Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations

The so-called ergodic approximation is a technique introduced in [32] to study the existence of solutions of the Hamilton-Jacobi equation¹¹1All solutions in the paper are meant in the viscosity sense. The definition will be provided later.

on the flat $d$-dimensional torus $M:=\mathbb{T}^{d}\simeq\mathbb{R}^{d}/\mathbb{Z}^{d}$, where the Hamiltonian $H$ is a continuous function on $T^{*}M$, coercive in the gradient variable, uniformly with respect to $x\in M$, and $c$ is a real number. Let $\lambda>0$ and $u_{\lambda}$ be the unique solution of

According to [32], the functions $-\lambda u_{\lambda}$ uniformly converge on $M$, as $\lambda\to 0^{+}$, to a constant $c_{0}$. Furthermore, the solutions $(u_{\lambda})_{\lambda>0}$ are equi-Lipschitz, yielding, by the Arzelá-Ascoli Theorem, that the functions $u_{\lambda}-\min_{x\in M}u_{\lambda}$ uniformly converge, along subsequences as $\lambda\to 0^{+}$, to a solution of (1) with $c$ equaling $c_{0}$. The constant $c_{0}$ is called critical value of $H$ and is characterized by the property of being the unique constant $c\in\mathbb{R}$ such that (1) admits solutions. At that time, it was not clear if different converging sequences yield the same limit. Some constraints on the possible limit solutions were subsequently found in [22, 29], but the breakthrough came with the work [12], where the authors proved that the unique solution $u_{\lambda}$ of

converges to a distinguished solution of

as $\lambda\rightarrow 0^{+}$ under the sole additional assumption that $H$ is convex in the gradient variable. The proof relies on techniques and tools issued from weak KAM Theory, in particular on the concept of Mather measure, and it works whenever $M$ is a closed Riemannian manifold. This kind of problem is also known as the vanishing discount problem. When the convexity condition on $H$ is dropped, the functions $u_{\lambda}$ may not converge, as it was pointed out in [44] through a counterexample posed on the 1-dimensional torus.

As a nonlinear generalization (see [6, 5, 23] and [39]), one can study the uniform convergence of the unique solution of

as $\lambda\rightarrow 0^{+}$, where $H_{\lambda}(x,p,u)$ is strictly increasing in $u$, and uniformly converges to $H(x,p)$ on compact sets as $\lambda\rightarrow 0^{+}$. This kind of problem is called the vanishing contact structure problem. The vanishing discount problem falls in this framework as a particular case by choosing $H_{\lambda}(x,p,u):=\lambda u+H(x,p)$.

The asymptotic convergence result has been subsequently established in many different situations. For the second order case, one can refer to [26, 27, 34, 43]. For the discrete case, one can refer to [11, 38, 42] and also [2] in the context of twist maps. For the similar problem in the mean field game theory, one can refer to [4]. For the weakly coupled Hamilton-Jacobi systems, one can refer to [17, 14, 24, 25]. For the non-compact setting, one can refer to [18, 28].

A natural and challenging question is to weaken the hypothesis on the monotonicity of the Hamiltonian. A first degenerate case was studied in [41], where the author considered the convergence of the solution of

as $\lambda\rightarrow 0^{+}$, where $a(x)\geqslant 0$ on $M$, and $a(x)>0$ on the projected Aubry set of $H$. Inspired by the works [5, 41], the authors studied in [7] the vanishing discount problem for contact Hamilton-Jacobi equations of the form (E_λ), where the positivity hypothesis on $a$ assumed in [41] is weakened and generalized by introducing the non-degeneracy integral condition (L5). This work highlights once more that the concept of Mather measure plays a central role in the convergence result.

It is worth pointing out that all works mentioned above required a global non-decreasing hypothesis of the Hamiltonians in $u$. If the discounted equation is not increasing in the unknown function $u$, solutions may even not exist, and, if they exist, they may be not unique. In [15, 40], the authors discussed the uniform convergence of the minimal solution of (${\textrm{HJ}}_{\lambda}$) as $\lambda\rightarrow 0^{-}$. For the non-monotone vanishing discount problem, the second author provided the first example in [35] of nonconvergence. In this example, there exist a convergent family of solutions and a divergent family of solutions at the same time. This phenomenon is new comparing with all the previous works in this direction. In the present paper, we show that the example in [35] is in fact a very general phenomenon when the Hamiltonian is continuous, convex and superlinear in the fibres.

Let us conclude by mentioning that the type of problems we study are also closely linked to optimization problems in economics. The discount factor then models the effect of time through interest rates or inflation. Negative interest rates or deflation have been studied by economists (see for instance [31]). Our results then give possible asymptotics in the presence of coexisting inflation and deflation.

We present here our main results. Section 2 contains our analysis on the asymptotic behavior of possible solutions of a general contact Hamilton-Jacobi equation of the form (E_λ) when the discount factor $\lambda$ goes to $0$. The Hamiltonian $G(x,p,u)$ is assumed convex and superlinear in $p$, and globally Lipschitz in $u$, see conditions (G1-3) in Section 2. It is also assumed to satisfy a $C^{1}$-type regularity condition in $u$ near $u=0$, see condition (G4) in Section 2. The latter is for instance satisfied when the map $u\mapsto G(x,p,u)$ is $C^{1}$ in a neighborhood of $u=0$ in the following sense:

• (G4^′)

  there exists $\varepsilon>0$ such that $\frac{\partial G}{\partial u}(x,p,u)$ exists for all $(x,p,u)\in T^{*}M\times(-\varepsilon,\varepsilon)$ and is continuous in $T^{*}M\times(-\varepsilon,\varepsilon)$.

Let us consider

and the limit equation

where $c_{0}$ is the critical value associated with $H:=G(\cdot,\cdot,0)$. We will furthermore assume the non-degeneracy integral condition (L5), where $\widetilde{\mathfrak{M}}$ denotes the set of Mather measures for $L:=L_{G}(\cdot,\cdot,0)$.

The main results contained in Section 2 can be summarized as follows, see Theorems 2.2 and 2.5.

We stress that all these three behaviors can happen at the same time for a properly chosen Hamiltonian $G$. Indeed, consider $G(x,p,u):=\sin(u)+\|p\|_{x}^{2}$. It is easlily seen that $G$ verifies all of the above conditions. The limit Hamiltonian is $H(x,p):=G(x,p,0)=\|p\|_{x}^{2}$, its critical constant is $c_{0}=0$ and the constant functions are the only solutions to $H(x,D_{x}u)=0$ in $M$. Moreover, for all $\lambda>0$, the three constant functions $u_{\lambda}=0$, and $u_{\lambda}^{\pm}=\pm\frac{\pi}{\lambda}$ are solutions to the discounted equation.

Theorem 1 shows in great generality that the only possible asymptotic behavior of families of solutions $(v_{\lambda})_{\lambda\in(0,\lambda_{0})}$ is, up to subsequences, either to uniformly diverge to $\pm\infty$, or to uniformly converge to a specific solution $u_{0}$ of (${\textrm{E}}_{0}$). We also provide two characterizations of $u_{0}$, see Theorems 2.2 and 2.3.

The general problem of existence and uniqueness of such solutions is addressed in Section 3. Here we consider the linear case $G(x,p,u):=a(x)u+H(x,p)$ under the minimal hypotheses on $a\in C(M)$ and $H\in C(T^{*}M)$ that guarantee that the conditions on $G$ and $L_{G}$ presented above are in force. The discounted equation is then

with limit equation

We prove the following existence and convergence result:

The previous theorem excludes the possibility of families of solutions that uniformly diverge to $+\infty$, but leaves open the possibility of families uniformly diverging to $-\infty$. By strengthening the non-degeneracy integral condition (L5) with a pointwise positivity condition on $a$ on the projected Aubry set $\mathcal{A}$, we are able to improve the statement of Theorem 1 as follows.

When the previous hypothesis is reinforced, we obtain a stronger conclusion:

Namely, in this last case, $(u_{\lambda})_{\lambda}$ is the only converging family of solutions.

Let us conclude this presentation by stressing that, combining this analysis with the results of [41], we fully understand the asymptotic behavior of solutions to (${\textrm{HJ}}_{\lambda}$) when $a>0$ on $\mathcal{A}$. The output is the following:

• (a)

  if $a\geqslant 0$ on the whole $M$, (this situation was considered in [41]) then, for all $\lambda>0$, there is a unique solution $u_{\lambda}$ to (${\textrm{HJ}}_{\lambda}$), and the family $(u_{\lambda})_{\lambda}$ converges as $\lambda\to 0^{+}$.

• (b)

  if there exist a point $x_{0}\in M$ such that $a(x_{0})<0$, (this situation is discussed in the present paper), then we can find a $\lambda_{0}>0$ small enough such that equation (${\textrm{HJ}}_{\lambda}$) admits at least two solutions for every $\lambda\in(0,\lambda_{0})$. The family $(u_{\lambda})_{\lambda\in(0,\lambda_{0})}$ of maximal solutions uniformly converges to $u_{0}$ as $\lambda\to 0^{+}$. Any family of other solutions $(v_{\lambda})_{\lambda}$ uniformly diverges to $-\infty$ as $\lambda\to 0^{+}$.

• $\circ$

  Throughout this paper, we assume that $M$ is a closed, connected and smooth Riemannian manifold.

• $\circ$

  We fix $g$ an auxiliary Riemannian metric on $M$. Let $d(x,y)$ be the distance between $x$ and $y$ in $M$ induced by $g$. By compactness of $M$ our results are independent on the choice of $g$.

• $\circ$

  Let diam$(M)$ be the diameter of $M$.

• $\circ$

  We denote by $TM$ and $T^{*}M$ the tangent and cotangent bundle over $M$ respectively. We denote by $(x,p)$ and $(x,v)$ points of $T^{*}M$ and $TM$ respectively.

• $\circ$

  Let $\pi:TM,T^{*}M\to M$ denote both canonical projections, the context will make it clear which one is considered.

• $\circ$

  We denote by $\|\cdot\|_{x}$ the norm on both $T_{x}M$ and $T_{x}^{*}M$ induced by $g$.

• $\circ$

  If $N$ is a smooth manifold, we will denote by $C(N)$ the Polish space of continuous functions from $N$ to $\mathbb{R}$ endowed with the metric of local uniform convergence on $N$. We will denote by $C_{c}(N)$ (resp. $C^{1}(N)$) the set of compactly supported (resp., $C^{1}$) functions from $N$ to $\mathbb{R}$.

• $\circ$

  We denote by $\mathscr{P}(TM)$ the space of Borel probability measures on $TM$ endowed with the weak-$*$ topology coming from the dual $\big{(}C_{c}(TM)\big{)}^{\prime}$.

• $\circ$

  We denote $C_{\ell}(TM)$ the set of continuous functions $g:TM\to\mathbb{R}$ with at most linear growth meaning that

  $$\sup\limits_{(x,v)\in TM}\frac{|g(x,v)|}{1+\|v\|_{x}}<+\infty.$$

  This last quantity defines a norm $\|g\|_{\ell}$ on the vector space $C_{\ell}(TM)$.

• $\circ$

  $\mathbb{N}$ denotes the set of positive integers.

We start by recalling the notion of viscosity solution.

In this paper, solutions, subsolutions, supersolutions will be always meant in the viscosity sense and implicitly assumed continuous. We recall that, if $u$ is $C^{1}$ on an open set $U$, then it is a viscosity solution (resp. subsolution, supersolution) in $U$ if and only if it is a pointwise solution (resp. subsolution, supersolution) in $U$.

The following stability result is well known, see for instance [3].

We assume $H:T^{*}M\to\mathbb{R}$ is a continuous Hamiltonian satisfying

• (H1)

  (Convexity) $H(x,p)$ is convex in $p$ for all $x\in M$.

• (H2)

  (Superlinearity) $\lim\limits_{\|p\|_{x}\to+\infty}H(x,p)/\|p\|_{x}=+\infty$.

Let $L:TM\to\mathbb{R}$ be the convex conjugate function of $H$, i.e.,

It is well known that the Lagrangian $L$ is a continuous function on $TM$ and it is convex and superlinear in $v$. The Fenchel inequality is a direct consequence of this definition:

Moreover, it can be proven that $H$ is itself the convex conjugate of $L$, i.e.,

Let $c_{0}\in\mathbb{R}$ denote the critical constant defined as follows:

We present here some facts that we will need about the critical equation, i.e.,

Solutions, subsolutions and supersolutions of (HJ₀) will be termed critical in the sequel.

Due to the convex character of $H$, the following holds, see for instance [3, 19].

More precisely, items (ii) and (iii) above require the convexity of $H$ in the momentum, while item (i) is a general fact.

Since we are assuming $H$ to be superlinear (hence coercive, which is enough), we also have the following characterization of critical subsolutions, see for instance [3, 19].

For every $t>0$, we define the minimal action function $h_{t}:M\times M\to\mathbb{R}$ as

where $\gamma:[-t,0]\to M$ is taken among all absolutely continuous curves²²2In the paper, even if not explicitly stated, all curves considered are at least absolutely continuous. satisfying $\gamma(-t)=x$ and $\gamma(0)=y$. The Peierls barrier is the function $h:M\times M\to\mathbb{R}$ defined by

It satisfies the following properties, see for instance [16]:

The projected Aubry set $\mathcal{A}$ is the closed set defined by

The following holds, see [19, 21]:

The last assertion directly follows from the definition of $c_{0}$, see (1.4).

We will say that a Borel probability measure $\tilde{\mu}$ on $TM$ is closed if it satisfies the following conditions:

• (a)

  $\displaystyle\int_{TM}\|v\|_{x}\,d\tilde{\mu}(x,v)<+\infty$;

• (b)

  for all function $f\in C^{1}(M)$, we have $\displaystyle\int_{TM}D_{x}f(v)\,d\tilde{\mu}(x,v)=0$.

We will denote by $\mathscr{P}_{0}$ the set of such measures.

We will furthermore denote by $\mathscr{P}_{\ell}$ the family of probability measures $\tilde{\mu}$ that satisfy condition (a) above. The inclusions $\mathscr{P}_{0}\subset\mathscr{P}_{\ell}\subset\big{(}C_{\ell}(TM)\big{)}^{\prime}$ hold. We will endow $\mathscr{P}_{\ell}$ with the weak-$*$ topology coming from the dual $\big{(}C_{\ell}(TM)\big{)}^{\prime}$. We refer the reader to [9] for more details on these families of measures.

Measures realizing the above minimum are called Mather measures for $L$. We denote by $\widetilde{\mathfrak{M}}$ the set of all Mather measures. This set is compact. The Mather set and the projected Mather set are defined as follows:

These sets are also compact, see [19] for a proof in the regular case.³³3For the present nonregular case, a proof of this can be found in Appendix A in the ArXiv version of [12]. Furthermore, the following holds, see [41, Proposition 3.13] for a proof in the nonregular case.

We conclude this paragraph by a technical lemma that will be of crucial use (see [9, Theorem 2-4.1.3.]).

Note that, as $L$ is bounded below, the quantity $\int_{TM}L(x,v)d\tilde{\mu}(x,v)\in\mathbb{R}\cup\{+\infty\}$ is well defined for any Borel probability measure. We also remark that in [9, Theorem 2-4.1.3.] the result is proved for a particular subclass of measures, however the proof makes no use of this fact and proves the above result.

We recall here known results that can be found in [36, 37] and the references therein. In this section, we consider a continuous Hamiltonian $G:T^{*}M\times\mathbb{R}\to\mathbb{R}$ which satisfies the following conditions

• (G1)

  (Lipschitz in $u$) $u\mapsto G(x,p,u)$ is $K$-Lipschitz continuous for some $K>0$, uniformly in $(x,p)\in T^{*}M$;

• (G2)

  (Convexity in $p$) $p\mapsto G(x,p,u)$ is convex for each $(x,u)\in M\times\mathbb{R}$;

• (G3)

  (Superlinearity in $p$) $p\mapsto G(x,p,u)$ is superlinear for each $(x,u)\in M\times\mathbb{R}$.

Let $L_{G}:TM\times\mathbb{R}\to\mathbb{R}$ be the convex conjugate function of $G$, i.e.,

Then it can be proven that $L_{G}$ verifies similar properties (see [5, Lemma 4.1] with easy adaptations):

• (L1)

  $u\mapsto L_{G}(x,v,u)$ is $K$-Lipschitz continuous uniformly in $(x,v)\in TM$;

• (L2)

  $v\mapsto L_{G}(x,v,u)$ is convex for each $(x,u)\in M\times\mathbb{R}$;

• (L3)

  $v\mapsto L_{G}(x,v,u)$ is superlinear for each $(x,u)\in M\times\mathbb{R}$;

By [37, Appendix D] and [36, Appendix A] we have

We then give an approximation result in this setting. It is an easy consequence of [20, Theorem 8.5].

Let $a\in C(M)$. Assume there are two points $x_{1}$ and $x_{2}$ such that $a(x_{1})>0$ and $a(x_{2})<0$. Let $c\in\mathbb{R}$ and consider the equation

This is a particular case of the previous setting for $G(x,p,u)=a(x)u+H(x,p)$. In this case, the associated Lagrangian is $L_{G}(x,v,u)=L(x,v)-a(x)u$. The following implicit Lax-Oleinik semigroup $(T^{-}_{t})_{t\geqslant 0}$ is a well defined semigroup of operators $T^{-}_{t}:C(M)\to C(M)$ that verify, for all $\varphi\in C(M)$,

where the infimum is taken among absolutely continuous curves $\gamma:[0,t]\rightarrow M$ with $\gamma(t)=x$. A similar property defines and characterizes the forward semigroup $(T^{+}_{t})_{t>0}$, where, for all $\varphi\in C(M)$,