The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

Dedicated to Italo Capuzzo Dolcetta with friendship, respect, and admiration on the occasion of his retirement.

1.   Introduction

We consider the weakly coupled $m$-system of Hamilton-Jacobi equations

where $m\in \mathbb{N}$, $\lambda$ is a nonnegative constant, called the discount factor in terms of optimal control. Here $\mathbb{T}^n$ denotes the $n$-dimensional flat torus, $H = (H_i)_{i\in \mathbb{I}}$ is a family of Hamiltonians given by

where $\mathbb{I} = \{1, \ldots, m\}$, $\Xi$ is a given compact metric space, $g = (g_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n \times \Xi, \mathbb{R}^n)^m$ and $L = (L_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n \times \Xi)^m$. The unknown in $\left({{{\rm{P}}_\lambda }} \right)$ is an $\mathbb{R}^m$-valued function $v^ \lambda = (v^ \lambda_i)_{i\in \mathbb{I}}$ on $\mathbb{T}^n$, $B \, :\, C(\mathbb{T}^n)^m \to C(\mathbb{T}^n)^m$ is a linear map represented by a matrix $B = (b_{ij})_{i, j\in \mathbb{I}}\in C(\mathbb{T}^n)^{m \times m}$, that is,

We use the abbreviated notation $H[v^ \lambda]$ to denote $(H_i(x, Dv_i^ \lambda(x))_{i\in \mathbb{I}}$. The system is called weakly coupled since the $i$-th equation depends on $Dv^ \lambda$ only through $Dv_i^ \lambda$ but not on $Dv_j^ \lambda$, with $j\not = i$. Problem $\left({{{\rm{P}}_\lambda }} \right)$ can be stated in the component-wise style as

We are mainly concerned with the asymptotic behavior of the solution $v^ \lambda$ of $\left({{{\rm{P}}_\lambda }} \right)$ as $\lambda \to 0+$. Asymptotic problems in this class are called the vanishing discount problem, in view that the constant $\lambda$ in $\left({{{\rm{P}}_\lambda }} \right)$ appears as a discount factor in the dynamic programming PDE in optimal control.

Recently, there has been a keen interest in the vanishing discount problem concerned with Hamilton-Jacobi equations and, furthermore, fully nonlinear degenerate elliptic PDEs. We refer to ^{[1,7,10,12,19,20,23,24,25,27]} for relevant work. The asymptotic analysis in these papers relies heavily on Mather measures or their generalizations and, thus, it is considered part of the Aubry-Mather and weak KAM theories. For the development of these theories we refer to ^[14,16,17] and the references therein.

We are here interested in the case of systems of Hamilton-Jacobi equations and, indeed, Davini and Zavidovique in ^[12] have established a general convergence result for the vanishing discount problem for $\left({{{\rm{P}}_\lambda }} \right)$. We establish a result (Theorem 9 below) similar to the main result of ^[12]. In establishing our convergence result, we adapt the argument in ^[23] (see also ^[18]) to the case of systems, especially, to construct generalized Mather measures for $\left({{{\rm{P}}_\lambda }} \right)$. Regarding the recent developments of the weak KAM theory and asymptotic analysis in its influence for systems, we refer to ^{[5,6,26,28,29,30,33]}.

The new argument, which is different from that of ^[12], makes it fairly easy to build a generalized Mather measure for systems in a wide class. One advantage of our argument is that it allows us to treat the case where the coupling matrix $B$ in $\left({{{\rm{P}}_\lambda }} \right)$ depends on the space variable $x\in \mathbb{T}^n$. As in ^[20,23], our approach is applicable to the system with nonlinear coupling of fully nonlinear second-order elliptic PDEs, but we restrict ourselves in this paper to the case of the linearly coupled system of first-order Hamilton-Jacobi equations. Another possible approach for constructing generalized Mather measures is the so-called adjoint method (see ^{[5,15,19,27,33]}).

This paper is part 1 of our study of the vanishing discount problem for weakly coupled systems of Hamilton-Jacobi equations and deals only with the linear coupling and with compact control sets $\Xi$. These restrictions make the presentation of our results clear and transparent. In part 2 ^[20], we remove these restrictions and establish a general convergence result extending Theorem 9 below. Sections 5 and 6 are devoted to the study of ergodic problems of the form $Bu+H[u] = c$, where $c\in \mathbb{R}^m$ is an unknown as well. Also, thanks to the linearity of the coupling, our results on the ergodic problems are applied to extend the scope of Theorem 9. On the other hand, the role of the ergodic problem, with general right-hand side $c$, is not clear at least for the author in the vanishing discount problem for the systems with the nonlinear coupling.

In this paper, we adopt the notion of viscosity solution to $\left({{{\rm{P}}_\lambda }} \right)$, for which the reader may consult ^[2,4,8,31].

To proceed, we give our main assumptions on the system $\left({{{\rm{P}}_\lambda }} \right)$.

We assume that $H$ is coercive, that is, for any $i\in \mathbb{I}$,

This is a convenient assumption, under which any upper semicontinuous subsolution of $\left({{{\rm{P}}_\lambda }} \right)$ is Lipschitz continuous on $\mathbb{T}^n$.

We assume that $B(x) = (b_{ij}(x))$ is a monotone matrix for every $x\in \mathbb{T}^n$, that is, it satisfies

This is a natural assumption that $\left({{{\rm{P}}_\lambda }} \right)$ should possess the comparison principle between a subsolution and a supersolution.

In what follows we set, for $\lambda\geq 0$,

and $\left({{{\rm{P}}_\lambda }} \right)$ can be written as

We use the symbol $u\leq v$ (resp., $u\geq v$) for $m$-vectors $u, v\in \mathbb{R}^n$ to indicate $u_i\leq v_i$ (resp., $u_i\geq v_i$) for all $i\in \mathbb{I}$.

The following theorem is well-known: see ^[13,22] for instance.

Theorem 1. Assume (C) and (M). Let $\lambda > 0$. Then the exists a unique solution $v^ \lambda\in \operatorname{Lip}(\mathbb{T}^n)^m$ of $\left({{{\rm{P}}_\lambda }} \right)$. Also, if $v = (v_i), w = (w_i)$ are, respectively, upper and lower semicontinuous on $\mathbb{T}^n$ and a subsolution and a supersolution of $\left({{{\rm{P}}_\lambda }} \right)$, then $v\leq w$ on $\mathbb{T}^n$.

Henceforth, let ${\bf{1}}$ denote the vector $(1, \ldots, 1)\in \mathbb{R}^m$.

Outline of proof. We follow the line of the arguments in ^[22]. Although ^[22] is concerned with the case when the domain is an open subset of a Euclidean space, the results in ^[22] is valid in the case when the domain is $\mathbb{T}^n$.

Choose a large constant $C > 0$ so that the constant functions $\pm C {\bf{1}}$ are a supersolution and a subsolution of $\left({{{\rm{P}}_\lambda }} \right)$, respectively. (See also (2.3) below.) According to [22,Theorems 3.3,Lemma 4.8], there is a function $v^ \lambda = (v_i^ \lambda)_{i\in \mathbb{I}} \, :\, \mathbb{T}^n \to \mathbb{R}^m$ such that the upper and lower semicontinuous envelopes $(v^ \lambda)^*$ and $v^ \lambda_*$ are a subsolution and a supersolution of $\left({{{\rm{P}}_\lambda }} \right)$, respectively. By the coercivity assumption (C), we find (see [9,Theorem I.14], [21,Example 1]) that the functions $(v_i^ \lambda)^*$ are Lipschitz continuous on $T^n$. Let $R_1 > 0$ be a Lipschitz bound of the functions $(v_i^ \lambda)^*$. To take into account the Lipschitz property of $(v_i^ \lambda)^*$, we modify the Hamiltonian $H$. Fix any $M > 0$ so that

and choose constants $N > 0$ and $R_2 > 0$ so that

and, in view of (1.1),

Define $G = (G_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n \times \mathbb{R}^n)^m$ by

By the choice of $R_1$, it is easy to see that $(v^ \lambda)^*$ is a subsolution of

Also, since $G\geq H$, $v_*^ \lambda$ is a supersolution of (1.2). Observe furthermore that, if $|p|\geq R_2$, then

the functions $G_i$ are uniformly continuous on $\mathbb{T}^n \times B_{R_2}$, and hence, for some continuous function $\omega$ on $[0, \, \infty)$, with $\omega(0) = 0$,

The last inequality above shows that $G$ satisfies [22,(A.2)], which allows us to apply [22,Theorem 4.7], to conclude that $(v^ \lambda)^*\leq v^ \lambda_*$ on $\mathbb{T}^n$ and, moreover, that $v^ \lambda\in \operatorname{Lip}(\mathbb{T}^n)^*$. Similarly, we deduce that the comparison assertion is valid. Thus, $v^ \lambda$ is a unique solution of $\left({{{\rm{P}}_\lambda }} \right)$.

Regarding the coercivity (C), the following proposition is well-knwon.

Proposition 2. The function given by (H) satisfies (C) if and only if there exists $\delta > 0$ such that

where $\, \operatorname{co}$ designates "convex hull" and $B_ \delta$ denotes the open ball with origin at the origin and radius $\delta$.

Outline of proof. Set $C(x, i) = \operatorname{co} \{g_i(x, \xi) \, :\, \xi\in \Xi\}.$ Assume that (1.3) is valid for some $\delta > 0$ and observe that

which shows that (C) holds.

Next, assume that (1.3) does not hold for any $\delta > 0$. Then there exists $(x_k, i_k)\in \mathbb{T}^n \times \mathbb{I}$ for each $k\in \mathbb{N}$ such that

For each $k\in \mathbb{N}$ select $q_k\in B_{1/k} \setminus C(x_k, i_k)$ and $r_k\in C(x_k, i_k)$ so that $r_k$ is the point of $C(x_k, i_k)$ closest to $q_k$. (Notice that $C(x_k, i_k)$ is a compact convex set.) Setting $\nu_k = (q_k-r_k)/|q_k-r_k|$, we find that

Sending $k\to\infty$ along an appropriate subsequence, say $(k_j)_{j\in \mathbb{N}}$, we find that there are a unit vector $\nu = \lim_{j\to\infty} \nu_{k_j}$ of $\mathbb{R}^n$, $r = \lim_{j\to\infty}r_{k_j}\in \mathbb{R}^n$ and $(x, i)\in \mathbb{T}^n \times \mathbb{I}$ such that

If $r\not = 0$, then we have $\nu = -r/|r|$, since $\lim_{k\to\infty}q_k = 0$, and the inequality above reads

These observations imply that for $t > 0$,

which shows that (C) does not hold. This completes the proof.

The rest of this paper is organized as follows. In Section 2, we recall some basic facts concerning monotone matrices. In Section 3, we study viscosity Green-Poisson measures for our system, which are crucial in our asymptotic analysis. We establish the main result for the vanishing discount problem in Section 4. We study the ergodic problem (P$_0$) in the cases when $B$ is irreducible, and $B$ is a constant matrix, respectively, in Sections 5 and 6, and combine the results with the analysis on the vanishing discount problem of Section 4.

2.   Monotone matrices

Here we are concerned with $m \times m$ real matrix $B = (b_{ij})_{i, j\in \mathbb{I}}$.

Let $e_i$ denote the vector $(e_{i1}, \ldots, e_{im})$, with $e_{ii} = 1$ and $e_{ij} = 0$ if $i\not = j$.

Lemma 3. Let $B = (b_{ij})$ be a real $m \times m$ matrix. It is monotone if and only if

We remark that if $B$ satisfies (2.1), then

Proof. We assume first that $B$ is monotone. Since

By the monotonicity of $B$, we have

Similarly, if $i\not = j$ and $t\geq 0$, then we have $1 = (e_i-te_j)_i = \max_{k\in \mathbb{I}}(e_i-te_j)_k$ and hence,

from which we find by sending $t\to \infty$ that

Hence, (2.1) is satisfied.

Next, we assume that (2.1) holds. Let $u\in \mathbb{R}^m$ satisfy

Then we observe that, since $u_k\geq u_j$ for all $j\in \mathbb{I}$,

Thus, $B$ is monotone.

Lemma 4. Let $u\in \mathbb{R}^m$ and $C\geq 0$ be a constant. Let $B$ be an $m \times m$ real monotone matrix. Then we have

Proof. Using Lemma 3, we see that

which states that $B {\bf{1}}\geq 0$. It is then obvious to compute that

and therefore,

3.   Viscosity Green-Poisson measures

For $\lambda\geq 0$ we write $\mathcal{F}(\lambda)$ for the set of all $(\phi, u)\in C(\mathbb{T}^n \times \Xi)^m \times C(\mathbb{T}^n)^m$ such that $u$ is a subsolution of

where $H_{\phi} = (H_{\phi, i})_{i\in \mathbb{I}}$ and

In the above, since $\phi$ is bounded on $\mathbb{T}^n \times \Xi$, if $H$ satisfies (C), then $H_\phi$ satisfies (C).

Lemma 5. The set $\mathcal{F}(\lambda)$ is a convex cone in $C(\mathbb{T}^n \times \Xi)^m \times C(\mathbb{T}^n)^m$ with vertex at the origin.

Proof. Recall [3,Remark 2.5] that for any $u\in Lip(\mathbb{T}^n)^m$, $u$ is a subsolution of

if and only if for any $i\in \mathbb{I}$,

and by the coercivity (C) that for any $(\phi, u)\in \mathcal{F}(\lambda)$, we have $u\in \operatorname{Lip}(\mathbb{T}^n)^m$.

Fix $(\phi, u), (\psi, v)\in \mathcal{F}(\lambda)$ and $t, s\in[0, \infty)$. Fix $i\in \mathbb{I}$ and observe that

which imply that there is a set $N\subset \mathbb{T}^n$ of Lebesgue measure zero such that

Multiplying the first and second by $t$ and $s$, respectively, adding the resulting inequalities and setting $w = tu+sv$, we obtain

which readily implies that $t(\phi, u)+s(\psi, v)\in \mathcal{F}(\lambda)$.

We refer the reader to [23,Lemma 2.2] for another proof of the above lemma.

We establish a representation formula for the solution of $\left({{{\rm{P}}_\lambda }} \right)$, with $\lambda > 0$, by modifying the argument in ^[23] (see also ^[18]).

For any nonnegative Borel measure $\nu$ on $\mathbb{T}^n \times \Xi$ and $\phi\in C(\mathbb{T}^n \times \Xi)$, we write

Similarly, for any collection $\nu = (\nu_i)_{i\in \mathbb{I}}$ of nonnegative Borel measures on $\mathbb{T}^n \times \Xi$ and $\phi = (\phi_i)\in C(\mathbb{T}^n \times \Xi)^m$, we write

Note that any collection $\nu = (\nu_i)_{i\in \mathbb{I}}$ of nonnegative Borel measures on $\mathbb{T}^n \times \Xi$ is regarded as a nonnegative Borel measure on $\mathbb{T}^n \times \Xi \times \mathbb{I}$ and vice versa.

We set

Note that

By assumption (M) and Lemma 3, we have $\rho_i\geq 0$ on $\mathbb{T}^n$ for all $i\in \mathbb{I}$.

Given a constant $\lambda > 0$, let $\operatorname{\mathbb{P}}_{B^ \lambda}$ denote the set of of nonnegative Borel measures $\nu = (\nu_i)_{i\in \mathbb{I}}$ on $\mathbb{T}^n \times \Xi \times \mathbb{I}$ such that

The last condition reads

where $|\nu_i|$ denotes the total mass of $\nu_i$ on $\mathbb{T}^n \times \Xi$. Note as well that $\operatorname{\mathbb{P}}_{B^ \lambda}$ can be identified with the space of Borel probability measures on $\mathbb{T}^n \times \Xi \times \mathbb{I}$ by the correspondence between $\nu = (\nu_i)_{i\in \mathbb{I}}$ and $\sum_{i\in \mathbb{I}}(\lambda+\rho_i)\nu_i\otimes \delta_i$, where $\otimes$ indicates the product of two measures and $\delta_i$ denotes the Dirac measure at $i$. If we set $\mu: = \sum_{i\in \mathbb{I}}(\lambda+\rho_i)\nu_i\otimes \delta_i$ and consider $\mu$ as a collection $(\mu_i)$ of measures on $\mathbb{T}^n \times \Xi$, then $\nu_i = (\lambda+\rho_i)^{-1}\mu_i$. We denote simply by $\operatorname{\mathbb{P}}$ the space of Borel probability measures on $\mathbb{T}^n \times \Xi \times \mathbb{I}$.

For $\lambda\geq 0$ and $(z, k)\in \mathbb{T}^n \times \mathbb{I}$ we set

and

Theorem 6. Assume (H), (C) and (M). Let $\lambda > 0$ and $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. Let $v^ \lambda\in C(\mathbb{T}^n \times \mathbb{I})$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$. Then there exists a $\nu^{z, k, \lambda} = (\nu^{z, k, \lambda}_i)_{i\in \mathbb{I}}\in \mathcal{G} \, ^\prime(z, k, \lambda)$ such that

We remark that for any $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$ we have $\langle{\nu, L}\rangle\geq v_k^ \lambda(z)\langle{\nu, B^ \lambda}\rangle = v_k^ \lambda(z)$ and, accordingly, in the theorem above, the measures $\nu^{z, k, \lambda}$ has the minimizing property:

We call any minimizing family $(\nu_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}_{B^ \lambda}$ of the optimization problem above a viscosity Green-Poisson measure for $\left({{{\rm{P}}_\lambda }} \right)$.

Proof. Note first that $(L, v^ \lambda)\in \mathcal{F}(\lambda)$ and hence, for any $\nu\in \mathcal{G} \, ^\prime (z, k, \lambda)$,

Next, we show that

Note that for $z\in \mathbb{T}^n$,

Hence, in order to prove (3.5), we only need to show that

We postpone the proof of (3.6) and, assuming temporarily that (3.5) is valid, we prove that there exists $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$ such that

which, together with (3.4), completes the proof.

To prove (3.7), we observe that $\operatorname{\mathbb{P}}_{B^ \lambda}$ and, by Lemma 5, $\mathcal{F}(\lambda)$ are convex,

is convex and continuous, in the topology of weak convergence of measures, for any $(\phi, u)\in \mathcal{F}(\lambda)$ and

is concave and continuous for any $\nu\in \operatorname{\mathbb{P}}_{B^ \lambda}$. Hence, noting moreover that $\mathbb{T}^n \times \Xi \times \mathbb{I}$ is a compact set, we apply the minimax theorem (^[34,32]), to find from (3.5) that

Observe by using the cone property of $\mathcal{F}(\lambda)$ that

This and (3.8) yield

which proves (3.7).

It remains to show (3.6). For this, we argue by contradiction and thus suppose that (3.6) does not hold. Accordingly, we have

for some $\varepsilon > 0$. We may select $(\phi, u)\in \mathcal{F}(\lambda)$ so that

That is, for any $\nu\in \operatorname{\mathbb{P}}_{B^ \lambda}$, we have

Plugging $\nu = (\lambda+\rho_i)^{-1} \delta_{(x, \xi, i)}\in \operatorname{\mathbb{P}}_{B^ \lambda}$, with any $(x, \xi, i)\in \mathbb{T}^n \times \Xi \times \mathbb{I}$, into the above, we find that

Hence, we have

This ensures that $u$ is a subsolution of

which implies that $u-(u_k(z)-v_k^ \lambda(z)- \varepsilon) {\bf{1}}$ is a subsolution of $\left({{{\rm{P}}_\lambda }} \right)$. By comparison (Theorem 1), we get

The $k$-th component of the above, evaluated at $x = z$, yields an obvious contradiction. Thus we conclude that (3.6) holds.

We have the following characterization of $\mathcal{G} \, ^\prime(z, k, \lambda)$.

Proposition 7. Assume (H), (C) and (M) hold. Let $\nu = (\nu_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}_{B^ \lambda}$ and $(z, k, \lambda)\in \mathbb{T}^n \times \mathbb{I} \times(0, \infty)$. Then we have $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$ if and only if

Proof. Assume first that $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$. Fix any $\psi = (\psi_i)_{i\in \mathbb{I}} \in C^1(\mathbb{T}^n)^m$ and define $\phi = (\phi_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n \times \mathbb{I})^m$ by

Observe that $u: = \pm\psi$ satisfy, respectively,

and, hence,

Since $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$, we have

respectively, which shows that (3.9) is valid.

Now, assume that (3.9) is satisfied. Fix any $(u, \phi)\in \mathcal{F}(\lambda)$. As noted in the proof of Theorem 1, we have $u\in \operatorname{Lip}(\mathbb{T}^n)$. By the standard mollification technique, given a positive constant $\varepsilon > 0$, we can approximate $u$ by a smooth function $u^ \varepsilon$ so that

The last inequality reads

Integrating the above by $\nu_i$, summing up in $i\in \mathbb{I}$ and using (3.9), we get

Sending $\varepsilon\to 0$ shows that $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$.

It is convenient to restate the theorem above as follows. For $\mu = (\mu_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}$ and $\lambda > 0$, consider $\nu = (\nu_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}_{B^ \lambda}$ given by

(Notice by the above definition that $\langle{\nu, B^ \lambda}\rangle = \langle{\mu, }\rangle = 1$.) Observe that for $\phi = (\phi_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n \times \Xi)^m$,

and that for any $(z, k)\in \mathbb{T}^n \times \mathbb{I}$, we have $\nu\in \mathcal{G} \, ^\prime(z, k, \lambda)$ if and only if

The condition above is stated in the spirit of Proposition 7 as

We define

The following proposition is an immediate consequence of Theorem 6.

Corollary 8. Assume (H), (C) and (M). Let $\lambda > 0$ and $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. Let $v^ \lambda\in C(\mathbb{T}^n \times \mathbb{I})$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$. Then there exists a $\mu^{z, k, \lambda} = (\mu^{z, k, \lambda}_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}(z, k, \lambda)$ such that

4.   A convergence result for the vanishing discount problem

We study the asymptotic behavior of the solution $v^ \lambda$ of $\left({{{\rm{P}}_\lambda }} \right)$, with $\lambda > 0$, as $\lambda\to 0$.

We make a convenient assumption on the system (P$_0$):

If $\rho_i > 0$ for all $i\in \mathbb{I}$, then Theorem 1 assures that there exists a unique solution $v_0$ of (E). In this situation, it is not difficult to show that the uniform convergence, as $\lambda \to 0+$, of $v^ \lambda$ to the unique solution $v_0$ on $\mathbb{T}^n$. In general, existence and uniqueness of a solution of (P$_0$) may fail. In fact, one can prove at least in the case when the $b_{ij}$ are constants (see Theorem 18) that there exists $c\in \mathbb{R}^m$ such that

has a solution $v_0\in \operatorname{Lip}(\mathbb{T}^n)$ and possibly multiple solutions. If such a $c = (c_i)$ exists, then the introduction of a new family of Hamiltonians,

allows us to view (4.1) as in the form of (P$_0$). The link between two vanishing discount problems for the original $\left({{{\rm{P}}_\lambda }} \right)$ and for $\left({{{\rm{P}}_\lambda }} \right)$, with $\widetilde H$ in place of $H$, is discussed in Sections 5 and 6.

Theorem 9. Assume (H), (C), (M) and (E). Let $v^ \lambda$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$ for $\lambda > 0$. Then there exists a solution $v^0\in \operatorname{Lip}(\mathbb{T}^n)^m$ of (P$_0$) such that the functions $v_i^ \lambda$ converge to $v_i^0$ uniformly on $\mathbb{T}^n$ as $\lambda\to 0$ for all $i\in \mathbb{I}$.

Lemma 10. Under the hypotheses of Theorem 9, there exists a constant $C_0 > 0$ such that for any $\lambda > 0$,

Proof. Let $v_0 = (v_{0, i})_{i\in \mathbb{I}}\in \operatorname{Lip}(\mathbb{T}^n)^m$ be the solution of (P$_0$). Choose a constant $C_0 > 0$ so that

and observe by the monotonicity of $B$ (Lemma 4) that $v_0+C_1 {\bf{1}}$ and $v_0-C_1 {\bf{1}}$ are a supersolution and a subsolution of (P$_0$), respectively. Noting that $v_0+C_1 {\bf{1}}\geq 0$ and $v_0-C_1 {\bf{1}}\leq 0$, we deduce that $v_0+C_1 {\bf{1}}\geq 0$ and $v_0-C_1 {\bf{1}}\leq 0$ are a supersolution and a subsolution of $\left({{{\rm{P}}_\lambda }} \right)$ for any $\lambda > 0$, respectively. By comaprison (Theorem 1), we see that, for any $\lambda > 0$, $v_0-C_1 {\bf{1}}\leq v^ \lambda\leq v_0+C_1 {\bf{1}}$ on $\mathbb{T}^n$ and, moreover, $-2C_1 {\bf{1}}\leq v^ \lambda\leq 2C_1 {\bf{1}}$ on $\mathbb{T}^n$. Thus, (4.2) holds with $C_0 = 2C_1$.

Lemma 11. Under the hypotheses of Theorem 9, the family $(v^ \lambda)_{ \lambda\in(0, \, 1)}$ is equi-Lipschitz continuous on $\mathbb{T}^n$.

Indeed, the family $(v^ \lambda)_{ \lambda > 0}$ is equi-Lipschitz continuous on $\mathbb{T}^n$, which we do not need here.

Proof. By Lemma 10, there is a constant $C_0 > 0$ such that

Hence, as $v^ \lambda$ is a solution of $\left({{{\rm{P}}_\lambda }} \right)$, we deduce by (C) that there exists a constant $C_1 > 0$ such that the $v_i^ \lambda$ are subsolutions of $|Du|\leq C_1$ in $\mathbb{T}^n$. It is a standard fact that the $v_i^ \lambda$ are Lipschitz continuous on $\mathbb{T}^n$ with $C_1$ as their Lipschitz bound.

In the proof of Theorem 9, Corollary 8 has a crucial role. We need also results for $\lambda = 0$ similar to the corollary.

We consider the condition for $\mu\in \operatorname{\mathbb{P}}$,

We denote by $\operatorname{\mathbb{P}}(0)$ the subset of $\operatorname{\mathbb{P}}$ consisting of those $\mu$ which satisfy (4.3).

Theorem 12. Assume (H), (C), (M) and (E). Assume that $\rho_i = 0$ on $\mathbb{T}^n$ for every $i\in \mathbb{I}$. Then there exists a $\mu^0 = (\mu_i^0)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}(0)$ such that

Proof. We fix a $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. By Corollary 8, for each $\lambda > 0$ there exists $\mu^ \lambda = (\mu_i^ \lambda)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}(z, k, \lambda)$ such that

Since $(\mu^ \lambda)_{ \lambda > 0}$ is a family of Borel probability measures on a compact space $\mathbb{T}^n \times \Xi \times \mathbb{I}$, there exists a sequence $(\lambda_j)_{j\in \mathbb{N}}\subset (0, \, 1)$ converging to zero such that the sequence $(\mu^{ \lambda_j})_{j\in \mathbb{N}}$ converges weakly in the sense of measures to a Borel probability measure $\mu^0$ on $\mathbb{T}^n \times \Xi \times \mathbb{I}$. It follows from (4.5) and Lemma 10 that

Observe that if $(\phi, u)\in \mathcal{F}(0)$, then, for any $\lambda > 0$, $u$ is a subsolution of

and hence, $(\psi, u) \in \mathcal{F}(\lambda)$, with $\psi(x, \xi) = \phi(x, \xi)+ \lambda u(x)$. Hence, the inclusion $\mu^ \lambda\in \mathcal{G} \, ^\prime(z, k, \lambda)$ yields

Multiplying the above by $\lambda$ and sending $\lambda = \lambda_j \to 0$, in view of Lemma 10, we get

This shows that $\mu^0\in \operatorname{\mathbb{P}}(0)$. These observations together with (4.3) for $\mu\in \operatorname{\mathbb{P}}(0)$ guarantee that

We state a characterization of $\operatorname{\mathbb{P}}(0)$ in the next, similar to Proposition 7, which we leave to the reader to verify.

Proposition 13. Assume (H), (C) and (M). Let $\mu = (\mu_i)_{i\in \mathbb{I}}\in \operatorname{\mathbb{P}}$. We have $\mu\in \operatorname{\mathbb{P}}(0)$ if and only if

We call any minimizer $\mu\in \operatorname{\mathbb{P}}(0)$ of the optimization problem (4.4) a viscosity Mather measure.

We denote by $\mathbb{M}_+$ the set of all Borel nonnegative measures $\mu = (\mu_i)_{i\in \mathbb{I}}$ on $\mathbb{T}^n \times \Xi \times \mathbb{I}$. We set

Theorem 14. Let $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. Assume (H), (C), (M) and (E). For any $\lambda > 0$, let $v^{ \lambda}$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$ and $\mu^ \lambda \in \operatorname{\mathbb{P}}(z, k, \lambda)$ be a minimizer of (3.11). Then there exists a subsequence of $(\lambda_j)$, which is denoted again by the same symbol, such that, as $j\to\infty$,

weakly in the sense of measures for some $\mu^0 = (\mu^0_i)_{i\in \mathbb{I}}\in \mathbb{M}_+(0)$, and $\mu^0$ satisfies

In particular,

Notice that the minimization problem (4.7) is trivial since $\mu^0 = 0$ is a minimizer.

Proof. The proof is similar to that of Theorem 12.

We fix a $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. For each $\lambda > 0$, we have

Observe that

Accordingly, since $\mathbb{T}^n \times \Xi \times \mathbb{I}$ is a compact metric space, the families $(\lambda(\lambda+\rho_i)^{-1}\mu_i^ \lambda)_{ \lambda = \lambda_j, j\in \mathbb{N}}$ have a common subsequence, along which all the families converge to some Borel nonnegative measures $\mu^0_i$ weakly in the sense of measures. We may assume by replacing the original sequence $(\lambda_j)$ by its subsequence that

weakly in the sense of measures. Combining this with (4.8) yields

It is obvious to see that $\mu^0\in \mathbb{M}_+$.

Let $(\phi, u)\in \mathcal{F}(0)$. As before, we have $(\psi, u) \in \mathcal{F}(\lambda)$, with $\psi(x, \xi) = \phi(x, \xi)+ \lambda u(x)$ and moreover

Multiplying the above by $\lambda$ and sending $\lambda = \lambda_j \to 0$, we get

This shows that $\mu^0\in \mathbb{M}_+(0)$.

Proof of Theorem 9. Let $\mathcal{V}$ denote the set of accumulation points $v = (v_i)\in C(\mathbb{T}^n)^m$ in the space $C(\mathbb{T}^n)^m$ of $v^ \lambda$ as $\lambda\to 0$. In view of the Ascoli-Arzela theorem, Lemmas 4.2 and 4.3 guarantee that the family $(v^ \lambda)_{ \lambda\in(0, \, 1)}$ is relatively compact in $C(\mathbb{T}^n)^m$. In particular, the set $\mathcal{V}$ is nonempty. Note by the stability of the viscosity property under uniform convergence that any $v\in \mathcal{V}$ is a solution of (P$_0$).

If $\mathcal{V}$ is a singleton, then it is obvious that the whole family $(v^ \lambda)_{ \lambda > 0}$ converges to the unique element of $\mathcal{V}$ in $C(\mathbb{T}^n)^m$ as $\lambda\to 0$.

We need only to show that $\mathcal{V}$ is a singleton. It is enough to show that for any $v, w\in \mathcal{V}$ and $(z, k)\in \mathbb{T}^n \times \mathbb{I}$, the inequality $w_k(z)\leq v_k(z)$ holds.

Fix any $v, w\in \mathcal{V}$ and $(z, k)\in \mathbb{T}^n \times \mathbb{I}$. Select sequences $(\lambda_j)$ and $(\delta_j)$ converging to zero so that

By Corollary 8, there exists a sequence $(\mu^j)_{j\in \mathbb{N}}$ such that

In view of Theorem 14, we may assume by passing to a subsequence if necessary that, as $j\to\infty$,

for all $i\in \mathbb{I}$ and for some $\mu^0 = (\mu^0_i)_{i\in \mathbb{I}}\in \mathbb{M}_+(0)$ and, moreover,

Since $(L- \lambda v^ \lambda, v^ \lambda)\in \mathcal{F}(0)$ and $\mu^0\in \mathbb{M}_+(0)$, in view of (4.10), we have

which yields after dividing by $\lambda > 0$ and then sending $\lambda \to 0$ along $\lambda = \delta_j$

Now, note that $w$ is a solution of

and thus, $(L+ \lambda w, w)\in \mathcal{F}(\lambda)$ and infer by (4.9) that

Sending $j\to\infty$ now yields

This together with (4.11) shows that $w_k(z)\leq v_k(z)$, which completes the proof.

5.   The ergodic problem for irreducible matrix $B$

We consider the problem of finding $c = (c_i)_{i\in \mathbb{I}}\in \mathbb{R}^m$ and $v = (v_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n)^m$ such that $v$ is a solution of

The pair of such $c$ and $v$ is also called a solution of (5.1). This problem is called the ergodic problem in this paper although the term, ergodic problem, should be used only when the condition that $\sum_{j\in \mathbb{I}}b_{ij}(x) = 0$ holds for some $(i, x)\in \mathbb{I} \times \mathbb{T}^n$.

Henceforth, $D(x)$ denotes the diagonal matrix

where, as before, $\rho_i(x) = \sum_{j\in \mathbb{I}}b_{ij}(x)$.

Throughout this section, we treat the case when

The irreducibility of $B(x)$ is stated as follows: for any nonempty subset $I$ of $\mathbb{I}$, which is not identical to $\mathbb{I}$, there exists a pair of $i\in I$ and $j\in \mathbb{I} \setminus I$ such that $b_{ij}(x)\not = 0$.

The following result has been established in Davini-Zavidovique [11,Theorem 2.10] (see also ^[6,30]).

Proposition 15. Assume (H), (C), (M), (5.2), and that

Then there exist $c_0\in \mathbb{R}$ and $v_0\in \operatorname{Lip}(\mathbb{T}^n)^m$ such that the pair $(c_0 {\bf{1}}, v_0)$ is a solution of (5.1).

We remark that (5.3) is satisfied if and only if $B(x) {\bf{1}} = 0$ for all $x\in \mathbb{T}^n$, which holds if and only if $\rho_i(x) = 0$ for all $(i, x)\in \mathbb{I} \times \mathbb{T}^n$.

The next theorem states the central result of this section.

Theorem 16. Assume (H), (C), (M), (5.2), and (5.3). Let $v^ \lambda$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$ for $\lambda > 0$. Then there exists a constant $c^0\in \mathbb{R}$ and a function $v^0\in \operatorname{Lip}(\mathbb{T}^n)^m$ such that the functions $v^ \lambda+ \lambda^{-1}c^0 {\bf{1}}$ converge to $v^0$ uniformly on $\mathbb{T}^n$ as $\lambda\to 0$. Moreover, the pair $(c^0 {\bf{1}}, v^0)$ is a solution of (5.1).

Proof. Thanks to Proposition 15, there exists a solution $(c_0, v_0)\in \mathbb{R}^m \times C(\mathbb{T}^n)^m$ of (5.1). We set $\widetilde H = H-c_0 {\bf{1}}$, and note that, since $B(x) {\bf{1}} = 0$ for all $x\in \mathbb{T}^n$, the function $w^ \lambda: = v^ \lambda+ \lambda^{-1}c_0 {\bf{1}}$ satisfies, in the viscosity sense,

By Theorem 9, there exists a solution $v^0\in \operatorname{Lip}(\mathbb{T}^n)^m$ of $Bv^0+ \widetilde H[v^0] = 0$ in $\mathbb{T}^n$ such that, as $\lambda\to 0+$, $w^ \lambda\to v^0$ in $C(\mathbb{T}^n)^m$. Noting that $(c_0 {\bf{1}}, v^0)$ is a solution of (5.1), we finish the proof.

The condition (5.3) in Proposition 15 can be removed and the following theorem is valid.

Theorem 17. Assume (H), (C), (M), and (5.2). Then there exist $c^0\in \mathbb{R}$ and $v^0 = (v^0_i)_{i\in \mathbb{I}}\in \operatorname{Lip}(\mathbb{T}^n)^m$ such that the pair $(c^0 {\bf{1}}, v^0)$ is a solution of (5.1).

Proof. For $x\in \mathbb{I} \times \mathbb{T}^n$, we set

and note that $B^0(x)$ is irreducible and (5.3) holds with $b_{ij}(x)$ replaced by $b^0_{ij}(x)$. Note also that $\rho_i(x)\geq 0$ for all $(i, x)\in \mathbb{I} \times \mathbb{T}^n$.

Thanks to Proposition 15, there exist $c^0\in \mathbb{R}$ and $v = (v_i)\in \operatorname{Lip}(\mathbb{T}^n)^m$ which solve

We choose a constant $C > 0$ so that $\max_{(i, x)\in \mathbb{I} \times \mathbb{T}^n}|v_i(x)|\leq C$ and set $v^\pm(x) = v(x)\pm C {\bf{1}}$, respectively. Observe that, since $v^+_i(x)\geq 0$ and $v^-_i(x)\leq 0$ for all $(i, x)\in \mathbb{I} \times \mathbb{T}^n$, the functions $u = v^+$ and $u = v^-$ are a supersolution and subsolution of

that is, $Bu+H[u] = c^0 {\bf{1}} \ \ \text{ in } \mathbb{T}^n$, respectively. In view of the Perron method, the function $v^0 = (v^0_i)_{i\in \mathbb{I}}\in \operatorname{Lip}(\mathbb{T}^n)$ given by

is a solution of (5.1), with $c = c^0 {\bf{1}}$.

Even without the assumption (5.3), it is immediate from Theorem 9 that, under the hypotheses of Theorem 17, if $c^0 = 0$, then the convergence holds for the whole family of the solutions $v^ \lambda$ of $\left({{{\rm{P}}_\lambda }} \right)$, with $\lambda > 0$. A typical case when $c^0 = 0$ is given by [6,Theorem 4.2] (see also ^[11,28]).

6.   The ergodic problem for constant matrix $B$

Throughout this section we assume that $B$ is a constant matrix, that is, independent of $x\in \mathbb{T}^n$.

The main results in this section are as follows.

Theorem 18. Assume (H), (C), (M), and that $B$ is a constant matrix. Then (5.1) has a solution $(c, v)\in \mathbb{R}^m \times C(\mathbb{T}^n)^m$.

Theorem 19. Under the same hypotheses of Theorem 18, let $(c, v_0)\in \mathbb{R}^m \times C(\mathbb{T}^n)^m$ be a solution of (5.1) and let $v^ \lambda$ be the unique solution of $\left({{{\rm{P}}_\lambda }} \right)$ for $\lambda > 0$. Then there exists a function $v^0\in C(\mathbb{T}^n)^m$ such that the functions $v^ \lambda+(\lambda I+B)^{-1}c$ converge to $v^0$ uniformly on $\mathbb{T}^n$ as $\lambda\to 0$. Moreover, the pair $(c, v^0)$ is a solution of (5.1).

Proof. It is well-known (and easily checked) that due to the monotonicity of $B$, $(\lambda I+B)$ is invertible for any $\lambda > 0$. We set $\widetilde H(x, p) = H(x, p)-c$ for $(x, p)\in \mathbb{T}^n \times \mathbb{R}^n$ and also $w^ \lambda(x) = v^ \lambda(x)+(\lambda I+B)^{-1}c$ for $x\in \mathbb{T}^n$. Observe that, in the viscosity sense,

It is clear that $\widetilde H$ satisfies (H) and (C) and that $v_0$ is a solution of $Bu+ \widetilde H[u] = 0$ in $\mathbb{T}^n$. By Theorem 9, we conclude that there exists a solution $v^0\in C(\mathbb{T}^n)^m$ of $Bu+ \widetilde H[u] = 0$ in $\mathbb{T}^n$ such that $w^ \lambda \to v^0$ in $C(\mathbb{T}^n)^m$ as $\lambda\to 0+$. Noting that $(c, v^0)$ is a solution of (5.1), we finish the proof.

For the proof of Theorem 18, we begin with a preliminary remark on the permutations.

For a given permutation $\pi \, :\, \mathbb{I} \to \mathbb{I}$, we define the $m \times m$ matrix $P$ by

where $\delta_{ij} = \delta_{i, j}: = 1$ if $i = j$ and $= 0$ otherwise. Note that $P^{-1} = (\delta_{i, \pi(j)})_{i, j\in \mathbb{I}} = P^ \mathrm{T}$ and that for any $u = (u_i)_{i\in \mathbb{I}}$,

The system of Hamilton-Jacobi equations

can be written component-wise as

By the use of $P$, the system above is expressed as

and furthermore, if $v = Pu$,

Set $A = (a_{ij})_{i, j\in \mathbb{I}} = PBP^ \mathrm{T}$ and observe that if $B$ is monotone, then

and

Consequently, if $B$ is monotone, then $PBP^ \mathrm{T}$ is monotone as well, and the system (6.2), by using the permutation matrix $P$, is converted to (6.3).

Proof of Theorem 18. It is well-known (see for instance [35,Section 2.3]) that, given a monotone matrix $B$, one can find a permutation $\pi \, :\, \mathbb{I}\to \mathbb{I}$ such that

where, $P$ is given by (6.1), $B^{(1)}$ is a diagonal matrix of order $r_1$ and, for $1 < i\leq p$, $B^{(i)}$ are irreducible matrices of order $r_i$. In view of the preliminary remark before this proof, to seek for a solution of (5.1), we may and do assume henceforth $B$ has the normal form of the right hand side of (6.4).

Set

Notice that $s_1 = 0$. If $r_1\geq 1$, then we first show that there exist an $r_1$-vector $c^{(1)} = (c^{(1)}_i)_{i\in \mathbb{I}_1}\in \mathbb{R}^{r_1}$ and a function $v^{(1)} = (v^{(1)}_i)_{i\in \mathbb{I}_1}\in C(\mathbb{T}^n)^{r_1}$ such that $v^{(1)}$ is a solution of

where $H^{(1)} = (H_i)_{i\in \mathbb{I}_1}$. The system is, in fact, a collection of single equations

and thus the existence of a solution $(c^{(1)}, v^{(1)})$ of (6.5) is a classical result. Indeed, for each $i\in \mathbb{I}_1$, if $b_{ii}^{(1)} > 0$, then (6.6) has a (unique) solution $v_i^{(1)}\in \operatorname{Lip}(\mathbb{T}^n)$ for any choice of $c_i^{(1)}$. If $b_{ii}^{(1)} = 0$, then (6.6) has a solution $(c_i^{(1)}, v_i^{(1)})\in \mathbb{R} \times \operatorname{Lip}(\mathbb{T}^n)$ (see unpublished work by Lions PL, Papanicolaou G, and Varadhan S: Homogenization of Hamilton-Jacobi equations). If $r_1 = m$, then we are done.

Next, assume that $r_1 < m$ (and equivalently, $1 < p$) and we show that there exist a vector $c^{(2)} = (c^{(2)}_i)_{i\in \mathbb{I}_2}\in \mathbb{R}^{r_2}$ and a function $v^{(2)} = (v^{(2)}_i)_{i\in \mathbb{I}_2}\in C(\mathbb{T}^n)^{r_2}$ such that $v^{(2)}$ is a solution of the system

where

According to Proposition 15, there exist $c^{(2)} = (c^{(2)}_i)_{i\in \mathbb{I}_2}\in \mathbb{R}^{r_2}$ and $v^{(2)} = (v^{(2)}_i)_{i\in \mathbb{I}_2}\in C(\mathbb{T}^n)^{r_2}$ which satisfy (6.7). This way (by induction), we find $c^{(1)}, \ldots, c^{(p)}$ and $v^{(1)}, \ldots, v^{(p)}$ such that

and $v^{(k)}$ satisfies

where

We define $c = (c_i)_{i\in \mathbb{I}}\in \mathbb{R}^m$ and $v = (v_i)_{i\in \mathbb{I}}\in C(\mathbb{T}^n)^m$ by setting

and observe that

This completes the proof.

Acknowledgments

The author would like to thank the anonymous referee for useful and critical comments on the original version of this paper, which have helped significantly to improve the presentation. This work is partially supported by the JSPS KAKENHI #16H03948, #18H00833, #20K03688, and #20H01817.

Conflict of interest

The author declares no conflicts of interest in this paper.