Topological and control theoretic properties of Hamilton-Jacobi equations via Lax-Oleinik commutators

Suppose $M$ is a smooth connected and compact manifold without boundary with $TM$ and $T^{*}M$ the tangent and cotangent bundle of $M$ respectively. Let $H$ be a Hamiltonian on $M$. The study of the forward Hamilton-Jacobi equation

and the backward Hamilton-Jacobi equation

plays an important rôle in many fields such as the calculus of variation and optimal control ([5, 16]), optimal transport ([26, 1]), Hamiltonian dynamical systems and PDE ([23, 19]). In principle, the solution $u(t,x)$ has a representation formula by Lax-Oleinik evolution, under very general regularity assumptions on $H$ and $u_{0}$. More precisely, for any $\phi:M\to\mathbb{R}$, we define the abstract Lax-Oleinik operators as

where $c_{t}(x,y):[0,+\infty)\times M\times M$ is the (dynamical) cost function. If $c_{t}(x,y)=A_{t}(x,y)$ is the action of the Lagrangian $L$ associated with $H$, then the solution of (1.1) has the representation $u^{-}(t,x)=T^{-}_{t}u_{0}(x)$ and the solution of (1.2) has the representation $u^{+}(t,x)=T^{+}_{T-t}u_{0}(x)$ and the uniqueness issues of (1.1) and (1.2) are well established. However, we will only touch the part for the case $c_{t}(x,y)=A_{t}(x,y)$, the fundamental solution of (1.1) and (1.2), in the current paper.

In this paper, we always suppose $H:T^{*}M\to\mathbb{R}$ is a Tonelli Hamiltonian and $L:TM\to\mathbb{R}$ is the associated Tonelli Lagrangian. We take the cost function $c_{t}(x,y)=A_{t}(x,y)$ with

where $c[H]$ is the Mañé’s critical value and $\Gamma^{t}_{x,y}$ is the set of absolutely continuous curves $\gamma:[0,t]\to M$ connecting $x$ to $y$. For convenience, we always suppose $c[H]=0$.

To understand the relation between the solutions of (1.1) and (1.2), we will study the Lax-Oleinik commutators $\{T^{-}_{t}\circ T^{+}_{t}\}_{t\geqslant 0}$ and $\{T^{+}_{t}\circ T^{-}_{t}\}_{t\geqslant 0}$ instead of the semigroups $\{T^{\pm}_{t}\}_{t\geqslant 0}$. A key observation is to characterize a function $\phi:M\to\mathbb{R}$ satisfying the relation

Our first result is

It is worth noting that Theorem 3.5 can be viewed as a controllability result for equation (1.1). More precisely, our result ensures that a function $\phi$ can be reached by some initial data $\psi$ for $T^{-}_{t}$ if and only if $\phi$ satisfies condition (1.3). We also provide a detailed description of the relation between this controllability result and underlying dynamics in Section 3.4. This kind of problems have been already addressed in the literature. For instance, in [6], for autonomous equations of the form

it shown that any solution $u$ is of class $C^{1}$ if solving the equation backward from $u(T,\cdot)$ one finds the value $u(0,\cdot)$ at $t=0$ (compare with Theorem 3.5). As for controllability, let us recall Proposition 9 in [3] describes a subset of $C(M,\mathbb{R})$ that is attainable for the negative Lax-Oleinik seigroup associated with the autonomous equation (1.4) (see also Proposition 5 in [2]) for a local version of this result that applies to more general Hamiltonians). A similar attainability result for (1.4) is obtained in [18] for nonsmooth Hamiltonians. In the case $t\ll 1$, (1.3) is closely related to a theorem by Marie-Claude Arnaud ([4]) on the evolution of the 1-graph of a semiconcave function under the Hamiltonian flow, as well as Lasry-Lions type regularization in the context of weak KAM theory as first studied by Patrick Bernard ([7]) (see Proposition 3.2).

The following result gives a new characterization of weak KAM solutions of the stationary Hamilton-Jacobi

In other words, weak KAM solutions are exactly those functions that are reachable for $T^{-}_{t}$ for all $t>0$.

We remark that $T^{-}_{t}\circ T^{+}_{t}\phi=\phi$ implies that $\phi$ is semiconcave automatically. If the equality $T^{-}_{t}\circ T^{+}_{t}\phi=\phi$ holds, then there exists $\psi:M\to\mathbb{R}$ which is semiconvex such that

Such a pair $(\phi,\psi)$ is also called an admissible Kantorovich pair (for the cost function $c_{t}(x,y)=A_{t}(x,y)$) in the theory of optimal transport (see [10, 11, 26]).

The second part of this paper is on the structure of the cut locus of weak KAM solutions to (1.5). Let $u$ be a weak KAM solution of (1.5). For any $x\in M$, the cut time function of $u$ is defined by

Moreover, $\tau(x)$ can be related to the commutators of the Lax-Oleinik semigroups as follows (see Section 3.3)

Then, we define the cut locus of $u$, $\mbox{\rm Cut}\,(u)$, and the Aubry set of $u$, $\mathcal{I}\,(u)$, as follows

In [14, 15] we proved that the complement of $\mathcal{I}\,(u)$ is homotopically equivalent to $\mbox{\rm Cut}\,(u)$.

Given a weak KAM solution $u$ of (1.5), we define

The following result characterizes the super/sub-level sets of the cut time function $\tau$ for $t>0$.

Finally, we remark that these results have essential applications to our recent work on the intrinsic construction of generalized characteristics and strict singular characteristics [12]. These results clarify the relations among certain features of Hamilton-Jacobi equations such as irreversibility, non-commutativity, and singularity.

The paper is organized as follows. In section 2, we give a brief introduction to weak KAM theory and Hamilton-Jacobi equations. In Section 3, we discuss equality $T^{-}_{t}\circ T^{+}_{t}\phi=\phi$ for both small time and arbitrary time $t$, from both functional level and underlying dynamics. We also discuss the long time behavior of the operators $T^{-}_{t}\circ T^{+}_{t}$. We finally analyze the commutator of $T^{+}_{t}$ and $T^{-}_{t}$ and its implications in the structure of the cut locus.

For any smooth connected and closed manifold $M$, we call $H=H(x,p):T^{*}M\to\mathbb{R}$ a Tonelli Hamiltonian if $H$ is of class $C^{2}$, the function $H(x,\cdot)$ is strictly convex and uniformly superlinear. The associated Lagrangian is defined by Legendre transformation:

As a real-valued function on $TM$, $L$ is called a Tonelli Lagrangian. That is $L$ is also of class $C^{2}$ and the function $L(x,\cdot)$ is strictly convex and uniformly superlinear. From classical calculus of variation, we say that a curve $\gamma:[a,b]\to M$ is an extremal if it satisfies the Euler-Lagrange equations locally. We denote by $\{\Phi^{t}_{H}\}_{t\in\mathbb{R}}$ the flow associated to the Hamiltonian vector field of $H$.

Given a Tonelli Hamiltonian $H$ with its associated Tonelli Lagrangian $L$. Mañé’s critical value of $H$ is the unique real number $c[H]$ such that the Hamilton-Jacobi equation

admits a viscosity solution. We always suppose $c[H]=0$ in this paper for convenience. For $\phi\in C(M)$, we define the Lax-Oleinik operators as

A function $\phi:M\to\mathbb{R}$ is called a negative (resp. positive) weak KAM solution if $T^{-}_{t}\phi=\phi$ (resp. $T^{+}_{t}\phi=\phi$) for all $t\geqslant 0$. It is known that weak KAM solutions and viscosity solutions coincide. We usually omit the adjective “negative” if we do not mention the positive one in the context. A pair of functions $(u^{-},u^{+})$ on $M$ is called a weak KAM pair if $u^{-}$ and $u^{+}$ are negative and positive weak KAM solutions, respectively, and they coincide on the projected Mather set (see more in [19, Section 5.1], [20]).

Any viscosity sub-solution of (2.1) is called a critical sub-solution. The following proposition characterizes critical sub-solutions.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded and open set, a function $\phi:\Omega\to\mathbb{R}$ is a semiconcave function (with linear modulus) if there is a constant $C>0$ such that

for any $x,y\in\Omega$ and $\lambda\in[0,1]$. Any constant $C$ that satisfies the above inequality is called a semiconcavity constant for $u$ in $\Omega$.

Let $\phi:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ be a continuous function. We recall that, for any $x\in\Omega$, the closed convex sets

are called the subdifferential and superdifferential of $\phi$ at $x$, respectively.

Let now $\phi:\Omega\to\mathbb{R}$ be locally Lipschitz. We recall that a vector $p\in\mathbb{R}^{n}$ is said to be a reachable (or limiting) gradient of $\phi$ at $x$ if there exists a sequence $\{x_{n}\}\subset\Omega\setminus\{x\}$, converging to $x$, such that $\phi$ is differentiable at $x_{k}$ for each $k\in\mathbb{N}$ and $\lim_{k\to\infty}D\phi(x_{k})=p$. The set of all reachable gradients of $\phi$ at $x$ is denoted by $D^{\ast}\phi(x)$.

Proposition 2.2 is useful to understand semiconcave functions and their superdifferential on manifolds. Suppose $M$ is endowed with a Riemannian metric $g$, with $d$ the associated Riemannian distance. We call $\phi:M\to\mathbb{R}$ semiconcave with constant $C$ if

where $\Gamma_{x,y}$ is the set of all geodesics $\gamma:[0,1]\to M$ connecting $x$ to $y$. For any $x\in M$, we say that $p_{x}\in D^{+}\phi(x)$ if $p_{x}$ is a one form such that

One can show that this definition of semiconcavity is independent of the choice of the Riemannian metric, even though semiconcavity constants may change. The reader can refer to [26, 25, 22] for more on semiconcave and convex functions on manifolds.

The following Moreau-Yosida-Lasry-Lions regularization type result in the context of weak KAM theory is due to Patrick Bernard.

In this section we consider the Hamilton-Jacobi equation