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Uniqueness structure of weakly coupled systems of ergodic problems of Hamilton-Jacobi equations

Here, we address a uniqueness structure of viscosity solutions for ergodic problems of weakly coupled Hamilton-Jacobi systems. In particular, we study comparison principle with respect to generalized Mather measures as a generalization of the result proved by Mitake and Tran, which addressed the case of a single equation. To get the main result, it is important to construct Mather measures effectively. We overcome this difficulty by nonlinear adjoint methods.


Introduction
In this paper, we consider the following weakly coupled Hamilton-Jacobi system: (A3) There exists C > 0 such that for all (x, p, i) ∈ T d ×R d ×I, |D x H(x, p, i)| C(1+|p| 2 ).
(A4) For all i, j ∈ I, c ij = c ji . It is known that there exists unique λ ∈ R such that (1.1) has viscosity solutions. Hence, by resetting the Hamiltonian, we can assume that the ergodic constant λ = 0 without loss of generality; see [5] and [18], for instance.
Weakly coupled Hamilton-Jacobi systems arise, for example, in the literature of optimal control problems with random switching costs, which are governed by specific Markov chains. These systems were discussed for a long time in the context of PDE theory; see [2], [12], and [21], for instance. In particular, [9] and [15] established the framework of viscosity solutions for these systems. To analyze the large time behavior of the solution for timedependent problems, the ergodic problems like (1.1) are derived; see [5] and [18].
First, we recall the case of the single equation, that is m = 1. It is known that (1.1) has multiple solutions, not even up to constant (for example, [16] Chapter 6). Therefore, it is important to investigate the structure of solutions for (1.1). In [3] and [4], the above nonuniqueness phenomena were studied in the context of weak KAM theory. For a development of weak KAM theory, many researchers studied the structure of solutions and the large time behavior of the associated time-dependent problems; see [3], [16] and references therein.
In the last decade, weakly coupled Hamilton-Jacobi systems were studied from a view point of weak KAM theory. For example, [5], [7], and [18] investigated the large time behavior of the solution for time-dependent problems. In [19], the authors studied homogenization for weakly coupled systems and the rate of convergence to matched solutions. On the other hand, [1] generalized the notion of Aubry sets for the case of systems and proved comparison principle with respect to their boundary data on Aubry sets. In [17], the authors characterized the subsolutions of the systems and showed explicit representation for subsolutions enjoying maximal property. We remark that [7] and [11] studied weakly coupled Hamilton-Jacobi system which is a different type from (1.1).
However, it is little known what kinds of conditions characterize uniqueness of solutions for (1.1). In the single case, one approach, studied in [3] and [4], is to find a uniqueness set; that is, if two solutions coincide on this set, they are totally equal on the domain. Recently, in [20], a new and simple way to find uniqueness sets was studied. In particular, the authors proved new comparison principle with respect to Mather measures in the single case. In this paper, we consider the above for the case of a weakly coupled system as a generalization of [20].
To present the main result, we recall the definition of a generalized Mather measure in [14]. Set L : Definition 1.1. We define a generalized Mather measure associated with (1.1) by a minimizer of the following minimizing problem: where F is the set of all Radon probability measures P (T d × R d × I) satisfying, We denote the set of all generalized Mather measures byM.
We remark that the infimum of (1.2) is zero because we set λ = 0. Indeed, we denote it later as Corollary 3.2.
The following is the main result of this paper: In this paper, we regard the index of m-components system as a variable i ∈ I. This is a successful setting to discuss the above comparison result. To prove this Theorem, we use nonlinear adjoint methods (established in [10]) which fit nicely with the system structure.
Another key point is to consider the Cauchy problem, not (1.1) itself, for the system with initial data being approximations of solutions to (1.1). It is important noting that solutions of Cauchy problems are still quite close to that of (1.1) (see Proposition 2.4). This way, we are able to use the large time averaging effect of the Cauchy problem to introduce the adjoint problems, and then construct Mather measures in our setting.
As well as [20], in light of Theorem 1.2, we can see that This paper is organized as follows. In Section 2, we provide some basic Lemmas. In Section 3, we prove the main Theorem using nonlinear adjoint methods. Finally, we show an example of a generalized Mather measure defined by above in Section 4.

Preliminaries
In this section, we study the related Cauchy problem and adjoint problem for (1.1) and give a priori estimate of solutions as preliminaries.
2.1. Some properties of Θ. Because c ij is symmetric, it holds the following identities.
where we used (A4) in the forth identity.
Proof. Using (A4) in the following third identity, we get (2.2): 2.2. Cauchy problem. Propositions in this subsection are obtained by standard arguments in the theory of viscosity solutions. However, we discuss them to make the paper selfcontained.
Proof. Applying (A2) and (2.2) to (1.1), we can estimate Hence, for k ∈ {1, 2, ..., d} and x ∈ T d , we get To get the latter estimate in (2.4), we calculate, for In this subsection, we consider the following Cauchy problems: Let u ε l (x, t, i) and w ε l (x, t, i) be the unique classical solution and viscosity solution of (2.6) and (2.7), respectively. It is obvious that the unique viscosity solution to (2.7) is w ε l = v l . First, we investigate the difference between u ε l (x, t, i) and w ε l (x, t, i).
Proof. To denote simply, we write u ε and w ε instead of u ε l and w ε l . Define Φ : for η > 0 and K > 0 to be fixed later.
Sending ρ → 0 yields a contradiction, which finishes the proof of Claim 1.
By the above claim, we get Hence, we have On the other hand, because Φ(y 0 , y 0 , 0, i 0 ) Φ(x 0 , y 0 , 0, i 0 ), we get |x 0 −y 0 | Cη. Combine the above two inequalities, to imply Setting η = ε 2 , it holds that By symmetry, we obtain the opposite inequality.
To prove the main result, Lipschitz bound for u ε is important. To get this, we prove the following Lemma.
Proof. Take k ∈ {1, 2, ..., d}. In this proof, we denote u ε instead of u ε l . Differentiate (2.6) with respect to x k to get

Multiplying by u ε
x k (·, i) and summing up with respect to k, we obtain Then, we can rewrite this as where ψ(x, t, i) = 1 2 |Du ε (x, t, i)| 2 . Take (x 0 , t 0 , i 0 ) ∈ T d × [0, 1] × I as a maximum point of ψ. In the case t 0 = 0, it holds that Hence, it suffices to prove the case t 0 ∈ (0, 1]. Then, Using Lemma 2.5, we get Applying the above inequality to (2.11), we get (2.12) Also, we can see Hence, we obtain In light of (A2), we get the conclusion.
2.3. Adjoint problem. Fix x 0 ∈ T d and k ∈ I. Then, consider where γ ik denotes the Kronecker delta and δ x0 is the Dirac delta mass at x 0 . Let σ ε (·, i) be the solution of this problem.
In this subsection, we recall that σ ε is nonnegative and preserves its total mass. Proposition 2.7. For all t ∈ [0, 1] and ε > 0, σ ε 0 and Proof. First, we prove that σ ε is nonnegative. Let z ε solve where s ∈ [0, T ], ψ(·, i) ∈ C ∞ (T d ) with ψ > 0. By comparison, z ε > 0 in T d ×[s, 1]×I. Next, we multiply (2.14) by σ ε and (2.13) by z ε , respectively. Adding each other and integrating over T d × I, we have On the other hand, integrate the above over [s, 1] to yield Hence, for each t ∈ [0, 1], For any µ ∈ F , we have where we used the property of F in the last equality. Sending δ → 0, we get (2.16).

proof of main theorem
Under the above estimates, we prove the main result. The following argument is introduced in [20] to study comparison principle with respect to Mather measures for a single equation.
Combining Proposition 2.8 and (3.7), we can see the following fact.

Example of Mather measure
The following weakly coupled PDE is an example introduced in [1], [5] and [18], for instance. Here, we use this to show an example of a generalized Mather measure. Let where v : T d × I → R is unknown. Note that in this setting the ergodic constant is zero. The above problem corresponds to (1.1), when H(x, p, i) = 1 2 |p| 2 − f (x, i). Proof. At first, we show that µ ∈ F . For φ(·, i) ∈ C 1 (T d ), we get 0 · φ(x 0 , i) = 0, and using (2.2), we can see Θφ(x 0 , i) = 0, which implies µ ∈ F . On the other hand, we have f (x 0 , i) = 0.
In view of Proposition 2.8, µ is a minimizer of (1.2).

Remark 4.2.
We recall the case of single equation, that is m = 1. For n ∈ N, let a k 0 be constants satisfying n k=1 a k = 1. Then, µ = n k=1 a k δ (x k ,0,1) is a Mather measure, where x k ∈ A 1 for k ∈ {1, 2, ..., n}. However, in the system case, because we generalize the holonomic condition for the minimizing problem (1.2), the above convex combinations are not generalized Mather measures in general.