∎

¹¹institutetext: Xifeng Su²²institutetext: Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
Tel: +86-15910594182
²²email: [email protected], [email protected] ³³institutetext: Jianlu Zhang ⁴⁴institutetext: Hua Loo-Keng Key Laboratory of Mathematics & Mathematics Institute, Academy of Mathematics and systems science, Chinese Academy of Sciences, Beijing 100190, China
Tel: +86-182-1038-3625
⁴⁴email: [email protected]

Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

Xifeng Su Jianlu Zhang

(Received: date / Accepted: date)

Abstract

For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$ , we construct a unique forward weak KAM solution of

H(x,d_{x}u)=c(H)

by a vanishing discount approach, where $c(H)$ is the Mañé critical value. We also discuss the dynamical significance of such a special solution.

Keywords:

discounted equation, weak KAM solution, Aubry Mather theory, Hamilton-Jacobi equation, viscosity solution

MSC:

35B40, 37J50, 49L25,37J51

1 Introduction

For a compact connected manifold $M$ without boundary, the Hamiltonian is usually mentioned as a continuous function defined on its cotangent bundle $T^{*}M$ . In LPV , the authors firstly proposed the ergodic approximation technique, to consider the existence of so called viscosity solutions to the Hamilton-Jacobi equation

H(x,d_{x}u)=c(H),\quad x\in M

(HJ₀)

for the Mañé critical value

c(H):=\inf\{c\in\mathbb{R}|\exists\ \omega\in C(M,\mathbb{R})\text{ such that }H(x,d_{x}\omega)\leq c,\quad a.e.\ x\in M\}.

The Hamiltonian they concerned satisfies

•

(Coercivity) $H(x,p)$ is coercive in $p\in T_{x}^{*}M$ , uniformly w.r.t. $x\in M$ .
•

(Convexity) $H(x,p)$ is convex in $p\in T_{x}^{*}M$ for all $x\in M$ .

They perturbed (HJ₀) by the following discounted equation

\displaystyle\lambda u+H(x,d_{x}u)=c(H),\quad x\in M,\lambda>0

(1)

of which the Comparison Principle is allowed. Therefore, the viscosity solution $u_{\lambda}^{-}$ of (1) is unique. In DFIZ , they established the convergence of $u_{\lambda}^{-}$ as $\lambda\rightarrow 0_{+}$ , to a specified viscosity solution $u_{0}^{-}$ of (HJ₀) which can be characterized by the combination of subsolutions of (HJ₀), or Peierls barrier

h^{\infty}:M\times M\rightarrow\mathbb{R}

w.r.t. the projected Mather measures $\mathfrak{M}$ of (HJ₀), see Appendix A for the relevant definitions of $\mathfrak{M},\ h^{\infty}$ , subsolutions etc.

In this paper, we consider a negative limit technique and try to find another specified solution of (HJ₀). Precisely, we consider

-\lambda u+H(x,d_{x}u)=c(H),\quad x\in M,\lambda>0

(HJ_λ)

of which a unique forward $\lambda-$ weak KAM solution $u_{\lambda}^{+}$ can be found (see Remark 2.3 for the definition). As $\lambda\rightarrow 0_{+}$ , we get the following conclusion:

Theorem 1.1

Let $H:T^{*}M\rightarrow\mathbb{R},(x,p)\mapsto H(x,p)$ be a continuous Hamiltonian coercive and convex in $p$ . For $\lambda>0$ , the unique forward $\lambda-$ weak KAM solution $u_{\lambda}^{+}$ of (HJ_λ) uniformly converges as $\lambda\rightarrow 0_{+}$ , to a unique forward $0-$ weak KAM solution¹¹1see Definition A.6 $u_{0}^{+}$ of (HJ₀), which can be interpreted as

\displaystyle u_{0}^{+}(x)=\inf\mathcal{F}_{+}

(2)

with

\displaystyle\mathcal{F}_{+}:=\Big{\{}w\text{ is a subsolution of (\ref{eq:hj}) }\Big{|}\int_{M}wd\mu\geq 0,\ \forall\mu\in\mathfrak{M}\Big{\}}

(3)

and

\displaystyle u_{0}^{+}(x)=-\inf_{\mu\in\mathfrak{M}}\int_{M}h^{\infty}(x,y)d\mu(y).

(4)

Remark 1.2

The novelty of this paper is that we adapt a symmetric Lagrangian skill to our $C^{0}-$ setting. The lack of regularity invalidates a bunch of important properties of the Mather measures, Peierls barrier etc., so we have to find substitutes in the weak sense.

Besides, we mention that $-u_{0}^{+}(x)$ is a viscosity solution of the symmetric equation

H(x,-d_{x}u(x))=c(H).

Comparing to the backward $0-$ weak KAM solutions, the notion of viscosity solutions is more familiar to PDE specialists, although both are proved to be equivalent in F .

1.1 Dynamic interpretation of $u_{0}^{+}$

Now the vanishing discount approach supplies us with a pair of solutions of (HJ₀):

\left\{\begin{aligned} &u_{0}^{-}(x)=\inf_{\mu\in\mathfrak{M}}\int_{M}h^{\infty}(y,x)d\mu(y),\\ &u_{0}^{+}(x)=-\inf_{\mu\in\mathfrak{M}}\int_{M}h^{\infty}(x,y)d\mu(y).\end{aligned}\right.

(5)

Definition 1.3 (Conjugated Pair)

A backward $0-$ weak KAM solution $u^{-}$ of (HJ₀) is conjugated to a forward $0-$ weak KAM solution $u^{+}$ , if

•

$u^{-}=u^{+}$ on the projected Mather set $\mathcal{M}$ (see Definition A.4).
•

$u^{-}\geq u^{+}$ on $M$ .

In F , above definition is actually proposed to $C^{2}-$ Tonelli Hamiltonian²²2 $H:(x,p)\in T^{*}M\rightarrow\mathbb{R}$ is called Tonelli, if it’s positive definite and superlinear in $p$ . However, we have no difficulty to reserve this concept to the $C^{0}-$ case by means of the evidence in DS . Moreover, For the following typical Hamiltonians, $(u_{0}^{-},u_{0}^{+})$ indeed forms a conjugated pair:

1.

(uniquely ergodic) Suppose $\mathfrak{M}$ consists of a uniquely ergodic projected Mather measure (generic for $C^{2}-$ Tonelli Hamiltonians, see Mn ), then

$d_{c}(x,y):=h^{\infty}(x,y)+h^{\infty}(y,x)\geq 0$

for all $x,y\in M$ , and ‘ $=$ ’ holds for $x,y\in\mathcal{M}$ (due to the definition of the Mather measure in Appendix A ). So $(u_{0}^{-},u_{0}^{+})$ is a conjugated pair.
2.

(mechanical system) For a mechanical Hamiltonian

$H(x,p)=\frac{1}{2}\langle p,p\rangle+V(x),$

we can easily get $c(H)=\max_{x\in M}V(x)$ , then the associated $L(x,v)+c(H)\geq 0$ on $TM$ . Consequently, $h^{\infty}:M\times M\rightarrow\mathbb{R}$ is nonnegative, so $u_{0}^{-}\geq u_{0}^{+}$ on $M$ . On the other side, due to the definition of $\mathfrak{M}$ , all the Mather measures are supported by equilibriums. So $u_{0}^{-}=u_{0}^{+}$ on $\mathcal{M}$ . In summary $(u_{0}^{-},u_{0}^{+})$ is a conjugated pair.
3.

(constant subsolution) Such a case is also discussed in DFIZ . If $H(x,0)\leq c(H)$ for all $x\in M$ , i.e. constant is a subsolution of (HJ₀), then due to the Young Inequality we get

$L(x,v)+c(H)\geq L(x,v)+H(x,0)\geq\langle v,0\rangle=0$

for all $(x,v)\in TM$ , by a similar analysis like the case of mechanical systems $(u_{0}^{-},u_{0}^{+})$ proves to be a conjugated pair.

1.2 Organization of the article.

In Sec. 2, we prove some variational properties for nonsmooth Lagrangians. In Sec. 3, we prove the convergence of $u_{\lambda}^{+}$ as $\lambda\rightarrow 0_{+}$ and give a representative formula for the limit. For the consistency and readability of the article, some preliminary materials are moved to Appendix.

Acknowledgement. J. Zhang is supported by the National Natural Science Foundation of China (Grant No. 11901560). X. Su is supported by the National Natural Science Foundation of China (Grant No. 11971060, 11871242).

2 Nonsmooth symmetric Lagrangians

With the same adaption as in DFIZ , without loss of generality we can assume $H(x,p)$ is superlinear in $p$ , i.e.

•

(Superlinearity) $\lim_{|p|\rightarrow+\infty}H(x,p)/|p|=+\infty$ , for any $x\in M$ .

In that case, by Fenchel’s formula (see 10), the Hamiltonian has an associated Lagrangian $L:(x,v)\in TM\rightarrow\mathbb{R}$ which is superlinear and convex in the fibers of the tangent bundle. Consequently, we can propose a symmetrical Lagrangian $\widehat{L}(x,v):=L(x,-v)$ , of which the following fundamental facts hold:

Lemma 2.1

(i)

The conjugated Hamiltonian $\widehat{H}:T^{*}M\rightarrow\mathbb{R}$ of $\widehat{L}(x,v)$ satisfies $\widehat{H}(x,p)=H(x,-p)$ for all $(x,p)\in T^{*}M$ . Therefore, $\widehat{H}$ is also continuous, superlinear and convex.
(ii)

$\widehat{H}(x,d_{x}\omega)\leq c\iff H(x,d_{x}(-\omega))\leq c$
(iii)

$c(\widehat{H})=c(H)$ .
(iv)

The projected Mather measure set $\widehat{\mathfrak{M}}$ (associated with $\widehat{H}(x,p)$ ) keeps the same with $\mathfrak{M}$ .
(v)

The Peierls barrier function associated with $\widehat{L}(x,v)$ satisfies $\widehat{h}^{\infty}(y,x)=h^{\infty}(x,y)$ for any $x,y\in M$ .

Proof

(i). Due to (10) and the definition of $\widehat{L}$ , we have

\begin{split}\widehat{H}(x,p)&=\max_{v\in T_{x}M}\{\langle p,v\rangle-\widehat{L}(x,v)\}\\ &=\max_{v\in T_{x}M}\{\langle p,v\rangle-L(x,-v)\}\\ &=\max_{w\in T_{x}M}\{\langle-p,w\rangle-L(x,w)\}=H(x,-p),\end{split}.

as is desired.

(ii). If $\omega$ is a subsolution of $\widehat{H}(x,d_{x}\omega)\leq c$ , then for any absolutely continuous $\gamma:[-T,T]\rightarrow M$ connecting $x,y\in M$ , we get

\omega(y)-\omega(x)\leq\int_{-T}^{T}\big{(}\widehat{L}(\gamma,\dot{\gamma})+c\big{)}dt.

Furthermore,

$\displaystyle-\omega(x)-(-\omega(y))$	$\displaystyle=$	$\displaystyle\omega(y)-\omega(x)$
	$\displaystyle\leq$	$\displaystyle\int_{-T}^{T}\big{(}\widehat{L}(\gamma,\dot{\gamma})+c\big{)}dt$
	$\displaystyle=$	$\displaystyle\int_{-T}^{T}\big{(}L(\gamma,-\dot{\gamma})+c\big{)}dt$
	$\displaystyle=$	$\displaystyle\int_{-T}^{T}\big{(}L(\widehat{\gamma},-\dot{\widehat{\gamma}})+c\big{)}dt$

with $\widehat{\gamma}(t):=\gamma(-t)$ for all $t\in[-T,T]$ . As $\gamma$ is arbitrarily chosen, so we get $-\omega\prec L+c$ ⁴⁴4see Definition A.5, then $H(x,-d_{x}\omega)\leq c$ for a.e. $x\in M$ . Similarly, $\omega\prec L+c$ indicates $-\omega\prec\widehat{L}+c$ .

(iii). As $c(\widehat{H})=\inf\{c\in\mathbb{R}|\exists\ \omega\in C(M,\mathbb{R})\text{ such that }\omega\prec\widehat{L}+c\}$ , then due to (ii), $c(\widehat{H})=c(H)$ .

(iv). Due to Proposition 2-4.3 of CI , for any measure $\widetilde{\mu}\in\widetilde{\mathfrak{M}}$ , there exists a sequence of closed measures $\widetilde{\mu}_{n}\in\mathbb{P}_{c}(TM)$ (defined in Appendix A), such that $\widetilde{\mu}_{n}$ weakly converges to $\widetilde{\mu}$ and

\lim_{n\rightarrow+\infty}\int_{TM}Ld\widetilde{\mu}_{n}=\int_{TM}Ld\widetilde{\mu}.

Moreover, for each $\widetilde{\mu}_{n}$ there exists an absolutely continuous curve $\gamma_{n}:t\in[-T_{n},T_{n}]\rightarrow M$ with $T_{n}\rightarrow+\infty$ as $n\rightarrow+\infty$ , such that

\int_{TM}gd\mu_{n}=\frac{1}{2T_{n}}\int_{-T_{n}}^{T_{n}}g(\gamma_{n}(t),\dot{\gamma}_{n}(t))dt,\quad\forall\\ g\in C_{c}(TM,\mathbb{R}).

Therefore, for $\widehat{\gamma}_{n}(t):=\gamma_{n}(-t)$ , we have

$\displaystyle-c(H)$	$\displaystyle=$	$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{2T_{n}}\int_{-T_{n}}^{T_{n}}L(\gamma_{n}(t),\dot{\gamma}_{n}(t))dt$
	$\displaystyle=$	$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{2T_{n}}\int_{-T_{n}}^{T_{n}}L(\widehat{\gamma}_{n}(-t),-\dot{\widehat{\gamma}}_{n}(-t))dt$
	$\displaystyle=$	$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{2T_{n}}\int_{-T_{n}}^{T_{n}}\widehat{L}(\widehat{\gamma}_{n}(-t),\dot{\widehat{\gamma}}_{n}(-t))dt$
	$\displaystyle=$	$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{2T_{n}}\int_{-T_{n}}^{T_{n}}\widehat{L}(\widehat{\gamma}_{n}(s),\dot{\widehat{\gamma}}_{n}(s))ds$
	$\displaystyle=$	$\displaystyle-c(\widehat{H}).$

That indicates $S^{*}\widetilde{\mu}$ is a Mather measure for $\widehat{L}(x,v)$ , where $S:TM\rightarrow TM$ is a diffeomorphism defined by $S(x,v)=(x,-v)$ . Namely, we have

\int_{TM}g(x,v)dS^{*}\widetilde{\mu}(x,v):=\int_{TM}g(x,-v)d\widetilde{\mu}(x,v),\quad\forall g\in C_{c}(TM,\mathbb{R}).

Due to (A) and $S\circ S=id$ ,

$\displaystyle\int_{M}f(x)d\mu(x)$	$\displaystyle=$	$\displaystyle\int_{TM}f\circ\pi(x,v)d\widetilde{\mu}(x,v)$
	$\displaystyle=$	$\displaystyle\int_{TM}f\circ\pi(x,-v)dS^{*}\widetilde{\mu}(x,v)$
	$\displaystyle=$	$\displaystyle\int_{M}f(x)d\pi^{}S^{}\widetilde{\mu}(x,v)$

for all $f\in C(M,\mathbb{R})$ , then $\pi^{*}S^{*}\widetilde{\mu}=\mu\in\mathfrak{M}$ . So $\widehat{\mathfrak{M}}=\mathfrak{M}$ .

(v). Due to the definition of the Peierls barrier function, we calculate

$\displaystyle\widehat{h}^{\infty}(y,x)$	$\displaystyle=$	$\displaystyle\liminf_{t\rightarrow+\infty}\left(\inf_{\begin{subarray}{c}\xi\in C^{ac}([0,t],M)\\ \xi(0)=y\\ \xi(t)=x\end{subarray}}\int_{0}^{t}\widehat{L}(\xi(s),\dot{\xi}(s))ds+c(H)t\right)$
	$\displaystyle=$	$\displaystyle\liminf_{t\rightarrow+\infty}\left(\inf_{\begin{subarray}{c}\gamma\in C^{ac}([0,t],M)\\ \gamma(t)=y\\ \gamma(0)=x\end{subarray}}\int_{0}^{t}L\big{(}\gamma(t-s),-\frac{d\gamma(t-s)}{ds}\big{)}ds+c(H)t\right)$
	$\displaystyle=$	$\displaystyle\liminf_{t\rightarrow+\infty}\left(\inf_{\begin{subarray}{c}\gamma\in C^{ac}([0,t],M)\\ \gamma(0)=x\\ \gamma(t)=y\end{subarray}}\int_{0}^{t}L(\gamma(\tau),\dot{\gamma}(\tau))d\tau+c(H)t\right)$
	$\displaystyle=$	$\displaystyle h^{\infty}(x,y),$

where $\xi(s):=\gamma(t-s)$ for $s\in[0,t]$ . ∎

Now we can propose a function

\displaystyle u_{\lambda}^{+}(x):=-\inf_{\begin{subarray}{c}\gamma\in C^{ac}([0,+\infty),M)\\ \gamma(0)=x\end{subarray}}\int_{0}^{+\infty}e^{-\lambda t}\big{(}L(\gamma(t),\dot{\gamma}(t))+c(H)\big{)}\ dt,

(6)

of which the following properties hold:

Proposition 2.2

(1)

For $\lambda\in(0,1]$ , $u_{\lambda}^{+}$ is uniformly Lipschitz and equi-bounded on $M$ , with the Lipschitz (resp. equi-bound) constant depending only on $L$ .
(2)

For every $x\in M$ , there exists a forward curve $\gamma_{\lambda,x}^{+}:[0,+\infty)\rightarrow M$ which achieves the minimum of (6).

(3)

(Domination) $u_{\lambda}^{+}$ is $\lambda$ -dominated by $L$ and is denoted by $u_{\lambda}^{+}\prec_{-\lambda}L+c(H)$ , i.e., for any $(x,y)\in M\times M$ and $b-a\geq 0$ , we have

e^{-\lambda b}u_{\lambda}^{+}(y)-e^{-\lambda a}u_{\lambda}^{+}(x)\leq\inf_{\begin{subarray}{c}\gamma\in C^{ac}([a,b],M)\\ \gamma(a)=x,\gamma(b)=y\end{subarray}}\int_{a}^{b}e^{-\lambda s}\Big{(}L(\gamma,\dot{\gamma})+c(H)\Big{)}ds

(4)

(Calibration) For any $t\geq 0$ ,

u_{\lambda}^{+}(x)=e^{-\lambda t}u_{\lambda}^{+}(\gamma_{\lambda,x}^{+}(t))+\int_{t}^{0}e^{-\lambda s}\Big{(}L(\gamma_{\lambda,x}^{+}(s),\dot{\gamma}_{\lambda,x}^{+}(s))+c(H)\Big{)}ds.

Such a curve $\gamma_{\lambda,x}^{+}$ is called a forward calibrated curve of $u_{\lambda}^{+}$ .

(5)

$-u_{\lambda}^{+}(x)$ is the viscosity solution of the following symmetrical H-J equation

$\displaystyle\lambda u+\widehat{H}(x,\partial_{x}u)=c(H),\quad\lambda>0.$ (7)

Remark 2.3 (forward $\lambda-$ weak KAM solution)

A continuous function $w:M\rightarrow\mathbb{R}$ is called a forward $\lambda-$ weak KAM solution of (HJ_λ) if it satisfies item (3) and (4) of Proposition 2.2. Due to Property (5) of Proposition 2.2, such a forward $\lambda-$ weak KAM solution is unique.

Proof of Proposition 2.2: (5) By a simple transformation, we can see

\displaystyle-u_{\lambda}^{+}(x)=\inf_{\begin{subarray}{c}\xi\in C^{ac}((-\infty,0],M)\\ \xi(0)=x\end{subarray}}\int^{0}_{-\infty}e^{-\lambda t}\big{(}\widehat{L}(\xi(t),\dot{\xi}(t))+c(\widehat{H})\big{)}\ dt.

Due to Appendix 2 of DFIZ , $-u_{\lambda}^{+}$ is a viscosity solution of (7), which is unique due to the Comparison Principle.

(1) For any viscosity solution $u_{0}(x)$ of (HJ₀), we get

\underline{u}_{0}(x):=u_{0}(x)-\|u_{0}\|\leq 0\leq u_{0}(x)+\|u_{0}\|:=\bar{u}_{0}(x),\quad\forall x\in M.

Consequently, $\underline{u}_{0}$ (resp. $\bar{u}_{0}$ ) is a subsolution (resp. supersolution) of (7). Due to the Comparison Principle again, for any $\lambda>0$ and any viscosity solution $\omega_{\lambda}$ of (7) satisfies $\underline{u}_{0}\leq\omega_{\lambda}\leq\bar{u}_{0}$ . So we get the equi-boundedness of $\{u_{\lambda}^{+}\}_{\lambda\in(0,1]}$ .

Let $\gamma:[0,d(x,y)]\rightarrow M$ be the geodesic joining $y$ to $x$ parameterized by the arc-length, where $d:M\times M\rightarrow\mathbb{R}$ is the Euclidean distance. For every absolute continuous curve $\xi:[0,+\infty)\rightarrow M$ with $\xi(0)=x$ , we define a curve

\eta(t)=\left\{\begin{aligned} &\gamma(t),\qquad\qquad\qquad t\in[0,d(x,y)],\\ &\xi(t-d(x,y)),\qquad t\in[d(x,y),+\infty).\end{aligned}\right.

(8)

Then we have

\begin{split}-u_{\lambda}^{+}(y)&\leq\int_{0}^{+\infty}e^{-\lambda t}\big{(}L(\eta(t),\dot{\eta}(t))+c(H)\big{)}\ dt\\ &\leq\int_{0}^{d(x,y)}e^{-\lambda t}\big{(}L(\gamma(t),\dot{\gamma}(t))+c(H)\big{)}\ dt\\ &\quad+\int_{d(x,y)}^{+\infty}e^{-\lambda t}\big{(}L(\xi(t-d(x,y))),\dot{\xi}(t-d(x,y)))+c(H)\big{)}\ dt\\ &\leq\int_{0}^{d(x,y)}e^{-\lambda t}\big{(}L(\gamma(t),\dot{\gamma}(t))+c(H)\big{)}\ dt\\ &\quad+e^{\lambda d(x,y)}\int_{0}^{+\infty}e^{-\lambda t}\big{(}L(\xi(t)),{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\dot{\xi}(t)})+c(H)\big{)}\ dt.\end{split}

By minimizing with respect to all $\xi\in C^{ac}\big{(}[0,+\infty)\big{)}$ with $\xi(0)=x$ , we obtain

-u_{\lambda}^{+}(y)\leq-e^{\lambda d(x,y)}u_{\lambda}^{+}(x)+\int_{0}^{d(x,y)}e^{-\lambda t}\big{(}L(\gamma(t),\dot{\gamma}(t))+c(H)\big{)}\ dt.

Therefore, we have

\begin{split}u_{\lambda}^{+}(x)-u_{\lambda}^{+}(y)&\leq\big{(}1-e^{\lambda d(x,y)}\big{)}u_{\lambda}^{+}(x)+\int_{0}^{d(x,y)}e^{-\lambda t}\big{(}L(\gamma(t),\dot{\gamma}(t))+c(H)\big{)}\ dt\\ &\leq\frac{1-e^{\lambda d(x,y)}}{\lambda}\left(|\lambda u_{\lambda}^{+}(x)|+C_{1}\right)\leq(C+C_{1})d(x,y),\end{split}

where $C:=\max\{|\max_{x\in M}L(x,0)+c(H)|,|\min_{(x,v)\in TM}L(x,v)+c(H)\}$ and $C_{1}:=\max\{L(x,v)~{}:~{}x\in M,\|v\|\leq 1\}$ . By exchanging the role of $x$ and $y$ , we get the other inequality, which shows that $u_{\lambda}^{+}$ is uniformly Lipschitz and the Lipschitz constant is independent of $\lambda$ .

(2), (3) & (4) By a similar analysis as Propositions 6.2–6.3 in DFIZ , all these three items can be easily proved. ∎

3 Discounted limit of forward $\lambda-$ weak KAM solutions

Recall that $\widehat{u}_{\lambda}^{-}:=-u_{\lambda}^{+}$ is the unique viscosity solution of (7),

\widehat{u}_{\lambda}^{-}(x)=\inf_{\gamma(0)=x}\int_{-\infty}^{0}e^{\lambda t}\big{(}\widehat{L}(\gamma,\dot{\gamma})+c(H)\big{)}dt.

So the following conclusion holds instantly:

Lemma 3.1

DFIZ As $\lambda\rightarrow 0_{+}$ , $\widehat{u}_{\lambda}^{-}$ converges to a unique function $\widehat{u}_{0}^{-}$ which is a viscosity solution of the following

\displaystyle\widehat{H}(x,\partial_{x}u)=c(H)

(9)

with the following two different interpretations:

•

$\widehat{u}_{0}^{-}$ is the maximal subsolution $w:M\rightarrow\mathbb{R}$ of (9) such that for any projected Mather measure $\widehat{\mu}\in\widehat{\mathfrak{M}}$ , $\int_{M}w\cdot d{\widehat{\mu}}\leq 0$ .
•

$\widehat{u}_{0}^{-}$ is the infimum of functions $\widehat{h}_{\mu}^{\infty}$ defined by

$\widehat{h}_{\mu}^{\infty}(x):=\int_{M}\widehat{h}^{\infty}(y,x)d\widehat{\mu}(y),\quad\widehat{\mu}\in\widehat{\mathfrak{M}}.$

Due to Lemma 2.1 and Lemma 3.1, we get

u_{0}^{+}:=\lim_{\lambda\rightarrow 0_{+}}u_{\lambda}^{+}=-\lim_{\lambda\rightarrow 0_{+}}\widehat{u}_{\lambda}^{-}=-\widehat{u}_{0}^{-}

which is uniquely established and interpreted as the following:

Lemma 3.2

$u_{0}^{+}$ is a forward $0-$ weak KAM solution of (HJ₀).

Proof

As $\widehat{u}_{0}^{-}$ is the viscosity solution of (9), then $\widehat{u}_{0}^{-}\prec\widehat{L}+c(H)$ due to Proposition 5.3 of DFIZ . On the other side, due to

U(x,t):=\inf_{\begin{subarray}{c}\gamma\in C^{ac}([0,t],M)\\ \gamma(0)=x\end{subarray}}\{\widehat{u}_{0}^{-}(\gamma(-t))+\int^{0}_{-t}\widehat{L}(\gamma(\tau),\dot{\gamma}(\tau))+c(H)\mbox{d}\tau\},\quad\forall t\geq 0,

is the unique viscosity solution of the Cauchy problem

\left\{\begin{aligned} &\partial_{t}u+\widehat{H}(x,d_{x}u)=c(H)\\ &u(x,0)=\widehat{u}_{0}^{-}(x),\quad t\geq 0,\end{aligned}\right.

whereas $\widehat{u}_{0}^{-}(x)$ is also a viscosity solution of the Cauchy problem. So it follows that $U(x,t)=\widehat{u}_{0}^{-}(x)$ for all $x\in M,t>0$ . Hence, by the same analysis as in the proof of Proposition 6.2 of DFIZ , for any $x\in M$ , there exists a curve $\gamma_{x}^{-}:(-\infty,0]\rightarrow M$ absolutely continuous and ending with $x$ , such that

\widehat{u}_{0}^{-}(\gamma_{x}^{-}(t))-\widehat{u}_{0}^{-}(\gamma_{x}^{-}(s))=\int_{s}^{t}\widehat{L}(\gamma_{x}^{-}(\tau),\dot{\gamma}_{x}^{-}(\tau))+c(H)\mbox{d}\tau

for all $s\leq t\leq 0$ . After all, $\widehat{u}_{0}^{-}$ has to be a backward $0-$ weak KAM solution of (9). Consequently, $u_{0}^{+}=-\widehat{u}_{0}^{-}$ has to be a forward $0-$ weak KAM solution of (HJ₀).∎

Proof of Theorem 1.1: It’s a direct corollary from Lemma 2.1, 3.1 and 3.2.∎

Appendix A Aubry-Mather theory of nonsmooth convex Hamiltonians

As is known, the continuous, superlinear, convex $H(x,p)$ has a dual Lagrangian

\displaystyle L(x,v):=\max_{p\in T_{x}^{*}\mathbb{T}^{n}}\{\langle p,v\rangle-H(x,p)\},\quad(x,v)\in TM

(10)

which is also continuous, superlinear and convex in $v$ . Consequently, for any $x,y\in M$ and $t>0$ , the action function

\displaystyle h^{t}(x,y):=\inf_{\begin{subarray}{c}\gamma\in C^{ac}([0,t],M)\\ \gamma(0)=x,\gamma(t)=y\end{subarray}}\int_{0}^{t}\big{(}L(\gamma,\dot{\gamma})+c(H)\big{)}ds

(11)

always attains its infimum at an absolutely continuous minimizing curve $\gamma_{\min}:[0,t]\rightarrow M$ due to the Tonelli Theorem. In Ma2 , the Peierls barrier function

\displaystyle h^{\infty}(x,y):=\liminf_{t\rightarrow+\infty}h^{t}(x,y)

(12)

is proved to be well-defined and continuous on $M\times M$ .

Definition A.1

Ma2 The projected Aubry set is defined by

\mathcal{A}:=\{x\in M:h^{\infty}(x,x)=0\}

Consider $TM$ (resp. $M$ ) as a measurable space and $\mathbb{P}(TM)$ (resp. $\mathbb{P}(M)$ ) by the set of all Borel probability measures on it. A measure on $TM$ is denoted by $\widetilde{\mu}$ , and we remove the tilde if we project it to $M$ . We say that a sequence $\{\widetilde{\mu}_{n}\}_{n}$ of probability measures weakly converges to a probability measure $\widetilde{\mu}$ if

\lim_{n\rightarrow+\infty}\int_{TM}f(x,v)d\widetilde{\mu}_{n}(x,v)=\int_{TM}f(x,v)d\widetilde{\mu}(x,v)

for any $f\in C_{c}(TM,\mathbb{R})$ . Accordingly, the deduced probability measure $\mu_{n}$ weakly converges to $\mu$ , i.e.

$\displaystyle\lim_{n\rightarrow+\infty}\int_{M}f(x)d\mu_{n}(x)$	$\displaystyle:=$	$\displaystyle\lim_{n\rightarrow+\infty}\int_{TM}f(x)d\pi^{*}\widetilde{\mu}_{n}(x)$
	$\displaystyle=$	$\displaystyle\lim_{n\rightarrow+\infty}\int_{TM}f\circ\pi(x,v)d\widetilde{\mu}_{n}(x,v)$
	$\displaystyle=$	$\displaystyle\int_{TM}f\circ\pi(x,v)d\widetilde{\mu}(x,v)$	(14)
	$\displaystyle=$	$\displaystyle\int_{M}f(x)d\pi^{*}\widetilde{\mu}(x)=:\int_{M}f(x)d\mu(x)$	(15)

for any $f\in C(M,\mathbb{R})$ .

Definition A.2

A probability measure $\widetilde{\mu}$ on $TM$ is closed if it satisfies:

•

$\int_{TM}|v|d\widetilde{\mu}(x,v)<+\infty$ ;
•

$\int_{TM}\langle\nabla\phi(x),v\rangle d\widetilde{\mu}(x,v)=0$ for every $\phi\in C^{1}(M,\mathbb{R})$ .

Let’s denote by $\mathbb{P}_{c}(TM)$ the set of all closed measures on $TM$ , then the following conclusion is proved in DFIZ :

Theorem A.3

$\min_{\widetilde{\mu}\in\mathbb{P}_{c}(TM)}\int_{TM}L(x,v)d\widetilde{\mu}(x,v)=-c(H)$ . Moreover, the minimizer is called a Mather measure and we denote by $\widetilde{\mathfrak{M}}$ the set of them. Similarly, we can project $\widetilde{\mathfrak{M}}$ to $\mathfrak{M}\subset\mathbb{P}(M)$ w.r.t. $\pi:TM\rightarrow M$ , which contains all the projected Mather measures.

Definition A.4

Ma The Mather set is defined by

\widetilde{\mathcal{M}}:=\overline{\bigcup_{\tilde{\mu}\in\widetilde{\mathfrak{M}}}supp(\tilde{\mu})}\subset TM

and the projected Mather set $\mathcal{M}:=\pi\widetilde{\mathcal{M}}$ is accordingly defined.

Definition A.5 (subsolution)

A function $u:M\rightarrow\mathbb{R}$ is called a viscosity subsolution, or subsolution for short of

H(x,d_{x}u)=c,\quad x\in M

(denoted by $u\prec L+c$ ), if $u(y)-u(x)\leq h^{t}(x,y)+(c-c(H))t$ for all $(x,y)\in M\times M$ and $t\geq 0$ .

Definition A.6

A function $u:M\rightarrow\mathbb{R}$ is called a backward (resp. forward) $0-$ weak KAM solution of (HJ₀) if it satisfies:

•

$u\prec L+c(H)$ , i.e. for any two points $(x,y)\in M\times M$ and any absolutely continuous curve $\gamma:[a,b]\rightarrow M$ connecting them, we have

$u(y)-u(x)\leq\int_{a}^{b}\big{(}L(\gamma,\dot{\gamma})+c(H)\big{)}dt.$

•

for any $x\in M$ there exists a curve $\gamma_{x}^{-}:(-\infty,0]\rightarrow M$ (resp. $\gamma_{x}^{+}:[0,+\infty)\rightarrow M$ ) ending with (resp. starting from) $x$ , such that for any $s<t\leq 0$ (resp. $0\leq s<t$ ),

u(\gamma_{x}^{-}(t))-u(\gamma_{x}^{-}(s))=\int_{s}^{t}\big{(}L(\gamma_{x}^{-},\dot{\gamma}_{x}^{-})+c(H)\big{)}dt

\bigg{(}resp.\quad u(\gamma_{x}^{+}(t))-u(\gamma_{x}^{+}(s))=\int_{s}^{t}\big{(}L(\gamma_{x}^{+},\dot{\gamma}_{x}^{+})+c(H)\big{)}dt\bigg{)}.

References

(1) Crandall, M. G., Evans, L. C., & Lions, P.-L. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), no. 2 487-502.
(2) Contreras G. & Iturriaga R., Global minimizers of autonomous Lagrangians, 22^∘ Colóquio Brasileiro de Matemática. In: 22nd Brazilian Mathematics Colloquium, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (1999).
(3) Gonzalo Contreras & Gabriel P. Paternain, Connecting orbits between static classes for generic Lagrangian systems, Topology 41 (2002), no. 4, 645-666.
(4) Contreras G, Iturriaga R, Paternain G.P. & Paternain M. Lagrangian graphs, minimizing measures and Mañé’s critical values. Geometric and Functional Analysis, 8(1998), no. 5, 788-809.
(5) Davini A, Fathi A, Itturiaga R. & Zavidovique M., Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math. 206 (2016), no. 1, 29-55.
(6) Davini A. & Siconolfi A., A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., Vol. 38, No.2, pp478-502, 2006.
(7) Fathi A., weak KAM theorems in Lagrangian dynamics, version 10 unpublished, 2008.
(8) Lions P.-L., Papanicolaou G. & Varadhan S. Homogenization of Hamilton-Jacobi equation, unpublished preprint, (1987)
(9) Mañé R., Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996), no. 2, 273-310.
(10) Mather J., Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991), no. 2, 169-207.
(11) Mather J., Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349-1386.

Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

Abstract

Keywords:

MSC:

1 Introduction

Theorem 1.1

Remark 1.2

1.1 Dynamic interpretation of u0+u_{0}^{+}

Definition 1.3 (Conjugated Pair)

1.2 Organization of the article.

2 Nonsmooth symmetric Lagrangians

Lemma 2.1

Proof

Proposition 2.2

Remark 2.3 (forward λ−\lambda-weak KAM solution)

3 Discounted limit of forward λ−\lambda-weak KAM solutions

Lemma 3.1

Lemma 3.2

Proof

Appendix A Aubry-Mather theory of nonsmooth convex Hamiltonians

Definition A.1

Definition A.2

Theorem A.3

Definition A.4

Definition A.5 (subsolution)

Definition A.6

References

1.1 Dynamic interpretation of $u_{0}^{+}$

Remark 2.3 (forward $\lambda-$ weak KAM solution)

3 Discounted limit of forward $\lambda-$ weak KAM solutions