 0 i (t, x , p ) H + i (t, x , p , p i ) = H i (t, x , p , p i ) if p i ≥ π 0 i (t, x , p ) H i (t, x , p , π 0 i (t, x , p )) if p i ≤ π 0 i (t, x , p ).
For λ ≥ min p i ∈R H i (t, x , p , p i ), the functions π + i and π+ i are defined by

π + i (t, x , p , λ) = inf{p i : H i (t, x , p , p i ) = H + i (t, x , p , p i ) = λ}, (1.7) π+ i (t, x , p , λ) = sup{p i : H i (t, x , p , p i ) = H + i (t, x , p , p i ) = λ} (1.8)
The function A 0 is defined for t, x , p ∈ R d by (1.3). We recall that

A 0 (t, x , p ) = max i=1,...,N min p i ∈R H i (t, x , p , p i ).
The functions p 0 i (t, x , p ) are defined as 2. Relaxed and flux-limited solutions 2.1. Test functions. In order to define relaxed and flux-limited solutions, the set of test functions is to be made precise.

p 0 i (t, x , p ) = π + i (t, x , p , A 0 (t, x , p )).

Definition 2.1 (Test functions). A function

φ : (0, T ) × J → R is a test function for (1.1) if it is continuous in (0, T ) × J, φ| (0,T )×J i is C 1 t ∩ C 1 x and φ| (0,T )×J * i is C 2 x .
We classically say that a function φ touches another function u at a point (t, x) from below (respectively from above) if u ≥ φ (respectively u ≤ φ) in a neighbourhood of (t, x) with equality at (t, x).

Relaxed solutions.

Definition 2.2 (L-relaxed solutions). A function u : (0, T ) × J → R is an L-relaxed subsolution (resp. L-relaxed super-solution) of (1.1) if it is upper semi-continuous (resp. lower semi-continuous) and for all test functions φ touching u from above (resp. from below) at (t, x) ∈ (0, T ) × J i , we have

φ t + F i (t, x, Dφ, D 2 φ) ≤ 0 at (t, x) (resp. φ t + F i (t, x, Dφ, D 2 φ) ≥ 0 at (t, x)) if x /
∈ Γ, and

either φ t + H i (t, x, Dφ) ≤ 0 at (t, x) for some i ∈ {1, . . . , N }, or L(-φ t , ∂ 1 φ, . . . , ∂ N φ, t, x, D φ) ≤ 0 at (t, x) resp. either φ t + H i (t, x, Dφ) ≥ 0 at (t, x) for some i ∈ {1, . . . , N }, or L(-φ t , ∂ 1 φ, . . . , ∂ N φ, t, x, D φ) ≥ 0 at (t, x) if x ∈ Γ.
The following observation is important for stability and the reduction of the set of test functions. The proof contained in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] can be easily extended to generalized junction conditions. We give a short proof for the reader's convenience. Lemma 2.3 ("Weak continuity" of relaxed sub-solutions). Assume (F) and (L). Let u : (0, T ) × J → R be an L-relaxed sub-solution of (1.1). Then for all i ∈ {1, . . . , N }, and x = (x , 0) ∈ Γ, u(t, x) = lim sup (s,y)→(t,x),y∈J * i u(s, y).

Proof. Let i ∈ {1, . . . , N }. Since u is upper semi-continuous, we have for all (t, x) ∈ (0, T ) × Γ,

u(t, x) ≥ lim sup (s,y)→(t,x),y∈J * i u(s, y) =: U i (t, x).
Remark that the function U i : (0, T ) × Γ → R is upper semi-continuous. In order to prove that u = U i in (0, T ) × Γ, we assume that there exists (t * , x * ) ∈ (0, T ) × Γ such that (2.1)

u(t * , x * ) ≥ U i (t * , x * ) + δ
for some δ > 0.

The density theorem [10, Theorem 3.1] can be applied to the restriction of -u to (0, T )×Γ around (t * , x * ). Roughly speaking, this theorem claims that the proximal subdifferential (which is a subset of the viscosity subdifferential) is nonempty in a dense set. This result even ensures that there exists a point (t 0 , x 0 ) ∈ (0, T ) × Γ such that (t 0 , x 0 , -u(t 0 , x 0 )) is as close as desired to (t * , x * , -u(t * , x * )) and there exists a viscosity subdifferential of -u at (t 0 , x 0 ). More precisely, for all ε > 0, there exists a C 1 function Ψ : (0, T ) × Γ → R and (t 0 , x 0 ) ∈ (0, T ) × Γ such that Ψ strictly touches u from above at (t 0 , x 0 ) ∈ B r (t * , x * ) ∩ (0, T ) × Γ for some r > 0 and

(t 0 , x 0 ) ∈ B ε (t * , x * ) and -u(t * , x * ) -ε ≤ -u(t 0 , x 0 ) ≤ -u(t * , x * ).
In particular, u(t 0 , x 0 ) ≥ u(t * , x * ). Moreover, since U i is upper semi-continuous, we can choose ε small enough in order to ensure that U i (t 0 , x 0 ) ≤ U i (t * , x * ) + δ/2.

We now get from (2.1) that

(2.2) u(t 0 , x 0 ) ≥ lim sup (s,y)→(t 0 ,x 0 ),y∈J * i u(s, y) + δ/2.
Since the test function strictly touches u at (t 0 , x 0 ), we have Ψ -u ≥ δ 1 > 0 in a neighbourhood (with respect to (0, T ) × J) of ∂B r (t 0 , x 0 ) ⊂ (0, T ) × Γ. We now consider the test function Φ(t, x) = Ψ(t, x ) + p j x j for x ∈ J j with p j > 0 if j = i and p i < 0. Thanks to (2.2), |p i | can be chosen arbitrarily large. We now use the coercivity of the F j (see (F3)) to show that for min j |p j | large enough, Φ touches u from above at (t 0 , x 0 ). But this implies that L(-∂ t Φ(t 0 , x 0 ), p 1 , . . . , p N , t 0 , x 0 , D Φ(t 0 , x 0 )) ≤ 0 which contradicts (L4) since the min k=1,...,N p k = p i → -∞. The proof is now complete.

Stability and existence.

The following results related to stability of relaxed suband super-solutions are expected; even more, relaxed solutions are defined in such a way that they satisfy such stability properties.

In order to state the first stability result, we recall the definition of upper semi-continuous envelope u * (resp. lower semi-continuous envelope u * ) of a function u : (0, T ) × J → R, u * (t, x) = lim sup (s,y)→(t,x) u(s, y), u * (t, x) = lim inf (s,y)→(t,x) u(s, y). Proposition 2.4 (Stability of relaxed solutions -I). Assume (F) and (L). If (u α ) α is a family of relaxed sub-solutions (resp. relaxed super-solutions) of (1.1) which is locally uniformly bounded from above (resp. from below), then the upper semi-continuous (resp. lower semi-continuous) envelope of sup α u α (resp. inf α u α ) is a relaxed sub-solution (resp. relaxed super-solution) of (1.1).

Proof. We only treat the sub-solution case since the super-solution one is similar. Let u denote the upper semi-continuous envelope of sup α u α . Consider a test function φ strictly touching u from above at (t, x). There then exist a sequence (t n , x n ) → (t, x) and α n such that φ touches u αn from above at (t n , x n ). Writing the viscosity inequalities and passing to the limit yields the desired result.

In order to state the second stability result, we recall the definition of upper semi-limit ū (resp. lower semi-limit u) of a family of functions

u ε : (0, T ) × J → R, ε > 0, ū(t, x) = lim sup (s,y)→(t,x),ε→0 u ε (s, y), u(t, x) = lim inf (s,y)→(t,x),ε→0 u ε (s, y).
Proposition 2.5 (Stability of relaxed solutions -II). Assume (F) and (L). If {u ε } ε>0 is a family of relaxed sub-solutions (resp. relaxed super-solutions) of (1.1) which is locally uniformly bounded from above (resp. from below), then the relaxed upper limit (resp. relaxed lower limit) of {u ε } ε>0 is a relaxed sub-solution (resp. relaxed super-solution) of (1.1).

Proof. We only treat the sub-solution case since the super-solution one is similar. Consider a test function φ strictly touching ū from above at (t, x). We can assume that the contact is strict. There then exist a sequence (t k , x k ) → (t, x) and ε k → 0 such that φ touches u n k from above at (t k , x k ) → (t, x) as k → +∞. Either there is a subsequence k p along which x kp ∈ J * i for some i ∈ {1, . . . , N } or x k ∈ Γ for large k's. Writing the viscosity inequalities in both cases and passing to the limit yields the desired result.

The stability properties satisfied by relaxed solutions ensure the existence of discontinuous relaxed solutions. Theorem 2.6 (Existence of discontinuous relaxed solutions). Assume (F) and (L) and consider u 0 uniformly continuous. Assume also that for all R > 0,

C R := sup{|F i (t, x, p, A)| : i ∈ {1, . . . , N }, t ∈ (0, T ), x ∈ J, |p| ≤ R, |A| ≤ R} < +∞.
There exists u such that its upper semi-continuous (resp. lower semi-continuous) envelope is a relaxed sub-solution (resp. relaxed super-solution) of (1.1) such that u(0, x) = u 0 (x) for x ∈ J.

Remark 2.7. This theorem states the existence of discontinuous solutions in the sense of Ishii [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF].

Proof. In view of the stability results, it is enough to construct a solution for some initial datum u 0 such that u i 0 = u 0 | J i are in C 1,1 . For such u 0 's, we can construct barriers in the classical way:

u ± (t, x) = u 0 (x) ± Ct. For C ≥ C R 0 with R 0 ≥ Du i 0 ∞ + D 2 u i 0 ∞
for all i = 1, . . . , N , the function u + is a relaxed super-solution while u -is a relaxed sub-solution. Indeed, as far as the equations in J i are concerned, it is classical; as far as the junction condition is concerned, the equation is satisfied up to Γ and thus u ± are relaxed semisolutions on Γ. We then consider W the set of all functions lying below u + whose upper semi-continuous envelope is a relaxed sub-solution. Then the supremum of w ∈ W is in W and it is maximal. Let w denote this maximal element. If the lower semi-continuous envelope is not a relaxed super-solution, there exists a test function φ and a point (t, x) such that φ touches w * from below at (t, x) without satisfying the corresponding viscosity inequality. This implies φ < (u + ) * in a neighbourdhood of (t, x) and we can prove that φ is a relaxed sub-solution in the same neighbourhood. Then we can construct a relaxed sub-solution w δ which is not below w, contradicting its maximality.

2.4. Flux-limited solutions. It is proved in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] that, in the special case where L = L A defined in (1.2) and for first order Hamilton-Jacobi equations, relaxed solutions satisfy the junction condition in a strong sense, which is made precise in the following definition.

Definition 2.8 (Flux-limited solutions). Given a function

A : (0, T ) × Γ × R d → R such that A ≥ A 0 ,
a function u : (0, T ) × J → R is a A-flux-limited sub-solution (resp. A-fluxlimited super-solution) of (1.1) if it is upper semi-continuous (resp. lower semi-continuous) and for any test function φ in the sense of Definition 2.1 touching u from above (resp. from below) at (t, x) ∈ (0, T ) × J i , we have

φ t + F i (t, x, Dφ, D 2 φ) ≤ 0 at (t, x) resp. φ t + F i (t, x, Dφ, D 2 φ) ≥ 0 at (t, x) if x /
∈ Γ, and

L A (-φ t , ∂ 1 φ, . . . , ∂ N φ, t, x , D φ) ≤ 0 at (t, x) resp. L A (-φ t , ∂ 1 φ, . . . , ∂ N φ, t, x , D φ) ≥ 0 at (t, x) if x ∈ Γ.
Remark 2.9. When proving that a function is a sub-solution or a super-solution of (1.1) at one given point of (0, T ) × Γ, it is enough to consider a reduced set of test functions associated with this specific point. It is thus interesting to consider sub-or super-solution of (1.1) at only one point of (0, T ) × Γ -see Theorem 3.2 about the reduction of the set of test functions.

The following proposition asserts that L A -relaxed solutions coincide with A-flux-limited solutions. It was proved in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] in the case of first order equations. We point out that the multidimensional proof of [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF] applies without any change to degenerate parabolic equations satisfying (F). Proposition 2.10 (L A -relaxed solutions are A-flux-limited solutions - [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF]). Assume (F) and (L). Then any L A -relaxed sub-solution (resp. super-solution) of (1.1) is an A-fluxlimited sub-solution (resp. super-solution) of (1.1).

Reduced set of test functions for flux-limited solutions

In this section, we explain why it is sufficient to consider a reduced set of test functions in order to check that a function is a flux-limited (sub/super)solutions of (1.1). Such a result is used in an essential way when proving Theorem 1.9.

Definition 3.1 (Reduced test functions). Consider a flux limiter

A ≥ A 0 and a point (t 0 , x 0 ) ∈ (0, T ) × Γ. A function ϕ : (0, T ) × J → R is a reduced test function for (1.1) at (t 0 , x 0 ) if there exists a function φ ∈ C 1 ((0, T ) × R d ) and N functions φ i ∈ C 1 ([0, +∞)), i = 1, . . . , N , such that ∀t ∈ (0, T ), ∀(x , x i ) ∈ J i , ϕ(t, (x , x i )) = φ(t, x ) + φ i (x i )
and, for all i = 1, . . . , N , φ i (0) = 0 and the slope p i = φ i (0) and the tangential gradient p = D φ(t 0 , x 0 ) satisfy

(3.1) H i (t 0 , x 0 , p , p i ) = H + i (t 0 , x 0 , p , p i ) = A(t 0 , x 0 , p ) that is to say p i ∈ [π +
i (t 0 , x 0 , p ), π+ i (t 0 , x 0 , p )]. Theorem 3.2 below generalizes the one contained in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. In order to state it, we need to consider the equation on each (open) branch i, i.e. away from the junction hyperplane Γ:

(3.2) u t + F i (t, x, Du, D 2 u) = 0, (t, x) ∈ (0, T ) × J * i .
We can now state and prove the following theorem.

Theorem 3.2 (Reducing the set of test functions). Assume (F) and consider a function

A : (0, T ) × Γ × R d → R such that A ≥ A 0 . Given a function u : (0, T ) × J → R, the following properties hold true. i) If, for all i ∈ {1, . . . , N }, u is a sub-solution of (3.2) and for (t, x) ∈ (0, T ) × Γ, (3.3) u(t, x) = lim sup s→t,y→x,y∈J * i u(s, y),
then u is an A 0 -flux limited sub-solution of (1.1) at (t, x). ii) If, for all i ∈ {1, . . . , N }, u is a sub-solution of (3.2) satisfying (3.3) and if for any reduced test function ϕ in the sense of Definition 3.1 touching u from above at (t, x) ∈ (0, T ) × Γ, we have

ϕ t (t, x) + A(x , D ϕ(t, x)) ≤ 0,
then u is an A-flux-limited sub-solution of (1.1) at (t, x).

iii) If, for all i ∈ {1, . . . , N }, u is a super-solution of (3.2) and if for any reduced test function ϕ in the sense of Definition 3.1 touching u from below at (t, x) ∈ (0, T ) × Γ we have ϕ t (t, x) + A(x, D ϕ(t, x)) ≥ 0, then u is an A-flux-limited super-solution of (1.1) at (t, x).

Remark 3.3. In the previous statement, functions are flux-limited solution of (1.1) at only one point of (0, T ) × Γ -see Remark 2.9 above.

Proof. The proof of [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Theorem 2.7] applies here without any change after proving the two lemmas 3.4 and 3.5 about critical normal slopes. Indeed, with such technical results in hands, the proof focuses on what happens on Γ and second derivatives do not appear any more.

Lemma 3.4 (Super-solution property for the critical normal slope on each branch). Let i ∈ {1, . . . , N } be fixed. Let u : (0, T ) × J i → R be a lower semi-continous super-solution of (3.2). Let φ be a test function touching u from below at some point (t 0 , x 0 ) ∈ (0, T ) × Γ. We consider

p i = sup{p ∈ R : ∃r > 0, φ(t, x) + px i ≤ u(t, x) for (t, x) ∈ B r (t 0 , x 0 ) ∩ (0, T ) × J i }.
If p i < +∞, then we have

φ t + H i (t, x, D φ, ∂ i φ + p i ) ≥ 0 at (t 0 , x 0 ) with p i ≥ 0.
Lemma 3.5 (Sub-solution property for the critical normal slope on each branch). Let i ∈ {1, . . . , N } be fixed. Let u : (0, T ) × J i → R be a sub-solution of (3.2). Let φ be a test function touching u from above at some point (t 0 , x 0 ) ∈ (0, T ) × Γ. We consider

p i = inf{p ∈ R : ∃r > 0, φ(t, x) + px i ≥ u(t, x) for (t, x) ∈ B r (t 0 , x 0 ) ∩ (0, T ) × J i }. If u satisfies (3.4) u(t 0 , x 0 ) = lim sup s→t 0 ,y→x 0 ,y∈J * i u(s, y),
then p i > -∞; moreover, we have in this case

φ t + H i (t, x, D φ, ∂ i φ + p i ) ≤ 0 at (t 0 , x 0 ) with p i ≤ 0.
We first prove Lemma 3.4.

Proof of Lemma 3.4. The proof follows the same lines of [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Lemma 2.8]. From the definition of p i , for all ε > 0 small enough, there exists δ = δ(ε) ∈ (0, ε) such that (3.5) u(s, y) ≥ φ(s, y) + (p i -ε)y i for all (s, y) ∈ B δ (t 0 , x 0 ) ∩ (0, T ) × J i and there exists (t ε , x ε ) ∈ B δ/2 (t 0 , x 0 ) such that

u(t ε , x ε ) < φ(t ε , x ε ) + (p i + ε)x i ε .
We choose a smooth function Ψ :

R d+2 → [-1, 0] such that Ψ = 0 in B 1/2 (t 0 , x 0 ) -1 outside B 1 (t 0 , x 0 ).
We define for (s, y) ∈ (0, T )

× J i , Φ(s, y) = φ(s, y) + 2εΨ δ (s, y) + (p i + ε)y i with Ψ δ (Y ) = δΨ Y δ . Remark that for (s, y) ∈ ∂(B δ (t 0 , x 0 ) ∩ (0, T ) × J i ), we have y i ≤ δ. In particular, -2εδ + (p i + ε)y i ≤ (p i -ε)y i for such (s, y). Hence (3.5) implies Φ(s, y) = φ(s, y) -2εδ + (p i + ε)y i ≤ u(s, y) for (s, y) ∈ ∂(B δ (t 0 , x 0 ) ∩ (0, T ) × J i ), Φ(s, x) ≤ φ(s, x) ≤ u(s, x) for (s, x) ∈ (t 0 -δ, t 0 + δ) × Γ and Φ(t ε , x ε ) = φ(t ε , x ε ) + (p i + ε)x i ε > u(t ε , x ε ). We conclude that there exists a point (t ε , x ε ) ∈ B δ (t 0 , x 0 ) ∩ ((0, T ) × J * i ) such that u -Φ reaches a minimum in B δ (t 0 , x 0 ) ∩ ([0, T ] × J i ).
We thus can write the viscosity inequality

Φ t + F i (t, x, DΦ, D 2 Φ) ≥ 0 at (t ε , x ε ) which reads (3.6) φ t (t ε , x ε ) + 2ε(Ψ δ ) t (t ε , x ε ) +F i (t ε , x ε , (D φ+2εD Ψ δ )(t ε , x ε ), ∂ i φ(t ε , x ε )+2ε∂ i Ψ δ (t ε , x ε )+p i +ε, D 2 φ+2εD 2 Ψ δ (t ε , x ε )) ≥ 0.
We now send ε → 0 in the above inequality; recall that δ ∈ (0, ε) and Ψ δ = δΨ(•/δ); in particular,

(3.7) ε(Ψ δ ) t (t ε , x ε ), εD Ψ δ (t ε , x ε ), ε∂ i Ψ δ (t ε , x ε ) → 0 as ε → 0.
As far as second derivatives are concerned, we have

|εD 2 Ψ δ | ≤ D 2 Ψ ∞ .
In particular,

(3.8) εD 2 Ψ δ (t ε , x ε )) → B ∈ S d+1 (R)
along a subsequence. Since (t ε , x ε ) → (t 0 , x 0 ), we finally get from (3.6), (3.7) and (3.8) that φ t (t 0 , x 0 ) + F i (t 0 , x 0 , D φ(t 0 , x 0 ), ∂ i φ(t 0 , x 0 ) + p i , D 2 φ(t 0 , x 0 ) + B) ≥ 0 which is the desired inequality since x 0 ∈ Γ and F i satisfies (F4). The proof is now complete.

We now turn to the proof of Lemma 3.5

Proof of Lemma 3.5. The main difference with the previous lemma is the claim that the critical normal slope is finite. This is the reason why we only explain this point. Here again, we follow closely [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. Let p ∈ (-∞, 0] be such that there exists r > 0 such that φ

+ px i ≥ u in B = B r (t 0 , x 0 ) ∩ (0, T ) × J i . Remark first that, replacing φ with φ + (t -t 0 ) 2 + |x -x 0 | 2 if necessary, we can assume that (3.9) u(t, x) < φ(t, x) + px i if (t, x) = (t 0 , x 0 ).
In particular, there exists δ > 0 such that φ + px i ≥ u + δ on ∂B \ Γ. Since u satisfies (3.4), there exists (t ε , x ε ) → (t 0 , x 0 ) such that x ε ∈ J * i and u(t 0 , x 0 ) = lim ε→0 u(t ε , x ε ).

We now introduce the following perturbed test function

Ψ(t, x) = φ(t, x) + px i + η
x i where η = η(ε) is a small parameter to be chosen later. Let (s ε , y ε ) realize the infimum of Ψ -u in B. In particular,

(3.10) (φ + px i -u)(s ε , y ε ) ≤ Ψ(s ε , y ε ) -u(s ε , y ε ) ≤ Ψ(t ε , x ε ) -u(t ε , x ε ) → 0 as ε → 0 as soon as η(ε) = o(x i ε ) with x ε = (x ε , x i ε ).
In particular, in view of (3.9), this implies that (s ε , y ε ) → (t 0 , x 0 ) as ε → 0. Since u is a sub-solution of (3.2), we know that

φ t (s ε , y ε ) + F i (s ε , y ε , D φ(s ε , y ε ), ∂ i φ(s ε , y ε ) + p - η (y i ε ) 2 , D 2 φ(s ε , y ε ) + 2η (y i ε ) 3 e d+1 ⊗ e d+1
) ≤ 0 (where (e 1 , . . . , e d+1 ) is an orthonormal basis of R d+1 and e d+1 is orthogonal to Γ). Use now (F2) in order to get

φ t (s ε , y ε ) + F i (s ε , y ε , D φ(s ε , y ε ), ∂ i φ(s ε , y ε ) + p - η (y i ε ) 2 , D 2 φ(s ε , y ε )) ≤ 2C i η y i ε . Remark now that (3.10) implies η y i ε - η x i ε ≤ p(x i ε -y i ε ) + (u -φ)(s ε , y ε ) -(u -φ)(t ε , x ε ) → 0 as ε → 0.
Recalling that η is chosen so that η/x i ε → 0 as ε → 0, we thus get η y i ε → 0 as ε → 0.

In particular, the coercivity of F i (see (F3)) implies that p -η (y i ε ) 2 is bounded as ε → 0. Hence we can pass to the limit as ε → 0 in the viscosity inequality and get φ t (t 0 , x 0 ) + H i (t 0 , x 0 , D φ(t 0 , x 0 ), ∂ i φ(t 0 , x 0 ) + p 0 ) ≤ 0 where p 0 ∈ (-∞, 0] is any accumulation point of p-η (y i ε ) 2 as ε → 0. The previous inequality and (F3) implies in particular that p 0 is bounded from below by a constant C which only depends on H i , φ t , Dφ at (t 0 , x 0 ). Indeed, (F3) implies in particular that lim |p|→+∞ inf (t,x )∈(0,T )×Γ H i (t, x , p) = +∞. But this also implies that p ≥ C and, in turn, p i ≥ C. The proof is now complete.

Proof of the main theorem

This section is devoted to the proof of the first main result, Theorem 1.9. Throughout this section, we do not write the (t, x , p ) dependence of A L , π + , π+ etc. (see (1.7) and (1.8) for a definition) in order to clarify the presentation and proofs.

The proof of Theorem 1.9 relies on properties and other representations of the effective flux limiter A L ; we gather them in the following preparatory proposition. Proof. Remark that p 0 i in Definition 1.7 coincides with π + i (A 0 ). In particular, if L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) ≤ 0 then Definition 1.7 says that A L = A 0 . This proves i).

. i) If L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) ≤ 0 then A L = A 0 . ii) If L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) > 0 then A L is
) = 0. iii) If L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) > 0, then A L = sup{λ ≥ A 0 : L(λ, π + 1 (λ), . . . , π + N (λ)) > 0} (4.1) A L = inf{λ ≥ A 0 : L(λ, π+ 1 (λ), . . . , π+ N (λ)) < 0}. (4.2) iv) Moreover, if L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) > 0, we also have L(A L , π + 1 (A L ), . . . , π + N (A L )) ≥ 0 (4.3) L(A L , π+ 1 (A L ), . . . , π+ N (A L )) ≤ 0. ( 4
We now assume that L(A 0 , π + 1 (A 0 ), . . . , π + N (A 0 )) > 0. Assumption (L5) implies that there exists λ > A 0 such that L( λ, π+ 1 ( λ), . . . , π+ N ( λ)) < 0. In particular, the two following quantities are finite,

S := sup{λ ≥ A 0 : L(λ, π + 1 (λ), . . . , π + N (λ)) > 0} I := inf{λ ≥ A 0 : L(λ, π+ 1 (λ), . . . , π+ N (λ)) < 0}. Using that π + i is left
* i ∈ [π + i (A L ), π+ i (A L )] such that L(A L , p * 1 , . . . , p * N ) = 0.
which contradicts (4.3). The reader may remark that the contradiction cannot be reached without the use of ε. We now consider

L A L -ε (-∂ t ϕ, ∂ 1 ϕ, . . . , ∂ N ϕ, x 0 , D ϕ) = ∂ t φ(t 0 , x 0 ) + max(A L -ε, max i H - i (π * i (A L -ε))).
where the derivatives of ϕ in the left hand side are computed at (t 0 , x 0 ). Remark now that H - i (π * i (A L -ε)) = min p i ∈R H i (p i ) and in particular max i H - i (π * i (A Lε)) = A 0 . Since A L -ε > A 0 and λ = -∂ t φ(t 0 , x 0 ), the previous equality and (4.8) yield

L A L -ε (-∂ t ϕ, ∂ 1 ϕ, . . . , ∂ N ϕ, x 0 , D ϕ) = -λ + A L (x 0 , p 0 ) -ε ≤ 0
which is the desired inequality. The proof is now complete.

The vanishing viscosity limit

This section is devoted to the study of the limit (as ε → 0) of the solution u ε of the following Hamilton-Jacobi equation posed on a multi-dimensional junction J, (5.1)

u ε t + H i (t, x, Du ε ) = ε∆u ε (t, x) ∈ (0, T ) × J * i , L(-u ε t , ∂ 1 u ε , . . . , ∂ N u ε , t, x , D u ε ) = 0 (t, x) ∈ (0, T ) × Γ subject to the initial condition (5.2) u(0, x) = u 0 (x), x ∈ J.
Notice that this equation is not of the form (1.1) since the diffusion does not degenerate along the junction hyperplane. In particular, Theorem 3.2 does not hold true anymore in this case since it uses the degeneracy along Γ in an essential way. Still, we can consider relaxed solutions as in Definition 2.2, even if we expect solutions to be classical -see Remark 5.2 below. As we shall see it, the solutions u ε converge towards the solution of (5.3)

u t + H i (t, x, Du) = 0 (t, x) ∈ (0, T ) × J * i , L(-u t , ∂ 1 u, . . . , ∂ N u, t, x , D u) = 0 (t, x) ∈ (0, T ) × Γ.
The first result applies to general junction functions L. Theorem 5.1 (Vanishing viscosity limit). Assume (L) and

     H i continuous ∀(t, x ) ∈ (0, T ) × Γ, λ ∈ R, {p = (p , p i ) ∈ R d+1 : H i (t, x , p) ≤ λ} convex lim |p|→+∞ inf (t,x )∈(0,T )×Γ H i (t, x , p) = +∞.
Let u 0 be uniformly continuous in J. Assume there exists a relaxed solution u ε of (5.1), (5.2) and a constant C such that |u ε (t, x) -u 0 (x)| ≤ Ct for all (t, x) ∈ (0, T ) × J. Then u ε converges locally uniformly towards the unique relaxed solution u of (5.3), (5.2).

Remark 5.2. Even if we will not discuss it, the existence of solutions whose restriction to

J i are C 1,1 (J i ) ∩ C 2 (J * i )
is expected in the case of (5.1). Some results are proved in [START_REF] Von | A maximum principle for semilinear parabolic network equations[END_REF][START_REF] Von | An existence result for semilinear parabolic network equations with dynamical node conditions[END_REF] on compact junctions and some others are announced in [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF].

= min p i H i (p i ) but in general A 0 ≥ min p i H i (p i ).
We now prove the following theorem, which is equivalent to Theorem 1.12.

Theorem 5.8 (The vanishing viscosity limit selects the maximal Ishii solution). Assume

     H i continuous {p ∈ R d+1 : H i (t, x , 0, p) ≤ λ} convex for all λ ∈ R, lim |p|→+∞ inf (t,x )∈(0,T )×Γ H i (t, x , p) = +∞.
Then the relaxed solution u ε of (5.1), (5.2) converges towards the unique

A - I -flux-limited solution of u t + H i (x, Du) = 0, x ∈ J * i u(0, x) = u 0 (x), x ∈ J.
Proof. Once again, the tangential variables (t, x , p ) are not shown in order to clarify the presentation.

In view of Corollary 5.4, we only have to prove that A e = A - I where A - I is given by Proposition 5.6.

If π 0 1 + π 0 2 ≥ 0, then we know on the one hand from Proposition 5.6 that A - I = A 0 and on the other hand, since p 0 1 + p 0 2 ≥ π 0 1 + π 0 2 ≥ 0, we know from Corollary 5.4 that A e = A 0 . We thus conclude that A e = A 0 = A - I in this case. We now assume that π 0 1 + π 0 2 ≤ 0. In particular, Proposition 5.6 implies that (5.5)

A - I = A + I = max(A 0 , A * ) with A * = max q∈[π 0 2 ,-π 0 1 ]
min(H 1 (-q), H 2 (q)).

Remark that the function H 2 is non-decreasing on the interval [π 0 2 , -π 0 1 ] and the function H1 (q) = H 1 (-q) is non-increasing. We are going to distinguish three cases as shown in Figure 5. Either the graphs of H 2 and H1 do not intersect on the interval [π 0 2 , -π 0 1 ] and H 2 is above (Case 1), or they do intersect (Case 2), or they do not intersect and H1 is above (Case 3). To distinguish cases, it is enough to compare the values of H1 and H 2 at the boundary of the interval.

It is useful to introduce

A 1 = min p 1 ∈R H 1 (p 1 ) and A 2 = min p 2 ∈R H 2 (p 2 ). Recall that A 0 = max(A 1 , A 2 ).
In Case 1, we have H 2 (π 0 2 ) = A 0 = A 2 ≥ H1 (π 0 2 ). It implies that H1 ≤ H 2 on the interval. In particular A * = H1 (π 0 2 ) ≤ A 0 . On the one hand, (5.5) implies that A - I = A 0 . On the In Case 2, there exists q I ∈ [π 0 2 , -π 0 1 ] such that A * = H 2 (q I ) = H 1 (-q I ) and A * ≥ A 0 . (5.5) implies that A - I = A * . But the fact that q I ≥ π 0 2 such that A * = H 2 (q I ) implies that q I = π + 2 (A * ); similarly, -q I = π + 1 (A * ); hence π + 1 (A * ) + π + 2 (A * ) = 0 with A * ≥ A 0 . We thus have from Corollary 5.4 that A e = A * . Hence A - I = A e in Case 2. In Case 3, A 0 = A 1 ≥ A * . (5.5) implies that A - I = A 0 . We also remark that -p 0 1 = -π 0 1 ≤ π + 2 (A 0 ) = p 0 2 (have a look at the picture). In particular, we have from Corollary 5.4 that A e = A 0 . We thus conclude that A - I = A e in Case 3. The proof is now complete.

A large deviation problem

In [START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF], the authors study large deviation problems related to diffusion processes whose drift is smooth on either side of a hyperplane. Their proofs rely on probability tools and ideas. Our goal in this section is to propose an analytical/PDE proof. Furthermore, by using the results of previous sections, the rate function is related to the maximal Ishii solution of a Hamilton-Jacobi equation.

Consider the stochastic differential equation in R d+1 , dX ε (t) = b(X ε (t))dt + ε 1/2 σ(X ε (t))dW (t), X ε (0) = x 0 , 0 ≤ t ≤ 1 (6. with a i = σ i σ T i . Corresponding Lagrangians L1 and L2 are related to Hamiltonians H1 and H2 by the following formula [START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF] Hi (x, p) = sup q∈R d+1

{-pq -Li (x, q)}.

Set Ω 1 = R d × (-∞, 0), Ω 2 = R d × (0, +∞), H = R d × {0}. where L0 is defined by L0 (x, p , q) = inf λ L1 (x, p , q 1 ) + (1 -λ) L2 (x, p , q 2 ), λ ∈ [0, 1], q 1 ≥ 0, q 2 ≤ 0, λq 1 + (1 -λ)q 2 = q .

Call Σ x 0 the set of all absolutely continuous function φ ∈ C([0, 1], R d+1 ) satisfying φ(0) = x 0 . For any φ ∈ Σ x 0 , we define the rate function I x 0 (φ) as follows, where L is defined as in (6.2). We first state the Laplace principle as presented in [START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF] Definition 6.1. Let {Y ε (t), ε > 0, 0 ≤ t ≤ 1} with Y ε (0) = x 0 be a family of random variables taking values in a Polish space Y and let I x 0 be a rate function defined as in (6.3). We say that {Y ε } satisfies a Laplace principle with the rate function I x 0 if, for every bounded continuous function h mapping Y into R, we have

lim ε→0 ε ln E x 0 exp - h(Y ε ) ε = -inf φ∈Σx 0
{h(φ(1)) + I x 0 (φ)}. (6.4) In [START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF], the following large deviation result is proved using probabilistic arguments. We will give a PDE proof. Theorem 6.2 ([8]). Assume that    b i is continuous, σ is continuous and such that σσ T ≥ cI with c > 0, (6.1) has a unique strong solution where I is the identity matrix. Then the family {X ε , ε > 0} satisfies the Laplace principle in C([0, 1], R d+1 ) with the rate function I x 0 as defined in (6.3).

Proof. Given a function h, let h ε denote exp( -h ε ). The function u ε given by

u ε (t, x) = E x (h ε (X ε (t)))
is a solution of

       ∂uε ∂t = ε 2 Trace(a(x)D 2 u ε ) + b(x)Du ε , t ∈ (0, 1), x ∈ Ω 1 Ω 2 1 2 ∂ d+1 u ε (t, x , 0 + ) = 1 2 ∂ d+1 u ε (t, x , 0 -), x ∈ H u ε (0, x) = h ε (x), x ∈ Ω 1 Ω 2
(where a = σσ T ) The function v ε = -ε ln(u ε ) satisfies

       ∂vε ∂t = ε 2 Trace(a(x)D 2 v ε ) -1 2 a(x)Dv ε , Dv ε + b(x)Dv ε , t ∈ (0, 1), x ∈ Ω 1 Ω 2 1 2 ∂ d+1 v ε (t, x , 0 + ) = 1 2 ∂ d+1 v ε (t, x , 0), x ∈ H v ε (0, x) = h(x), x ∈ Ω 1 Ω 2 .
Moreover, in view of the definition of u ε and v ε , we have

v ε (t, x) = -ε ln E x exp -h(X ε (t)) ε .
Hence, our goal is to prove that where I x is defined in (6.3). We know from Theorem 1.12 that v ε converges locally uniformly towards the maximal Ishii solution U + of (6.5) ∂U + ∂t + Hi (x, DU + ) = 0, x ∈ Ω i , t ∈ (0, 1) U + (0, x) = h(x),

x ∈ Ω 1 Ω 2 .

It thus remains to prove that {pq -l i (x, q)} with l i (x, -q) = Li (x, q), here l i corresponds to the running costs considered in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Section 6]. In view of the definition of L0 recalled above, we have L0 (x, q , 0) = inf λ L1 (x, q , q 1 ) + (1 -λ) L2 (x, q , q 2 ), 0 ≤ λ ≤ 1, q 1 ≥ 0, q 2 ≤ 0, λq 1 + (1 -λ)q 2 = 0 = inf λl 1 (x, q , v 1 ) + (1 -λ)l 2 (x, q , v 2 ), 0 ≤ λ ≤ 1, v 1 ≤ 0, v 2 ≥ 0, λv 1 + (1 -λ)v 2 = 0 .

Hence, the formula of U + given in [START_REF] Imbert | Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF][START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF] coincides with (6.6). The proof is now complete.

Figure 1 .

 1 Figure 1. A parabolic equation posed on a multi-dimensional junction.Here there are 3 branches (N = 3) and the tangential dimension is 1 (d = 1). We did not illustrate the junction condition L = 0 on the junction hyperplane Γ (which is a line in this example).

  y, p, B + λe d+1 ⊗ e d+1 ) ≥ F i (s, y, p, B) -C i,R λ|y i | 2 . (F3). For all R > 0, lim |p|→+∞ inf t∈(0,T ),|x|+|B|≤R F i (t, x, p, B) = +∞.

Figure 2 .

 2 Figure 2. This figure illustrates Example 1.11 where N = 1 (number of branches) and d = 0 (dimension of the tangential space). The effective flux limiter A is determined in each case by looking at the points where the vertical line {p = 0} intersects the graph of the Hamiltonian H = H 1 ; the variable p reduces here to p 1 in the general setting.

Figure 3 .

 3 Figure 3. Non-increasing part H - i of a Hamiltonian H i . The Hamiltonian is in black, the monotone part in red. The tangent variables (t, x , p ) are not shown. In this example, the minimum of H i is lower than A 0 .

Figure 4 .

 4 Figure 4. Non-decreasing part H + i of a Hamiltonian H i . The Hamiltonian is in black, the monotone part in red. The tangent variables (t, x , p ) are not shown.

Proposition 4 . 1 (

 41 Representations of A L ). Let A L be the effective flux limiter given by Definition 1.7

  continuous and π+ i is right continuous, we have L(S, π + 1 (S), . . . , π + N (S)) ≥ 0 (4.5) L(I, π+ i (I), . . . , π+ N (I)) ≤ 0. (4.6) Proving ii), iii) and iv) (apart from uniqueness in ii)) reduces to proving that S = I. Indeed, if S = I then: iii) is proved with A L = I = S; (4.3) and (4.4) are satisfied with A L = I = S; the continuity of L (see (L1)) and the two previous inequalities imply the existence of p

Figure 5 . 2 (

 52 Figure 5. Three cases: Case 1 (left), Case 2 (center), Case 3 (right)

  1) with b(x) = b 1 (x) if x d+1 < 0 b 2 (x) if x d+1 > 0 and σ(x) = σ 1 (x) if x d+1 < 0 σ 2 (x) if x d+1 > 0In order to introduce the rate function, we have to define first Hamiltonians and Lagrangians. Hamiltonians are defined in[START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF] by Hi (x, p) = 1 2 a i (x)p, p -b i (x)p, x, p ∈ R d+1

  , p), x ∈ Ω 1 , L2 (x, p), x ∈ Ω 2 , L0 (x, p), x ∈ H,

lim ε→0 v ε ( 1 ,

 1 x) = inf φ∈Σx {h(φ(1)) + I x (φ)}

(6. 6 )

 6 U + (1, x) = inf φ∈Σx {h(φ(1)) + I x (φ)}.In view of the definition of Lagrangians and Hamiltonians from[START_REF] Boué | Large deviations for small noise diffusions with discontinuous statistics[END_REF] recalled above, we have Hi (x, p) = sup q∈R d+1

  x , p ) is the only real number λ ≥ A 0 (t, x , p ) such that there exists p + i ≥ p 0 i with H i (t, x , p , p + Remark 1.8. We will give in Section 4 other representations of A L -see Proposition 4.1. We note that if L satisfies (L) then λ is unique. But the p + i are not (in general) -see the case on the right at the top of Figure2in Example 1.11 below.

i ) = λ and L(λ, p + 1 , . . . , p + N , t, x , p ) = 0.

  well defined: there exists a unique λ * ∈ R and there exist p * i ∈ [π + i (λ * ), π+ i (λ * )] (not necessarily unique) such that L(λ * , p * 1 , . . . , p * N

  .4)

	Remark 4.2. We point out that p * i ∈ [π + i (λ), π+ i (λ * ) is equivalent to p * i ≥ p 0 i and L(λ * , p * 1 , . . . , p * N ) =
	0.

  (t, x , p , p d+1 ), H 1 (t, x , p , -p d+1 )

	where
	A * (t, x , p ) = min(H 2 and I(t, x , p ) = [min(-π 0 max p d+1 ∈I(t,x ,p ) 1 (t, x , p ), π 0 2 (t, x , p )), max(-π 0 1 (t, x , p ), π 0 2 (t, x , p ))].
	Remark 5.7. The functions p 0 i and π 0 i are different. The Hamiltonian H i achieves its minimum at π 0 i and it reaches the value A 0 at p 0 i . The only case where these functions
	coincide is when A 0

Acknowledgement. The authors thank Russell Schwab for fruitful discussions in early stage of this work and for suggesting to address the large deviation problem. They also thank Guy Barles for stimulating discussions about the vanishing viscosity limit. The authors are also indebted to one of the referrees who read very carefully two successive versions of this work and made valuable recommandations about both presentation and proofs.

If I < S, then π+ i (I) < π + i (S) for all i ∈ {1, . . . , N }; but (L3) and (4.5) then imply that L(I, π+ 1 (I), . . . , π+ N (I)) > L(S, π + 1 (S), . . . , π + N (S)) ≥ 0 which contradicts (4.6). Then S ≤ I.

If S < I then the definitions of S and I imply that for all λ * ∈]S, I[, L(λ * , π + 1 (λ * ), . . . , π + N (λ * )) ≤ 0 L(λ * , π+ 1 (λ * ), . . . , π+ N (λ * )) ≥ 0. But using the continuity of L (see (L1)), this implies that for all λ * ∈]S, I[, there exist

. But this cannot be true for two different λ * 's because of (L3). Hence S = I. Notice that we can prove in the same way uniqueness in ii). The proof is now complete.

We now prove the main theorem.

Proof of Theorem 1.9. Let A L be the effective flux limiter in the sense of Definition 1.7. It is well defined thanks to Proposition 4.1. Since A L ≥ A 0 , the coercivity is clear: lim |p|→+∞ inf x ∈Γ A(t, x , p ) = +∞. The proof of the continuity of A L and the convexity of sublevel sets is the same as in [17, Proof of Theorem 2.13].

We only deal with the sub-solution case since the super-solution case is very similar. If A L (t, x , p ) = A 0 (t, x , p ), then Lemma 2.3 and Theorem 3.2 imply that any L-relaxed sub-solution of (1.1) is an A 0 -flux limited sub-solution of (1.1).

We now consider the case where there exists (t, x , p ) such that A L (t, x , p ) > A 0 (t, x , p ). Let u be an L-relaxed sub-solution of (1.1) and let us prove that it is an (A L -ε)-fluxlimited sub-solution of (1.1) at (t, x ) ∈ (0, T ) × Γ for all ε > 0 such that A L -ε > A 0 (at (t, x , p )). We use here the fact that Theorem 3.2 is local in the sense that it asserts that a function is a flux-limited solution at one given point (t, x ) ∈ (0, T ) × Γ. In view of Lemma 2.3 and Theorem 3.2, we only have to consider a reduced test function ϕ touching u from above at (t, x ) ∈ (0, T ) × Γ. We recall that

In order to emphasize the interval in which φ i (0) lies, we write π * i (A L -ε) := φ i (0). By definition of relaxed solutions, we have either

We claim that (4.8) always holds true. We argue by contradiction by assuming that (A L -ε) > λ. In particular A L > λ and π + i (A L ) > π * i (A L -ε) for i = 1, . . . , N . Using (L3) and (4.7) successively, we have

As we previously mentioned it, a special case of the theorem is proved in [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF].

Proof of Theorem 5.1. By discontinuous stability, the relaxed upper limit ū of u ε is an Lrelaxed sub-solution of (5.3), i.e. an A L -flux-limited sub-solution of (5.3) (by Theorem 1.9). The relaxed lower limit u is an L-relaxed super-solution of (5.3), i.e. an A L -flux-limited super-solution of (5.3) (by Theorem 1.9 again). Moreover, the fact that |u ε (t, x) -u 0 (x)| ≤ Ct holds true for all (t, x) ∈ (0, T ) × J implies that ū(0, x) = u 0 (x) = u(0, x) for all x ∈ J. By comparison principle [17, Theorem 1.3], we conclude that ū ≤ u which yields the local uniform convergence towards the unique A L -flux-limited solution of (5.3), (5.2) which coincides with the relaxed solution (by Theorem 1.9).

Problem (1.4) can be translated into the junction framework as follows,

x ∈ J with H 1 (x, p , p d+1 ) = H1 (x, p , -p d+1 ) and H 2 (x, p , p d+1 ) = H2 (x, p , p d+1 ). In view of Theorem 5.1, u ε converges towards the relaxed solution (5.4)

x ∈ J associated with the generalized flux function L e (p 0 , p 1 , p 2 , t, x , p ) = -p 1 -p 2 .

Corollary 5.4 (The vanishing viscosity limit for the Kirchoff condition). The solution u ε of (5.1), (5.2) converges towards the A e -flux-limited solution of (5.4) where A e (t, x , p ) is determined as follows: if p 0 1 (t, x , p ) + p 0 2 (t, x , p ) ≥ 0 then A e (t, x , p ) = A 0 (t, x , p ); else A e (t, x , p ) is the unique λ ≥ A 0 (t, x , p ) such that there exist p