Global propagation of singularities for discounted Hamilton-Jacobi equations

Cui Chen and Jiahui Hong and Kai Zhao School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China [email protected] Department of Mathematics, Nanjing University, Nanjing 210093, China [email protected] School of Mathematical Sciences, Fudan University, Shanghai 200433, China zhao

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[email protected]

Abstract.

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation

\lambda v(x)+H(x,Dv(x))=0,\quad x\in\mathbb{R}^{n}.

(HJ_λ)

We reduce the problem for equation (HJ_λ) into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of (HJ_λ) propagate along locally Lipschitz singular characteristics which can extend to $+\infty$ . We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation (HJ_λ).

Key words and phrases:

Hamilton-Jacobi equation, viscosity solutions, singularities

1. introduction

It is commonly accepted that, in optimal control, a crucial role is played by the Hamilton-Jacobi equation

\begin{cases}D_{t}u(t,x)+H(t,x,D_{x}u(t,x))=0\quad&(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{n},\\ u(0,x)=u_{0}(x)&x\in\mathbb{R}^{n}.\end{cases}

(HJ_e)

It is well known that the singularities of such solutions propagate locally along generalized characteristics. The evidence of irreversibility for the Hamilton-Jacobi equation is the propagation of singularities. Once a singularity is created, it will propagate forward in time up to $+\infty$ . For a comprehensive survey of this topic, the readers can refer to [7].

The theory of local propagation of the singularities of the viscosity solutions of (HJ_e) has been established in [2] by introducing the notion of generalized characteristics (see also [14], [26]). Other progress on the local propagation includes the strict singular characteristics ([22], see also [24]). A recent remarkable result by Cannarsa and Cheng established the relation between generalized characteristics and strict singular characteristics on $\mathbb{R}^{2}$ ([6]).

In the paper [4], Cannarsa and Cheng introduced an intrinsic method and obtained a global propagation result for time-independent Hamiltonian (see also [12]). By a procedure of sup-convolution with the kernel the fundamental solutions of associated autonomous Hamilton-Jacobi equations, they constructed a global singular arc $\mathbf{y}_{x}(t):[0,t_{0}]\to\mathbb{R}^{n}$ from an initial singular point $x$ and $t_{0}$ is independent of the initial point $x$ . The uniformness of such $t_{0}$ holds because of uniform conditions (L1)-(L3) in [4] . In [5], they ask the following problem $\mathbf{A3}$ :

\displaystyle\text{Can we drop the uniformness requirement of such }t_{0}\text{ to obtain a global result ?}

The first task of this paper is to drop the uniformness of $t_{0}$ which can not be guaranteed by, e.g., the so called Fathi-Maderna conditions ([21]) which we will use for our purpose. In this paper, we showed that the answer to problem $\mathbf{A3}$ is affirmative for time-dependent case and discounted case.

There is a very natural connection between the discounted Hamilton-Jacobi equation (HJ_λ) and the evolutionary Hamilton-Jacobi equation (HJ_e) using a conformal Hamiltonian (see, for instance, [23]) or a contact Hamiltonian (see, for instance, [15]). More precisely, if $v$ is the unique viscosity solution of (HJ_λ), we define

\displaystyle u(t,x)=e^{\lambda t}v(x),\qquad x\in\mathbb{R}^{n},t>0.

Then $u(t,x)$ is a viscosity solution of (HJ_e) with a time-dependent Hamiltonian in the form $e^{\lambda t}H(x,p/e^{\lambda t})$ . Notice that $v$ and $u$ share the singularity. Thus, we can discuss the problem of propagation of singularities for equation (HJ_e) instead of equation (HJ_λ). We developed the intrinsic method in [4] adapt to our problem which has more technical difficulty comparing to the time-independent case ([9]).

Now we introduce the associated Lagrangian as

\displaystyle L(s,x,v)=\sup_{p\in\mathbb{R}^{n}}\{p\cdot v-H(s,x,p)\},\qquad s>0,x\in\mathbb{R}^{n},v\in\mathbb{R}^{n}.

To deal with evolutionary Hamilton-Jacobi equation (HJ_e), we suppose $L=L(s,x,v):\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is of class $C^{2}$ and satisfies the following assumptions:

(L1)

$L(s,x,\cdot)$ is strict convex on $\mathbb{R}^{n}$ for all $s\in[0,\infty]$ , $x\in\mathbb{R}^{n}$ .
(L2)

For any fixed $T>0$ , there exist $c_{T}>0$ and two superlinear and nondecreasing function $\overline{\theta}_{T},\theta_{T}:[0,+\infty)\to[0,+\infty)$ , such that $\overline{\theta}_{T}(|v|)\geqslant L(s,x,v)\geqslant\theta_{T}(|v|)-c_{T},$ for all $(s,x,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}.$
(L3)

There exists $\widetilde{C}_{1},\widetilde{C}_{2}:\mathbb{R}\rightarrow[0,+\infty)$ such that $|L_{t}(s,x,v)|\leqslant\widetilde{C}_{1}(T)+\widetilde{C}_{2}(T)L(s,x,v)$ for all $(s,x,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ .

We say that a curve $\gamma:[a,b]\to\mathbb{R}^{n}$ is $\lambda$ -calibrated curve for equation (HJ_λ) if

e^{\lambda b}u(\gamma(b))-e^{\lambda a}u(\gamma(a))=\int_{a}^{b}e^{\lambda t}L(\gamma(t),\dot{\gamma}(t))\ dt,

(1.1)

and a curve $\gamma:[a,b]\to\mathbb{R}^{n}$ is calibrated curve for equation (HJ_e) if

u(b,\gamma(b))-u(a,\gamma(a))=\int_{a}^{b}L(t,\gamma(t),\dot{\gamma}(t))\ dt.

(1.2)

A point $x\in\mathbb{R}^{n}$ is a cut point of $u$ if no backward $\lambda$ -calibrated curve of equation (HJ_λ) with Hamilton $H$ ending at $x$ can be extended beyond $x$ . A point $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{n}$ is a cut point of $u$ if no backward calibrated curve of equation (HJ_e) with Hamilton $H$ ending at $(t,x)$ can be extended beyond $(t,x)$ . In both cases, we denote by $\mbox{\rm{Cut}}\,(u)$ the set of cut points of $u$ . If $u$ is a viscosity solution of (HJ_λ) or (HJ_e), a singularity of $u$ is a point where $u$ is not differentiable. We denote by $\mbox{\rm{Sing}}(u)$ the set of singularities of $u$ . It is well known that $\mbox{\rm{Sing}}\,(u)\subset\mbox{\rm{Cut}}\,(u)\subset\overline{\mbox{\rm{Sing}}\,(u)}$ .

Our main result for the time-dependent case is: Let $L$ be a Lagrangian which satisfies (L1)-(L3) and let $H$ be the associated Hamiltonian. Suppose $u_{0}=u(0,\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a Lipschitz continuous function. Then for any fixed $(t_{0},x)\in\mbox{\rm{Cut}}(u)\subset\mathbb{R}^{+}\times\mathbb{R}^{n}$ , there exists a curve $\mathbf{x}:\mathbb{R}\rightarrow\mathbb{R}^{n}$ with $\mathbf{x}(t_{0})=x$ , such that $(s,\mathbf{x}(s))\in\mbox{\rm{Sing}}(u)$ for all $s\in[t_{0},+\infty)$ . Moreover, If condition (A) (see Section 3) holds, then for any $T>0$ , $\mathbf{x}(s)$ is a Lipschitz curve on $s\in[t_{0},T]$ .

Similarly, for the discounted equation (HJ_λ) we denote by $L$ the associated Lagrangian of $H$ . We suppose $L=L(x,v):\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is of $C^{2}$ class and satisfying the following assumptions:

(L1’)

$L(x,\cdot)$ is strictly convex for all $x\in\mathbb{R}^{n}$ .

(L2’)

There exist $c_{1},c_{2}\geqslant 0$ and two superlinear functions $\theta_{1},\theta_{2}:[0,+\infty)\to[0,+\infty)$ such that

\displaystyle\theta_{2}(|v|)+c_{2}\geqslant L(x,v)\geqslant\theta_{1}(|v|)-c_{1},\qquad\forall(x,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}.

Our main result for the discounted case is: Let $L$ be a Lagrangian which satisfies (L1’)-(L2’) and $H$ be the associated Hamiltonian and $\lambda>0$ . Suppose $v:\mathbb{R}^{n}\to\mathbb{R}$ is a Lipschitz continuous semiconcave viscosity solution of (HJ_λ). Then

(1)

for any fixed $x\in\mbox{\rm{Cut}}(v)$ , there exists a locally Lipschitz curve $\mathbf{x}:[0,+\infty)\rightarrow\mathbb{R}^{n}$ with $\mathbf{x}(0)=x$ , such that $\mathbf{x}(s)\in\mbox{\rm{Sing}}(v)$ for all $s\in[0,+\infty)$ ,

(2)

the inclusions

\mbox{\rm{Sing}}(v)\subset\mbox{\rm{Cut}}(v)\subset\Big{(}\mathbb{R}^{n}\backslash\mathcal{I}(v)\Big{)}\cap\overline{\mbox{\rm{Sing}}(v)}\subset\mathbb{R}^{n}\backslash\mathcal{I}(v)

are all homotopy equivalences and the spaces $\mbox{\rm{Sing}}\,(u)$ and $\mbox{\rm{Cut}}\,(u)$ are all locally contractible.

It worth noting that the construction of the homotopy equivalence here we used is very similar to what used in [8], [11] and [9]. The general notion of the cut locus of $u$ for contact type Hamilton-Jacobi equation was studied in [16] recently for smooth initial data.

This paper is organized as follows. In Sect. 2, we introduce Lax-Oleinik operator associated to (HJ_e) and give our global result on the propagation of singularities along local Lipschitz curves under an extra condition (A). In Sect.3, we discuss the global propagation of singularities for discounted Hamiltonian (HJ_λ) and give homotopy equivalence results as an application.This paper contains three appendices which include some background materials and useful conclusions. In Appendix A, we collect some relevant regularity results with respect to the fundamental solution of (HJ_e). In Appendix B and Appendix C, we give the proof of Lemma 2.2, Lemma 2.6 and Lemma 3.8.

Acknowledgements. Cui Chen is partly supported by National Natural Science Foundation of China (Grant No. 11801223, 11871267).

2. Global propagation of singularities for time-dependent Hamiltonian

In this section, we will discuss the connection between sup-convolution, singularities and generalized characteristics for the following time-dependent Hamilton-Jacobi equation:

\begin{cases}D_{t}u(t,x)+H(t,x,D_{x}u(t,x))=0\quad&(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{n},\\ u(0,x)=u_{0}(x)&x\in\mathbb{R}^{n}.\end{cases}

(HJ_e)

Let $L(s,x,v)$ be the associated Lagrangian of $H(s,x,p)$ . We assume that $L(s,x,v):[0,+\infty)\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is a $C^{2}$ -smooth function which satisfies the following standard assumptions:

(L1)

$L_{vv}(s,x,v)>0$ for any $(s,x,v)\in[0,+\infty)\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ .

(L2)

For any fixed $T>0$ , there exist $c_{T}>0$ and two superlinear and nondecreasing functions $\overline{\theta}_{T},\theta_{T}:[0,+\infty)\to[0,+\infty)$ , such that

\displaystyle\overline{\theta}_{T}(|v|)\geqslant L(s,x,v)\geqslant\theta_{T}(|v|)-c_{T},\quad(s,x,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}.

(L3)

There exist two locally bounded functions $\widetilde{C}_{1},\widetilde{C}_{2}:[0,+\infty)\rightarrow[0,+\infty)$ such that for any $T>0$ ,

|L_{t}(s,x,v)|\leqslant\widetilde{C}_{1}(T)+\widetilde{C}_{2}(T)L(s,x,v),\quad(s,x,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}.

For any $0\leqslant s<t<+\infty$ and $x,y\in\mathbb{R}^{n}$ , we define the fundamental solution of the Hamilton-Jacobi equation (HJ_e) as follows:

A_{s,t}(x,y)=\inf_{\xi\in\Gamma^{s,t}_{x,y}}\int_{s}^{t}L(\tau,\xi(\tau),\dot{\xi}(\tau))d\tau.

(2.1)

where

\Gamma^{s,t}_{x,y}=\{\xi\in W^{1,1}([0,t],\mathbb{R}^{n}):\gamma(s)=x,\gamma(t)=y\}

We call $\xi\in\Gamma^{s,t}_{x,y}$ a minimizer for $A_{s,t}(x,y)$ if $A_{s,t}(x,y)=\int_{s}^{t}L(\tau,\xi(\tau),\dot{\xi}(\tau))d\tau.$ The existence of minimizers in (2.1) is a well known result in Tonelli’s theory (see, for instance, [17]). Moreover, we have the following proposition:

Proposition 2.1.

Suppose $L$ satisfies (L1)-(L3). Then for any $0\leqslant s<t<+\infty$ and $x,y\in\mathbb{R}^{n}$ , there exists $\xi\in\Gamma^{s,t}_{x,y}$ such that $\xi$ is a minimizer for $A_{s,t}(x,y)$ and the following properties hold:

(1)

$\xi$ is of class $C^{2}$ and satisfies

\frac{d}{ds}L_{v}(\tau,\xi(\tau),\dot{\xi}(\tau))=L_{x}(\tau,\xi(\tau),\dot{\xi}(\tau)),\quad\forall\tau\in[s,t].

(2)

Let $p(\tau)=L_{v}(\tau,\xi(\tau),\dot{\xi}(\tau))$ for $\tau\in[s,t]$ . Then $(\xi,p)$ satisfies

\begin{cases}\dot{\xi}(\tau)=H_{p}(\tau,\xi(\tau),p(\tau)),\\ \dot{p}(\tau)=-H_{x}(\tau,\xi(\tau),p(\tau)),\end{cases}\qquad\forall\tau\in[s,t].

(2.2)

In Appendix A, we collect some relevant regularity results with respect to the fundamental solution $A_{s,t}(x,y)$ . The proofs of these regularity results are similar to those in [4] for autonomous case.

2.1. Semiconcave functions

Let $\Omega\subset\mathbb{R}^{n}$ be a convex open set. We recall that a function $u:\Omega\to\mathbb{R}$ is said to be semiconcave (with linear modulus) if there exists a constant $C>0$ such that

\lambda u(x)+(1-\lambda)u(y)-u(\lambda x+(1-\lambda)y)\leqslant\frac{C}{2}\lambda(1-\lambda)|x-y|^{2},\quad\forall x,y\in\Omega,\lambda\in[0,1].

For any continuous function $u:\mathbb{R}^{n}\to\mathbb{R}$ and $x\in\mathbb{R}^{n}$ , we denote

	$\displaystyle D^{-}u(x)=\Big{\{}p\in T^{*}_{x}M:\lim\inf_{y\to x}\frac{u(y)-u(x)-\langle p,y-x\rangle}{\|y-x\|}\geqslant 0\Big{\}},$
	$\displaystyle D^{+}u(x)=\Big{\{}p\in T^{*}_{x}M:\lim\sup_{y\to x}\frac{u(y)-u(x)-\langle p,y-x\rangle}{\|y-x\|}\leqslant 0\Big{\}},$

which are called the subdifferential and superdifferential of $u$ at $x$ , respectively. Let now $u:\mathbb{R}^{n}\to\mathbb{R}$ be locally Lipschitz and $x\in\mathbb{R}^{n}$ . We call $p\in\mathbb{R}^{n}$ a reachable gradient of $u$ at $x$ if there exists a sequence $\{x_{k}\}$ such that $u$ is differentiable at $x_{k}$ for all $k\in\mathbb{N}$ and

\lim_{k\to\infty}x_{k}=x,\qquad\lim_{k\to\infty}Du(x_{k})=p.

The set of all reachable gradients of $u$ at $x$ is denoted by $D^{*}u(x)$ .

2.2. Lax-Oleinik operator in time-dependent case and a priori estimate

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a Lipschitz function. For any $0\leqslant t_{1}<t_{2}<+\infty$ and $x_{1},x_{2}\in\mathbb{R}^{n}$ , we define the Lax-Oleinik operator

T^{-}_{t_{1},t_{2}}f(x_{2}):=\inf_{z\in\mathbb{R}^{n}}\{f(z)+A_{t_{1},t_{2}}(z,x_{2})\},

(2.3)

T^{+}_{t_{1},t_{2}}f(x_{1}):=\sup_{y\in\mathbb{R}^{n}}\{f(y)-A_{t_{1},t_{2}}(x_{1},y)\},

(2.4)

and denote

\begin{split}Z(f,t_{1},t_{2},x_{2})=\{z\in\mathbb{R}^{n}:T^{-}_{t_{1},t_{2}}f(x_{2})=f(z)+A_{t_{1},t_{2}}(z,x_{2})\},\\ Y(f,t_{1},x_{1},t_{2})=\{y\in\mathbb{R}^{n}:T^{+}_{t_{1},t_{2}}f(x_{1})=f(y)-A_{t_{1},t_{2}}(x_{1},y)\}.\end{split}

(2.5)

From Appendix B, we have the following a priori estimates:

Lemma 2.2.

(proved in Appendix B) Suppose $L$ satisfies (L1)-(L3) and $f$ is a Lipschitz function on $\mathbb{R}^{n}$ . Then for any fixed $T>0$ , there exists a constant $\lambda_{1}(T,\mbox{\rm{Lip}}[f])>0$ such that for any $0\leqslant t_{1}<t_{2}\leqslant T$ and $x_{1},x_{2}\in\mathbb{R}^{n}$

(1)

$Z(f,t_{1},t_{2},x_{2})\neq\emptyset$ , and for any $z_{t_{1},t_{2},x_{2}}\in Z(f,t_{1},t_{2},x_{2})$ ,

$|z_{t_{1},t_{2},x_{2}}-x_{2}|\leqslant\lambda_{1}(T,\mbox{\rm{Lip}}[f])(t_{2}-t_{1}).$
(2)

$Y(f,t_{1},x_{1},t_{2})\neq\emptyset$ , and for any $y_{t_{1},t_{2},x_{1}}\in Y(f,t_{1},x_{1},t_{2})$ ,

$|y_{t_{1},t_{2},x_{2}}-x_{1}|\leqslant\lambda_{1}(T,\mbox{\rm{Lip}}[f])(t_{2}-t_{1}).$

where $\lambda_{1}(T,K)=\theta_{T}^{*}(K+1)+c_{T}+\overline{\theta}_{T}(0)$ for $T>0$ and $K\geqslant 0$ .

For $0\leqslant t_{1}<t_{2}<+\infty$ and $x_{1},x_{2}\in\mathbb{R}^{n}$ , denote

	$\displaystyle\Gamma^{t_{1},t_{2}}_{\cdot,x_{2}}=\{\xi\in W^{1,1}([t_{1},t_{2}],\mathbb{R}^{n}):\xi(t_{2})=x_{2}\},$
	$\displaystyle\Gamma^{t_{1},t_{2}}_{x_{1},\cdot}=\{\xi\in W^{1,1}([t_{1},t_{2}],\mathbb{R}^{n}):\xi(t_{1})=x_{1}\}.$

Lemma 2.3.

Suppose $L$ satisfies (L1)-(L3), $f$ is a Lipschitz function on $\mathbb{R}^{n}$ and $0\leqslant t_{1}<t_{2}<+\infty$ , $x_{1},x_{2}\in\mathbb{R}^{n}$ .

(1)

If $\xi\in\Gamma^{t_{1},t_{2}}_{\cdot,x_{2}}$ is a minimizer for $T_{t_{1},t_{2}}^{-}f(x_{2})$ , then $p(t_{1})=L_{v}(t_{1},\xi(t_{1}),\dot{\xi}(t_{1}))\in D^{-}f(\xi(t_{1})).$
(2)

If $\xi\in\Gamma^{t_{1},t_{2}}_{x_{1},\cdot}$ is a maximizer for $T_{t_{1},t_{2}}^{+}f(x_{1})$ , then $p(t_{2})=L_{v}(t_{2},\xi(t_{2}),\dot{\xi}(t_{2}))\in D^{+}f(\xi(t_{2})).$

From now on, suppose $u_{0}$ is a Lipschitz function on $\mathbb{R}^{n}$ and denote

u(t,x)=T^{-}_{0,t}u_{0}(x)=\inf_{z\in\mathbb{R}^{n}}\{u_{0}(z)+A_{0,t}(z,x)\},\quad(t,x)\in[0,+\infty)\times\mathbb{R}^{n}.

(2.6)

Actually, we also have the following representation:

u(t,x)=\inf_{\xi\in\Gamma^{0,t}_{\cdot,x}}\big{\{}u_{0}(\xi(0))+\int_{0}^{t}L(\tau,\xi(\tau),\dot{\xi}(\tau))d\tau\big{\}}.

(2.7)

Proposition 2.4.

[13] The following properties hold.

(1)

$u(t,x)$ is a viscosity solution of (HJ_e).
(2)

$u(t,x)$ is locally linear semiconcave on $(0,+\infty)\times\mathbb{R}^{n}$ . More precisely, for any $0<T_{1}<T_{2}$ and $R>0$ , there exists $C(T_{1},T_{2},R)>0$ such that $u(t,x)$ is linearly semiconcave on $(T_{1},T_{2})\times B(0,R)$ with semiconcavity constant $C(T_{1},T_{2},R)$ . Moreover, $C(T_{1},T_{2},R)$ is continuous with respect to $R$ .
(3)

For any $(t,x)\in(0,\infty)\times\mathbb{R}^{n}$ and any minimizer $\xi\in\Gamma^{0,t}_{\cdot,x}$ of (2.7), $u$ is differentiable at $(\tau,\xi(\tau))$ for all $\tau\in(0,t)$ .

Moreover, we have the following result

Proposition 2.5.

[13, Thm 6.4.9] For any $(t,x)\in(0,+\infty)\times\mathbb{R}^{n}$ , $(q,p)\in D^{*}u(t,x)$ if and only if there exists a minimizer $\gamma\in\Gamma^{0,t}_{\cdot,x}$ of (2.7) such that and $p=L_{v}(t,x,\dot{\gamma}(t))$ and $q=-H(t,x,p)$ .

Lemma 2.6.

(proved in Appendix B) For any fixed $T>0$ , there exists $F_{0}(T)\geqslant 0$ such that $u$ is a Lipschitz function on $(0,T]\times\mathbb{R}^{n}$ and $\mbox{\rm Lip}\,[u]\leqslant F_{0}(T)$ .

Due to Lemma 2.6 and Lemma 2.2, we obtain the following estimation:

Corollary 2.7.

For any $T>0$ , $0\leqslant t_{1}<t_{2}\leqslant T$ and $x_{1}\in\mathbb{R}^{n}$ , we have

\displaystyle Y(u(t_{2},\cdot),t_{1},x_{1},t_{2})\neq\emptyset,

and for any $y_{t_{1},t_{2},x_{1}}\in Y(u(t_{2},\cdot),t_{1},x_{1},t_{2})$ ,

|y_{t_{1},t_{2},x_{1}}-x_{1}|\leqslant\lambda_{2}(T)(t_{2}-t_{1}).

(2.8)

where $\lambda_{2}(T):=\lambda_{1}(T,F_{0}(T))$ for $T>0$ with $\lambda_{1}$ defined in Lemma 2.2 and $F_{0}$ defined in Lemma 2.6.

2.3. Propagation of singularities

Recall that $x\in\mathbb{R}^{n}$ is a cut point of $u$ if no backward calibrated curve ending at $x$ can be extended beyond $x$ . We denote by $\mbox{\rm{Cut}}\,(u)$ the set of cut points of $u$ . It is well known that $\mbox{\rm{Sing}}\,(u)\subset\mbox{\rm{Cut}}\,(u)\subset\overline{\mbox{\rm{Sing}}\,(u)}$ . In the following proposition 2.8, we construct a singular arc starting from any cut point of $u(t,x)$ .

Proposition 2.8.

Suppose $L$ is a Tonelli Lagrangian satisfying (L1)-(L3), $H$ is the associated Hamiltonian and $u_{0}:\mathbb{R}^{n}\to\mathbb{R}$ is a Lipschitz function.Then for any fixed $x\in\mathbb{R}^{n}$ and $0<t_{0}<T$ , there exist $t_{x,T}\in(0,T)$ which only depends on $x,T$ such that for all $t_{0}\leqslant t_{1}\leqslant T-t_{x,T}$ and $x_{1}\in B(x,\lambda_{2}(T)t_{1})$ , there is a unique maximum point $y_{t_{1},t,x_{1}}$ of $u(t,\cdot)-A_{t_{1},t}(x_{1},\cdot)$ for $t\in[t_{1},t_{1}+t_{x,T}]$ and the curve

\mathbf{y}_{t_{1},t_{1}+t_{x,T},x_{1}}(t):=\begin{cases}x_{1}\quad&\mbox{\rm{if}}\ \ t=t_{1},\\ y_{t_{1},t,x_{1}}\quad&\mbox{\rm{if}}\ \ t\in(t_{1},t_{1}+t_{x,T}].\end{cases}

(2.9)

satisfies $\mathbf{y}_{t_{1},t_{1}+t_{x,T},x_{1}}(t)\in B(x,\lambda_{2}(T)T)$ for any $t\in[t_{1},t_{1}+t_{x,T}]$ , where $\lambda_{2}$ is defined in Corollary 2.7. Moreover, if $(t_{1},x_{1})\in\mbox{\rm{Cut}}(u)$ , then $(t,\mathbf{y}(t))\in\mbox{\rm{Sing}}(u)$ for all $t\in[t_{1},t_{1}+t_{x,T}]$ .

Proof.

For any fixed $x\in\mathbb{R}^{n},T>0$ , by proposition 2.4 (2), there exists $C(x,T)>0$ such that it is a semiconcavity constant for $u$ on $[t_{0},t_{0}+T]\times B(x,\lambda_{2}(T)T)$ . By (3) of Proposition A.2, there exists $C_{2}(x,T)>0$ such that it is a uniformly convexity constant for $A_{t_{1},t}(x_{1},\cdot)$ on $[t_{0},t_{0}+T]\times B(x,\lambda_{2}(T)T)$ .

Therefore, $u(t,\cdot)-A_{t_{1},t}(x_{1},\cdot)$ is strictly concave on $\overline{B}(x,\lambda_{2}(T)t)$ for all $t\in[t_{1},t_{1}+t_{x,T}]$ provided that we further restrict $t_{x,T}$ in order to have

t_{x,T}:=\frac{C_{2}(x,T)}{2C(x,T)}.

We now proof that $y_{t_{1},t,x_{1}}$ is a singular point of $u$ for every $t\in(t_{1},t_{1}+t_{x,T}]$ . Let $\xi_{t_{1},t,x_{1}}\in\Gamma^{t_{1},t}_{x_{1},y_{t_{1},t,x_{1}}}$ be the unique minimizer for $A_{t_{1},t}(x_{1},y_{t_{1},t,x_{1}})$ and let

p_{t_{1},t,x_{1}}(s):=L_{v}(s,\xi_{t_{1},t,x_{1}}(s),\dot{\xi}_{t_{1},t,x_{1}}(s)),\quad s\in[t_{1},t_{1}+t_{x,T}],

be the associated dual arc. We claim that

p_{t_{1},t,x_{1}}(t)\in D^{+}u(t,y_{t_{1},t,x_{1}})\backslash D^{*}u(t,y_{t_{1},t,x_{1}})

(2.10)

which in turn yields $y_{t_{1},t,x_{1}}\in\mbox{\rm{Sing}}(u)$ . Indeed, if $p_{t_{1},t,x_{1}}(t)\in D^{*}u(t,y_{t_{1},t,x_{1}})$ , then by Proposition 2.5, there would exist a $C^{2}$ curve $\gamma_{t_{1},t,x_{1}}:(-\infty,t]\rightarrow\mathbb{R}^{n}$ solving the minimum problem

\min_{\gamma\in W^{1,1}([\tau,t];\mathbb{R}^{n})}\left\{\int_{\tau}^{t}L(s,\gamma(s),\dot{\gamma}(s))\ ds+u(\gamma(\tau)):\gamma(t)=y_{t_{1},t,x_{1}}\right\}

(2.11)

for all $\tau\leqslant t$ . It is easily to checked that $\gamma_{t_{1},t,x_{1}}$ and $\xi_{t_{1},t,x_{1}}$ coincide on $[t_{1},t]$ since both of them are extremal curves for $L$ and satisfy the same endpoint condition at $\gamma_{t_{1},t,x}$ i.e.

L_{v}(t,\xi_{t_{1},t,x_{1}}(t),\dot{\xi}_{t_{1},t,x_{1}}(t))=p_{t_{1},t,x_{1}}(t)=L_{v}(t,\gamma_{t_{1},t,x}(t),\dot{\gamma}_{t_{1},t,x}(t)).

This leads to a contradiction since $(t_{1},x_{1})\in\mbox{\rm{Cut}}(u)$ while $u$ should be smooth at $(t_{1},\gamma_{t_{1},t,x_{1}}(t_{1}))$ and $\gamma_{t_{1},t,x}(\tau)$ is a backward calibrated curve for $\tau\in[t_{1},t]$ . Thus, (2.10) holds true and $(t,\mathbf{y}_{t_{1},t,x_{1}}(t))\in\mbox{\rm{Sing}}(u)$ for all $t\in(t_{1},t_{1}+t_{x,T}]$ . ∎

By Proposition 2.8, for any $t>0$ and $x^{\prime}\in\mathbb{R}^{n}$ , there exist a $t^{\prime}>t$ which depends on $x^{\prime}$ such that $\arg\sup_{y\in\mathbb{R}^{n}}\{u(s,y)-A_{t,s}(x^{\prime},y)\}$ is singleton for any $t<s\leqslant t^{\prime}$ . We can denote that

Y(t,t^{\prime},x^{\prime}):=Y\big{(}u(t^{\prime},\cdot),t,t^{\prime},x^{\prime}\big{)}=\arg\sup_{y\in\mathbb{R}^{n}}\{u(t^{\prime},y)-A_{t,t^{\prime}}(x^{\prime},y)\}=y_{t,t^{\prime},x^{\prime}}.

By (2.8), it implies that

|Y(t,t^{\prime},x^{\prime})-x^{\prime}|\leqslant\lambda_{2}(T)(t^{\prime}-t)\quad\forall\ t_{0}<t\leqslant t^{\prime}\leqslant T.

(2.12)

Theorem 2.9.

Suppose $L$ is a Tonelli Lagrangian satisfying (L1)-(L3), $H$ is the associated Hamiltonian and $u_{0}:\mathbb{R}^{n}\to\mathbb{R}$ is a Lipschitz function. Then for any fixed $(t_{0},x)\in\mbox{\rm{Cut}}(u)$ and $T>t_{0}$ , there exists a curve $\mathbf{x}:[t_{0},+\infty)\to\mathbb{R}^{n}$ with $\mathbf{x}(t_{0})=x$ , such that $(s,\mathbf{x}(s))\in\mbox{\rm{Sing}}(u)$ for all $s\in[t_{0},+\infty)$ .

Proof.

For any fixed $(t_{0},x)\in\mbox{\rm{Sing}}(u)$ and $T>0$ , we denote that

\{\Omega_{n}\}_{n\in\mathbb{N}}:=B(x,\lambda_{2}(nT)\ nT),

with $\overline{\Omega}_{n}\subset\Omega_{n+1}$ and $\overline{\Omega}_{n}$ is compact for all $n$ , in addition, $\mathbb{R}^{n}=\bigcup_{n}\overline{\Omega}_{n}$ .

$\mathbf{Step\ I}$ :Uniform Lipschitz estimation of connections of $Y$ .

For any $s>0$ , there are a sequence of points $\{x_{j}\}_{1\leqslant j\leqslant n}$ and time $\{s_{j}\}_{1\leqslant j\leqslant n}$ with $x_{j}=Y(s_{j-1},s_{j},x_{j-1})$ for any $1\leqslant j\leqslant n$ and $x_{0}=x$ , for $t_{0}\leqslant s_{1}\leqslant s_{2}\leqslant\dots\leqslant s_{n}\leqslant s\leqslant\big{\lceil}\frac{s}{T}\big{\rceil}T$ .

By (2.12) , we have that

	$\displaystyle\|Y(s_{n},s,x_{n})-x\|\leqslant$	$\displaystyle\,\|Y(s_{n},s,x_{n})-x_{n}\|+\sum_{j=1}^{n}\|Y(s_{j-1},s_{j},x_{j-1})\|$
	$\displaystyle\leqslant$	$\displaystyle\,\lambda_{2}\left(\big{\lceil}\frac{s}{T}\big{\rceil}T\right)\left(s-s_{n}+\sum^{n}_{j=1}(s_{j}-s_{j-1})\right)$
	$\displaystyle=$	$\displaystyle\,\lambda_{2}\left(\big{\lceil}\frac{s}{T}\big{\rceil}T\right)(s-t_{0}),$

which means that for any $1\leqslant j\leqslant n$ ,

Y(s_{j},s,x_{j})\in\Omega_{\lceil\frac{s}{T}\rceil},\quad s\in[s_{j},s_{j+1}],

i.e. $|Y(s_{j},s,x_{j})-x|\leqslant\lambda_{2}\left(\big{\lceil}\frac{s}{T}\big{\rceil}T\right)(s-t_{0})$ for any $1\leqslant j\leqslant n$ .

$\mathbf{Step\ II}$ : Construction of curve $\mathbf{x}$ .

For $x_{1}:=x\in\mbox{\rm{Cut}}(u)\cap\Omega_{1}$ without loss of generality, then there exists $t_{1}:=t_{x,T}>0$ such that $Y(t_{0},\cdot,x)$ is defined on $[t_{0},t_{0}+t_{1}]$ by Proposition 2.8. One can extend $Y$ by induction.

For $Y(t_{0},t_{0}+t_{1},x)\in\Omega_{1}$ , then we define $Y(t_{0},s,x)=Y(t_{0}+t_{1},s,Y(t_{0},t_{0}+t_{1},x))$ for all $s\in[t_{0}+t_{1},t_{0}+2t_{1}]$ ; inductively, if $Y(t_{0},\cdot,x)$ is defined on $[t_{0},t_{0}+kt_{1}]$ such that $Y(t_{0},t_{0}+kt_{1},x)\in\Omega_{1}$ , then we define that $Y(t_{0},s,x)=Y(t_{0}+kt_{1},s,Y(t_{0},t_{0}+kt_{1},x))$ for all $s\in[t_{0}+kt_{1},t_{0}+(k+1)t_{1}]$ . Now, let

k_{1}=\Big{\lfloor}\frac{T-t_{0}}{t_{1}}\Big{\rfloor},

(2.13)

then $t_{0}+k_{1}t_{1}\leqslant T$ which implies $Y(t_{0},s,x)\in\Omega_{1}$ for any $s\in[t_{0},t_{0}+k_{1}t_{1}]$ by Step I .

In a similar way, $x_{2}:=Y(t_{0},t_{0}+k_{1}t_{1},x)\in\Omega_{1}\cap\mbox{\rm{Sing}}(u)\subset\Omega_{2}\cap\mbox{\rm{Sing}}(u)$ , we define

k_{2}=\Big{\lfloor}\frac{2T-k_{1}t_{1}-t_{0}}{t_{2}}\Big{\rfloor},

where $t_{2}:=t_{x,2T}\leqslant t_{1}$ is determined by applying Proposition 2.8 to $\Omega_{2}$ .We also conclude that $t_{0}+k_{1}t_{1}+k_{2}t_{2}\leqslant 2T$ which implies that $Y(t_{0},s,x)\in\Omega_{2}$ for all $s\in[t_{0},t_{0}+k_{1}t_{1}+k_{2}t_{2}]$ by Step I.

Therefore, by induction, for any $i\in\mathbb{N}$ , there exists $t_{i}:=t_{x,iT}\leqslant t_{i-1}$ is determined by applying Proposition 2.8 to $\Omega_{i}$ with $0<t_{i}\leqslant t_{i-1}$ , let

k_{i}=\Big{\lfloor}\frac{iT-\sum_{j=1}^{i-1}k_{j}t_{j}-t_{0}}{t_{i}}\Big{\rfloor},

(2.14)

We also conclude that $t_{0}+\sum_{j=1}^{i}k_{j}t_{j}\leqslant iT$ which implies that $Y(t_{0},s,x)\in\Omega_{i}$ for all $s\in[t_{0},t_{0}+\sum_{j=1}^{i}k_{j}t_{j}]$ by Step I.

Denote that $x_{i}:=Y(t_{0},t_{0}+\sum_{j=0}^{i-1}k_{j}t_{j},x)\in\Omega_{i-1}\cap\mbox{\rm{Sing}}(u)\subset\Omega_{i}\cap\mbox{\rm{Sing}}(u).$ This makes us to define an arc $\mathbf{x}:[0,\overline{t})\rightarrow\mathbb{R}^{n}$ by

\mathbf{x}(s):=Y(t_{0},s,x)=Y(t_{0}+\sum_{j=0}^{i-1}k_{j}t_{j},s,x_{i})\quad\forall s\in\Big{[}t_{0}+\sum_{j=0}^{i-1}k_{j}t_{j},t_{0}+\sum_{j=0}^{i}k_{j}t_{j}\Big{]}

where $\overline{t}:=t_{0}+\sum^{\infty}_{j=0}k_{j}t_{j}$ . It is clear that $\mathbf{x}$ is a generalized characteristic defined on $[0,\overline{t})$ and $\mathbf{x}(s)\in\mbox{\rm{Sing}}(u)$ for all $s\in[0,\overline{t})$ , by Proposition 2.8.

$\mathbf{Step\ III}$ : Estimation of time $\overline{t}$ .

To finish the proof, we only need to show that $\overline{t}=\infty$ . Indeed, since $T>t_{1}\geqslant t_{2}\geqslant t_{3}\geqslant\dots$ , we have that

	$\displaystyle\overline{t}>$	$\displaystyle\,t_{0}+\sum^{n}_{j=1}k_{j}t_{j}=t_{0}+\sum^{n-1}_{j=1}k_{j}t_{j}+\Big{\lfloor}\frac{nT-\sum_{j=1}^{n-1}k_{j}t_{j}-t_{0}}{t_{n}}\Big{\rfloor}t_{n}$
	$\displaystyle\geqslant$	$\displaystyle\,t_{0}+\sum^{n-1}_{j=1}k_{j}t_{j}+\left(\frac{nT-\sum_{j=1}^{n-1}k_{j}t_{j}-t_{0}}{t_{n}}-1\right)t_{n}$
	$\displaystyle=$	$\displaystyle\,nT-t_{n}\geqslant nT-t_{1}\rightarrow\infty\quad\text{as}\ n\rightarrow\infty.$

Therefore $\overline{t}=\infty$ . ∎

Remark 2.10.

Example 5.6.7 of [13] showed that there exists a counterexample for global propagation of singularities without condition (L2).

2.4. Local Lipschitz of singular curve

(A)

For any given $T\in\mathbb{R}^{+}$ , there exists $K(T)\in\mathbb{R}^{+}$ such that viscosity solution $u(t,x)$ of (HJ_e) is differentiable on $(t,x)\in[0,T]\times\mathbb{R}^{n}$ and

|D_{t}u(t,x)-D_{s}u(s,y)|\leqslant K(T)\Big{(}|t-s|+|x-y|\Big{)},\quad\forall x,y\in\mathbb{R}^{n},s,t\in[0,T].

Theorem 2.11.

If condition (A) holds, then for any $T>0$ and $\mathbf{x}(s)$ (see Theorem 2.9) is a Lipschitz curve on $s\in[0,T]$ .

The proof of the theorem above is a direct consequence of following Lemma.

Lemma 2.12.

For any $T>0$ and $(t_{0},x)\in[0,T]\times\mathbb{R}^{n}$ , let $t_{x,T}$ and $\mathbf{y}:[t_{0},t_{0}+t_{x,T}]\to\mathbb{R}^{n}$ be given by Proposition 2.8. If condition (A) holds, then $\mathbf{y}$ is Lipschitz on $[t_{0},t_{0}+t_{x,T}]$ .

Proof.

Without loss of generality, we assume that $t_{0}=0$ . Let $\xi_{t}:=\xi_{0,t,x}\in\Gamma_{x,\mathbf{y}(t)}^{0,t}$ , $\xi_{s}:=\xi_{0,s,x}\in\Gamma_{x,\mathbf{y}(s)}^{0,s}$ and $\eta:=\xi_{0,t,x}\in\Gamma_{x,\mathbf{y}(s)}^{0,t}$ be minimizers for $A_{0,t}(x,\mathbf{y}(t)),A_{0,s}(x,\mathbf{y}(s))$ and $A_{0,t}(x,\mathbf{y}(s))$ respectively. Setting $p_{t}=L_{v}(t,\xi_{t}(t),\dot{\xi}_{t}(t)),p_{s}=L_{v}(s,\xi_{s}(s),\dot{\xi}_{s}(s))$ and $p=L_{v}(t,\eta(t),\dot{\eta}(t))$ , we have $(q_{s},p_{s})\in D^{+}u(s,\mathbf{y}(s))$ and $(q_{t},p_{t})\in D^{+}u(t,\mathbf{y}(t))$ . Hence, by Proposition A.2, there exists $C_{3}(x,T)>0$ such that

		$\displaystyle\,\frac{C_{2}(x,T)}{t-s}\|\mathbf{y}(t)-\mathbf{y}(s)\|^{2}$
	$\displaystyle\leqslant$	$\displaystyle\,\langle p_{t}-p,\mathbf{y}(t)-\mathbf{y}(s)\rangle=\langle p_{t}-p_{s},\mathbf{y}(t)-\mathbf{y}(s)\rangle+\langle p_{s}-p,\mathbf{y}(t)-\mathbf{y}(s)\rangle$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{C_{3}(x,T)}{t-s}\cdot\|\mathbf{y}(t)-\mathbf{y}(s)\|\cdot\|t-s\|+\langle p_{t}-p_{s},\mathbf{y}(t)-\mathbf{y}(s)\rangle$
		$\displaystyle\,\quad\quad\quad\quad+\langle q_{t}-q_{s},t-s\rangle-\langle q_{t}-q_{s},t-s\rangle$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{C_{3}(x,T)}{t-s}\cdot\|\mathbf{y}(t)-\mathbf{y}(s)\|\cdot\|t-s\|+C_{1}(x,T)(\|\mathbf{y}(t)-\mathbf{y}(s)\|^{2}+\|t-s\|^{2})$
		$\displaystyle\,\quad\quad\quad\quad\quad\quad\quad\quad-\langle q_{t}-q_{s},t-s\rangle$

where $C_{2}(x,T)>0$ is a uniformly convexity constant in Proposition A.2 (3).

Actually, by condition (A), $u(t,x)$ is differentiable with respect to $t$ , and

|q_{t}-q_{s}|=|D_{t}u(t,x)-D_{s}u(s,x)|\leqslant K(T)\big{(}|t-s|+|\mathbf{y}(t)-\mathbf{y}(s)|\big{)}.

Therefore,

	$\displaystyle\frac{C_{2}(x,T)}{t-s}\|\mathbf{y}(t)-\mathbf{y}(s)\|^{2}\leqslant$	$\displaystyle\,\frac{C_{3}(x,T)}{t-s}\cdot\|\mathbf{y}(t)-\mathbf{y}(s)\|\cdot\|t-s\|+C_{1}(x,T)(\|\mathbf{y}(t)-\mathbf{y}(s)\|^{2}+\|t-s\|^{2})$
		$\displaystyle\,\quad\quad\quad\quad+K(T)\cdot\|\mathbf{y}(t)-\mathbf{y}(s)\|\cdot\|t-s\|+K(T)\|t-s\|^{2}.$

That is,

\displaystyle\Big{(}\frac{C_{2}(x,T)}{t-s}-C_{1}(x,T)\Big{)}\Big{|}\frac{\mathbf{y}(t)-\mathbf{y}(s)}{t-s}\Big{|}^{2}-\Big{(}\frac{C_{3}(x,T)}{t-s}+K(T)\Big{)}\Big{|}\frac{\mathbf{y}(t)-\mathbf{y}(s)}{t-s}\Big{|}\leqslant C_{1}+K(T).

Let $t-s$ be sufficiently small such that

t-s\leqslant\frac{C_{2}(x,T)}{2C_{1}(x,T)},

then there exists a constant $C_{4}$ which only depends on $x,T$ such that

\Big{|}\frac{\mathbf{y}(t)-\mathbf{y}(s)}{t-s}\Big{|}\leqslant C_{4}(x,T).

More precisely, we can take $C_{4}(x,T):=\frac{C_{3}(x,T)}{C_{2}(x,T)}+\frac{K(T)+\sqrt{C_{1}(x,T)+K(T)}}{2C_{1}(x,T)}$ . ∎

3. Global propagation of singularities for discounted Hamiltonian

3.1. Global propagation of singularities for discounted Hamiltonian

For $\lambda>0$ , we consider the Hamilton-Jacobi equation with discounted factor

\lambda v(x)+H(x,Dv(x))=0,\quad x\in\mathbb{R}^{n}.

(HJ_λ)

where $H$ is a Tonelli Hamiltonian.

Lemma 3.1.

[15, Proposition 3.3] $v(x)$ is a viscosity solution of (HJ_λ) if and only if $u(t,x)=e^{\lambda t}v(x)$ is a viscosity solution of the Hamilton-Jacobi equation

\begin{cases}D_{t}u+\widehat{H}(t,x,D_{x}u)=0,&(t,x)\in(0,+\infty)\times\mathbb{R}^{n}\\ u(0,x)=v(x),&x\in\mathbb{R}^{n},\end{cases}

(3.1)

where $\widehat{H}(t,x,p)=e^{\lambda t}H(x,e^{-\lambda t}p)$ . Moreover, for any $(t,x)\in(0+\infty)\times\mathbb{R}^{n}$ , we have

x\in\mbox{\rm{Sing}}(v)\Leftrightarrow(t,x)\in\mbox{\rm{Sing}}(u),\quad x\in\mbox{\rm{Cut}}(v)\Leftrightarrow(t,x)\in\mbox{\rm{Cut}}(u).

(3.2)

Remark 3.2.

Since $\gamma:[a,b]\to\mathbb{R}^{n}$ is a calibrated curve of equation (HJ_λ) with Hamiltonian $H$ if and only if $\gamma:[a,b]\to\mathbb{R}^{n}$ is a calibrated curve of equation (3.1) with Hamiltonian $\widehat{H}$ , then by the definition of $\mbox{\rm{Cut}}(u)$ in Definition 3.4, $x\in\mbox{\rm{Cut}}(v)$ and $(t,x)\in\mbox{\rm{Cut}}(u)$ are equivalent.

Theorem 3.3.

Let $H$ be a Tonelli Hamiltonian and $\lambda>0$ . Suppose $v:\mathbb{R}^{n}\to\mathbb{R}$ is the Lipschitz continuous viscosity solution of (HJ_λ). Then for any fixed $x\in\mbox{\rm{Cut}}(v)$ , there exists a locally Lipschitz curve $\mathbf{x}:[0,+\infty)\rightarrow\mathbb{R}^{n}$ with $\mathbf{x}(0)=x$ , such that $\mathbf{x}(\tau)\in\mbox{\rm{Sing}}(v)$ for all $\tau\in[0,+\infty)$ .

Proof.

By using the variable transformation $s=\tau+1\in[1,+\infty)$ , we only need to find a locally Lipschitz curve $\mathbf{x}:[1,+\infty)\rightarrow\mathbb{R}^{n}$ with $\mathbf{x}(1)=x$ , such that $\mathbf{x}(s)\in\mbox{\rm{Sing}}(v)$ for all $s\in[1,+\infty)$ . Due to (3.2), $x\in\mbox{\rm{Cut}}(v)$ implies $(1,x)\in\mbox{\rm{Cut}}(u)$ . It is easy to check that $\widehat{L}(t,x,v)=e^{\lambda t}L(x,v)$ satisfies (L1)-(L3). Therefore, by Theorem 2.9, there exists a curve $\mathbf{x}:[1,+\infty)\rightarrow\mathbb{R}^{n}$ with $\mathbf{x}(1)=x$ , such that $(s,\mathbf{x}(s))\in\mbox{\rm{Sing}}(u)$ for all $s\in(1,+\infty)$ , that is, $\mathbf{x}(\tau)\in\mbox{\rm{Sing}}(v)$ for all $\tau\in(0,+\infty)$ .

It remains to show that the curve $\mathbf{x}(\tau):[0,+\infty)\rightarrow\mathbb{R}^{n}$ is locally Lipschitz. Notice that $u(t,x)=e^{\lambda t}v(x)$ is differentiable with respect to $t$ and

\displaystyle D_{t}u(t,x)=\lambda e^{\lambda t}v(x)=\lambda u(t,x),\qquad(t,x)\in(0,+\infty)\times\mathbb{R}^{n}.

For any $(t,x)\in(0,+\infty)\times\mathbb{R}^{n}$ and $(s,y)\in(0,+\infty)\times\mathbb{R}^{n}$ , by Lemma 2.6, we have

\displaystyle|D_{t}u(t,x)-D_{s}u(s,y)|=|\lambda u(t,x)-\lambda u(s,y)|\leqslant F_{0}(T)(|t-s|+|x-y|),

which implies condition (A) holds. Hence $\mathbf{x}:[0,+\infty)\rightarrow\mathbb{R}^{n}$ is locally Lipschitz by Lemma 2.12. ∎

3.2. Homotopy equivalence

Now, suppose $u:\mathbb{R}^{n}\to\mathbb{R}$ is the Lipschitz viscosity solution of

\lambda u(x)+H(x,du(x))=0,\qquad x\in\mathbb{R}^{n},

(HJ_λ)

where $H$ is a Tonelli Hamiltonian and $\lambda>0$ .

Definition 3.4.

(Aubry set): We define $\mathcal{I}(u)$ ¹¹1If $M$ is compact, $\mathcal{I}(u)$ is not empty and can be characterized by conjugate pairs for contact Hamiltonian systems with increasing condition in [25]. For noncompact case, the question is still open if $\mathcal{I}(u)\neq\emptyset$ . , the Aubry set of $u$ , as

\displaystyle\mathcal{I}(u)=\{x\in\mathbb{R}^{n}:\text{ there exists a calibrated curve }\gamma:(-\infty,+\infty)\to\mathbb{R}^{n}\text{ with }\gamma(0)=x\}

In general we have the following inclusions:

\mbox{\rm{Sing}}\,(u)\subset\mbox{\rm{Cut}}\,(u)\subset\mathbb{R}^{n}\setminus\mathcal{I}(u),\quad\mbox{\rm{Sing}}\,(u)\subset\mbox{\rm{Cut}}\,(u)\subset\overline{\mbox{\rm{Sing}}\,(u)}.

Theorem 3.5.

The inclusions

\mbox{\rm{Sing}}\,(u)\subset\mbox{\rm{Cut}}\,(u)\subset\Big{(}\mathbb{R}^{n}\setminus\mathcal{I}(u)\Big{)}\cap\overline{\mbox{\rm{Sing}}\,(u)}\subset\mathbb{R}^{n}\setminus\mathcal{I}(u)

are all homotopy equivalences.

This theorem obviously implies the following corollary (see, for instance, [18])

Corollary 3.6.

For every connected component $C$ of $\mathbb{R}^{n}\setminus\mathcal{I}(u)$ , these three intersections $\mbox{\rm{Sing}}\,(u)\cap C$ , $\mbox{\rm{Cut}}\,(u)\cap C$ and $\overline{\mbox{\rm{Sing}}\,(u)}\cap C$ are path connected.

Theorem 3.7.

[8, Thm. 1.3] The spaces $\mbox{\rm{Sing}}\,(u)$ and $\mbox{\rm{Cut}}\,(u)$ are locally contractible.

The proof of Theorem 3.5 and 3.7 needs the following Lemma.

Lemma 3.8.

There exists a continuous homotopy $F:\mathbb{R}^{n}\times[0,+\infty)\to\mathbb{R}^{n}$ with the following properties:

(a)

for all $x\in\mathbb{R}^{n}$ , we have $F(x,0)=x$ ;
(b)

if $F(x,s)\notin\mbox{\rm{Sing}}(u)$ for some $s>0$ and $x\in\mathbb{R}^{n}$ , then the curve $\sigma\mapsto F(x,\sigma)$ is calibrated on $[0,s]$ ;
(c)

if there exists a calibrated curve $\gamma:[0,s]\to\mathbb{R}^{n}$ with $\gamma(0)=x$ , then $\sigma\mapsto F(x,\sigma)=\gamma(\sigma)$ , for every $\sigma\in[0,s]$ .

The proof of Lemma 3.8 is in Appendix C. These properties imply:

Lemma 3.9.

(1)

$F(\mbox{\rm Cut}\,(u)\times(0,+\infty))\subset\mbox{\rm{Sing}}(u)$ ;
(2)

if $F(x,s)\notin\mbox{\rm{Sing}}\,(u)$ for all $s\in[0,+\infty)$ , then $x\in\mathcal{I}(u)$ and $s\mapsto F(x,s)$ , $s\in[0,+\infty)$ is a forward calibrated curve with $F(x,0)=x$ ;
(3)

if $x\notin\mathcal{I}(u)$ , then $F(x,s)\notin\mathcal{I}(u)$ for every $s\in[0,+\infty)$ .

Now, for $x\in\mathbb{R}^{n}$ , we define $\tau(x)$ to be the supremum of the $t\geqslant 0$ such that there exists a calibrated curve $\gamma:[0,t]\to\mathbb{R}^{n}$ with $\gamma(0)=x$ .

Lemma 3.10.

(i)

$\tau(x)=0$ if and only if $x\in\mbox{\rm Cut}\,(u)$ ;
(ii)

$\tau(x)=+\infty$ if and only if $x\in\mathcal{I}(u)$ ;
(iii)

the function $\tau$ is upper semi-continuous.

Proof.

(i) and (ii) follows directly from the definition of $\mbox{\rm Cut}\,(u)$ and $\mathcal{I}(u)$ . It remains to prove (iii). Indeed, we only need to prove that for any $\tau^{\prime}>0$ the set $\{x\in\mathbb{R}^{n}:\tau(x)\geqslant\tau^{\prime}\}$ is closed. Take any sequence $x_{i}$ such that $\tau(x_{i})\geqslant\tau^{\prime}$ and $x_{i}\to x_{0}$ , and let $\gamma_{i}:[0,\tau^{\prime}]\to\mathbb{R}^{n}$ , $\gamma_{i}(0)=x_{i}$ be the associated calibrated curves. By taking a subsequence, we can assume that

\displaystyle\lim_{i\to\infty}Du(x_{i})=p_{0}\in D^{*}u(x_{0}).

Notice that $\gamma_{i}:[0,\tau^{\prime}]\to\mathbb{R}^{n}$ is the solution of (2.2) with initial condition $\gamma_{i}(0)=x_{i}$ , $p_{i}(0)=Du(x_{i})$ . Let $\gamma_{0}:[0,\tau^{\prime}]\to\mathbb{R}^{n}$ be the solution of (2.2) with initial condition $\gamma_{0}(0)=x_{0}$ , $p_{0}(0)=p_{0}$ . It follows that $\gamma_{i}$ converges to $\gamma_{0}$ in $C^{2}$ topology. Thus, we have

	$\displaystyle e^{\lambda\tau^{\prime}}u(\gamma_{0}(\tau^{\prime}))$	$\displaystyle=\lim_{i\to\infty}e^{\lambda\tau^{\prime}}u(\gamma_{i}(\tau^{\prime}))=\lim_{i\to\infty}u(\gamma_{i}(0))+\int_{0}^{\tau^{\prime}}e^{\lambda t}L(\gamma_{i}(t),\dot{\gamma}_{i}(t))\ dt$
		$\displaystyle=u(\gamma_{0}(0))+\int_{0}^{\tau^{\prime}}e^{\lambda t}L(\gamma_{0}(t),\dot{\gamma}_{0}(t))\ dt.$

This implies $\gamma_{0}:[0,\tau^{\prime}]\to\mathbb{R}^{n}$ is a calibrated curve and $\tau(x_{0})\geqslant\tau^{\prime}$ . Therefore, the set $\{x\in\mathbb{R}^{n}:\tau(x)\geqslant\tau^{\prime}\}$ is closed and the function $\tau$ is upper semi-continuous. ∎

Proof of Theorem 3.5.

By Lemma 3.10, the function $\tau$ is upper semi-continuous and finite on $\mathbb{R}^{n}\setminus\mathcal{I}(u)$ . Thus, by Proposition 7.20 in [3], we can find a continuous function $\alpha:\mathbb{R}^{n}\setminus\mathcal{I}(u)\to(0,+\infty)$ with $\alpha>\tau$ on $\mathbb{R}^{n}\setminus\mathcal{I}(u)$ . We now define $G:(\mathbb{R}^{n}\setminus\mathcal{I}(u))\times[0,1]\to\mathbb{R}^{n}\setminus\mathcal{I}(u)$ by

G(x,s)=F(x,s\alpha(x)).

Due to Lemma 3.8, Lemma 3.9 and the continuity of $\alpha$ , the map $G(x,s)$ is a homotopy of $\mathbb{R}^{n}\setminus\mathcal{I}(u)$ into itself, such that $G(\mathbb{R}^{n}\setminus\mathcal{I}(u),1)\subset\mbox{\rm Sing}\,(u)$ and $G(\mbox{\rm Cut}\,(u),(0,1])\subset\mbox{\rm Sing}\,(u)$ . Therefore, the time one map of $G$ gives a homotopy inverse for each one of the inclusions

\displaystyle\mbox{\rm Sing}\,(u)\subset\mbox{\rm Cut}\,(u)\subset\overline{\mbox{\rm Sing}\,(u)}\cap(\mathbb{R}^{n}\setminus\mathcal{I}(u))\subset\mathbb{R}^{n}\setminus\mathcal{I}(u).

∎

3.3. genuine propagation of singularities

To study genuine propagation of singularities, we have to check that the singular arc $\mathbf{x}$ in Theorem 3.3 is not a fixed point. As we show below, the following condition about strong critical point (see, for instance [14],[4]) can be useful for this purpose.

Definition 3.11.

We say that $x\in\mathbb{R}^{n}$ is a strong critical point of a viscosity solution $v$ of (HJ_λ) if

0\in\lambda v(x)+H_{p}(x,D^{+}v(x)).

Corollary 3.12.

Let $\mathbf{x}:[0,+\infty)\to\mathbb{R}^{n}$ be the singular curve in Theorem 3.3. If $x$ is not a strong critical point of $v$ , then there exists $t>0$ such that $\mathbf{x}(s)\neq x_{0}$ for all $s\in(0,t]$ .

3.4. Existence of global Lipschitz viscosity solution of (HJ_λ)

We assume $L=L(x,v):\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is a function of class $C^{2}$ satisfying:

(L1’)

$L(x,\cdot)$ is strictly convex for all $x\in\mathbb{R}^{n}$ .

(L2’)

There exist $c_{1},c_{2}\geqslant 0$ and two superlinear functions $\theta_{1},\theta_{2}:[0,+\infty)\to[0,+\infty)$ such that

\displaystyle\theta_{2}(|v|)+c_{2}\geqslant L(x,v)\geqslant\theta_{1}(|v|)-c_{1},\qquad\forall(x,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}.

The associated Hamiltonian $H:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ is defined by

\displaystyle H(x,p)=\sup_{v\in\mathbb{R}^{n}}\{\langle p,v\rangle-L(x,v)\},\qquad(x,p)\in\mathbb{R}^{n}\times\mathbb{R}^{n}.

Theorem 3.13.

Suppose $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ satisfies (L1’)-(L2’) and $\lambda>0$ . Then there exists $u_{0}:\mathbb{R}^{n}\to\mathbb{R}$ such that $u_{0}$ is the unique bounded and Lipschitz viscosity solution of (HJ_λ) on $\mathbb{R}^{n}$ .

Remark 3.14.

In Theorem 3.3, we suppose $u:\mathbb{R}^{n}\to\mathbb{R}$ is a globally Lipschitz continuous viscosity solution of equation (HJ_λ). Actually, the viscosity solutions of equation (HJ_λ) are not always globally Lipschitz continuous. There is a counterexample as follows:

u(x)+\frac{1}{2}|Du(x)|^{2}=0,\quad x\in\mathbb{R}^{n},

(3.3)

Obviously, $u_{1}(x)=-\frac{1}{2}x^{2}$ and $u_{2}(x)\equiv 0$ are both viscosity solutions of (3.3). But $u_{1}(x)=-\frac{1}{2}x^{2}$ is not globally Lipschitz on $\mathbb{R}^{n}$ and $u_{2}(x)\equiv 0$ is the unique globally Lipschitz viscosity solution of 3.3.

For any function $u:\mathbb{R}^{n}\to\mathbb{R}$ and $t>0$ , we define the Lax-Oleinik operator (See [10])

T_{t}^{-}u(x)=\inf_{\xi\in\Gamma_{\cdot,x}^{0,t}}\Big{\{}e^{-\lambda t}u(\xi(0))+\int_{0}^{t}e^{\lambda(s-t)}L(\xi,\dot{\xi})\ ds\Big{\}},\qquad x\in\mathbb{R}^{n}.

(3.4)

Recall some properties of $T_{t}^{-}$ as follows:

Lemma 3.15.

(1)

For any $u:\mathbb{R}^{n}\to\mathbb{R}$ and $t_{1},t_{2}>0$ , we have

$\displaystyle T_{t_{1}}^{-}T_{t_{2}}^{-}u=T_{t_{1}+t_{2}}^{-}u.$
(2)

Set $u_{i}:\mathbb{R}^{n}\to\mathbb{R},\ i=1,2$ . If $u_{1}\leqslant u_{2}$ , then there holds

$\displaystyle T_{t}^{-}u_{1}\leqslant T_{t}^{-}u_{2},\qquad\forall t>0$

(3)

Suppose $u_{i}:\mathbb{R}^{n}\to\mathbb{R},\ i=1,2$ are bounded on $\mathbb{R}^{n}$ . Then we have

\displaystyle\|T_{t}^{-}u_{1}-T_{t}^{-}u_{2}\|_{\infty}\leqslant e^{-\lambda t}\|u_{1}-u_{2}\|_{\infty},\qquad\forall t>0.

(4)

For any $t>0$ , $u\leqslant T_{t}^{-}u$ if and only if for any absolutely continuous curve $\gamma:[a,b]\to\mathbb{R}^{n}$ , there holds

e^{\lambda b}u(\gamma(b))\leqslant e^{\lambda a}u(\gamma(a))+\int_{a}^{b}e^{\lambda t}L(\gamma(t),\dot{\gamma}(t))\ dt.

(3.5)

Lemma 3.16.

There exists positive constants $K_{1}=c_{1}/\lambda$ , $K_{2}=(\theta_{2}(0)+c_{2})/\lambda$ such that

-K_{1}\leqslant T_{t}^{-}(-K_{1})(x)\leqslant K_{2},\qquad\forall t>0,x\in\mathbb{R}^{n}.

Proof.

On the one hand, for any $t>0$ , $x\in\mathbb{R}^{n}$ , consider the curve $\xi(s)\equiv x$ in (3.4), then we obtain that

	$\displaystyle T_{t}^{-}(-K_{1})(x)\leqslant$	$\displaystyle\,-e^{-\lambda t}K_{1}+\int_{0}^{t}e^{\lambda(s-t)}L(x,0)\ ds$
	$\displaystyle\leqslant$	$\displaystyle\,\int_{0}^{t}e^{(s-t)\lambda}(\theta_{2}(0)+c_{2})\ ds$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{\theta_{2}(0)+c_{2}}{\lambda}=K_{2}.$

On the other hand, let $\eta\in\Gamma_{\cdot,x}^{0,t}$ be a minimizer for (3.4). It follows that

	$\displaystyle T_{t}^{-}(-K_{1})(x)=$	$\displaystyle\,-e^{-\lambda t}K_{1}+\int_{0}^{t}e^{\lambda(s-t)}L(\eta,\dot{\eta})\ ds$
	$\displaystyle\geqslant$	$\displaystyle\,-e^{-\lambda t}K_{1}+\int_{0}^{t}e^{\lambda(s-t)}\big{(}\theta_{1}(\|\dot{\eta}(s)\|)-c_{1}\big{)}\ ds$
	$\displaystyle\geqslant$	$\displaystyle\,-e^{-\lambda t}K_{1}-c_{1}\int_{0}^{t}e^{\lambda(s-t)}\ ds$
	$\displaystyle=$	$\displaystyle\,-e^{-\lambda t}K_{1}-(1-e^{-\lambda t})K_{1}=-K_{1}.$

This completes our proof. ∎

Similar to [1, lemma 2.2], we show that bounded subsolutions is Lipschitz on $\mathbb{R}^{n}$ as follows:

Lemma 3.17.

Suppose $u:\mathbb{R}^{n}\to\mathbb{R}$ is bounded on $\mathbb{R}^{n}$ and $u\leqslant T_{t}^{-}u$ for any $t>0$ . Then $u$ is Lipschitz on $\mathbb{R}^{n}$ .

Proof.

For $x,y\in\mathbb{R}^{n}$ , $x\neq y$ , consider the curve $\xi(s)=x+s\frac{(y-x)}{|y-x|},\ s\in[0,|y-x|]$ . Then $u\leqslant T_{t}^{-}u$ implies that

u(y)\leqslant e^{-\lambda|y-x|}u(x)+\int_{0}^{|y-x|}e^{\lambda(s-|y-x|)}L\Big{(}x+s\frac{(y-x)}{|y-x|},\frac{(y-x)}{|y-x|}\Big{)}\ ds,

(3.6)

It follows that

		$\displaystyle\,u(y)-u(x)$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{1-e^{-\lambda\|y-x\|}}{\lambda}\cdot(-\lambda u(x))+\int_{0}^{\|y-x\|}e^{\lambda(s-\|y-x\|)}L\Big{(}x+s\frac{(y-x)}{\|y-x\|},\frac{(y-x)}{\|y-x\|}\Big{)}\ ds$
	$\displaystyle=$	$\displaystyle\,\int_{0}^{\|y-x\|}e^{\lambda(s-\|y-x\|)}\Bigg{[}L\Big{(}x+s\frac{(y-x)}{\|y-x\|},\frac{(y-x)}{\|y-x\|}\Big{)}-\lambda u(x)\Bigg{]}\ ds$
	$\displaystyle\leqslant$	$\displaystyle\,\int_{0}^{\|y-x\|}e^{\lambda(s-\|y-x\|)}\Big{(}\theta_{2}(1)+c_{2}+\lambda\\|u\\|_{\infty}\Big{)}\ ds$
	$\displaystyle\leqslant$	$\displaystyle\,(\theta_{2}(1)+c_{2}+\lambda\\|u\\|_{\infty})\|y-x\|.$

Similarly, there holds $u(x)-u(y)\leqslant(\theta_{2}(1)+c_{2}+\lambda\|u\|_{\infty})|y-x|$ . Therefore,

\displaystyle|u(y)-u(x)|\leqslant(\theta_{2}(1)+c_{2}+\lambda\|u\|_{\infty})|y-x|,\qquad\forall x,y\in\mathbb{R}^{n}.

∎

Following Fathi ([20, 19]), $u\in C^{0}(\mathbb{R}^{n},\mathbb{R})$ is called a weak-KAM solution of (HJ_λ) if

\displaystyle T_{t}^{-}u=u,\qquad\forall t>0.

Lemma 3.18.

Suppose $u:\mathbb{R}^{n}\to\mathbb{R}$ is bounded on $\mathbb{R}^{n}$ . Then $u$ is a weak-KAM solution of (HJ_λ) if and only if $u$ is a viscosity solution of (HJ_λ).

Proof of Theorem 3.13.

Due to Lemma 3.16, we have that

\displaystyle-K_{1}\leqslant T_{t}^{-}(-K_{1})(x)\leqslant K_{2},\qquad\forall t>0,x\in\mathbb{R}^{n},

and $T_{t}^{-}(-K_{1})$ is non-decreasing for $t>0$ by (1) and (4) of Lemma 3.15. It follows that $\|T_{1}^{-}(-K_{1})-(-K_{1})\|_{\infty}\leqslant K_{1}+K_{2}$ . Using Lemma 3.15 (3), we obtain that for any $k\in\mathbb{N}$ ,

	$\displaystyle\\|T_{k+1}^{-}(-K_{1})-T_{k}^{-}(-K_{1})\\|_{\infty}\leqslant$	$\displaystyle\,e^{-k\lambda}\\|T_{1}^{-}(-K_{1})-(-K_{1})\\|_{\infty}$
	$\displaystyle\leqslant$	$\displaystyle\,e^{-k\lambda}(K_{1}+K_{2}),$

which implies

\displaystyle\sum_{k=1}^{+\infty}\|T_{k+1}^{-}(-K_{1})-T_{k}^{-}(-K_{1})\|_{\infty}\leqslant\sum_{k=1}^{+\infty}e^{-k\lambda}(K_{1}+K_{2})=\frac{e^{-\lambda}(K_{1}+K_{2})}{1-e^{-\lambda}}<+\infty.

Therefore, there exists a unique $u_{0}:\mathbb{R}^{n}\to\mathbb{R}$ such that

\lim_{t\to+\infty}\|T_{t}^{-}(-K_{1})-u_{0}\|_{\infty}=0

(3.7)

with

-K_{1}\leqslant u_{0}(x)\leqslant K_{2},\qquad\forall x\in\mathbb{R}^{n}.

(3.8)

For any $t^{\prime}>0$ , by (3.7) and Lemma 3.15 (3) we have

T_{t^{\prime}}^{-}u_{0}=T_{t^{\prime}}^{-}\lim_{t\to+\infty}T_{t}^{-}(-K_{1})=\lim_{t\to+\infty}T_{t^{\prime}}^{-}T_{t}^{-}(-K_{1})=u_{0}.

(3.9)

(3.8), (3.9) and Lemma 3.17 implies $u_{0}$ is Lipschitz on $\mathbb{R}^{n}$ . Now we know that $u_{0}$ is a weak-KAM solution of (HJ_λ) which is bounded and Lipschitz on $\mathbb{R}^{n}$ . By Lemma 3.18, $u_{0}$ is also a viscosity solution of (HJ_λ). The uniqueness of $u_{0}$ is a direct consequence of Lemma 3.15 (3). This completes the proof of Theorem 3.13. ∎

Appendix A Regularity properties of fundamental solutions

Here we collect some relevant regularity results with respect to the fundamental solution of (HJ_e). The proofs of these regularity results are similar to those in [4] in autonomous case.

Proposition A.1.

Suppose L satisfies condition (L1)-(L3). Then for any $T>0$ , $0\leqslant s<t\leqslant T$ , $x,y\in\mathbb{R}^{n}$ , and any minimizer $\xi\in\Gamma^{s,t}_{x,y}$ for $A_{s,t}(x,y)$ , we have

\displaystyle\sup_{\tau\in[s,t]}|\dot{\xi}(\tau)|\leqslant\kappa(T,\frac{|x-y|}{t-s}).

where $\kappa:(0,+\infty)\times(0,+\infty)\rightarrow(0,+\infty)$ is nondecreasing.

Now, for $(s,x)\in[0,+\infty)\times\mathbb{R}^{n}$ , $\lambda>0$ and $\tau>0$ , let

\displaystyle S_{\lambda}(s,x,\tau)=\{(t,y)\in\mathbb{R}\times\mathbb{R}^{n}:s<t<s+\tau,|y-x|<\lambda(t-s)\}.

Proposition A.2.

Suppose L satisfies condition (L1)-(L3).Then for any fixed $T>0$ , $R>0$ and $\lambda>0$ , there exists $t_{0}(s,x,T,R,\lambda)>0$ such that for any $(s,x)\in[0,T]\times\bar{B}(0,R)$

(1)

The function $(t,y)\mapsto A_{s,t}(x,y)$ is semiconcave on the cone $S_{\lambda}(s,x,t_{0}(T,R,\lambda))$ and there exists $C_{0}(T,R,\lambda)>0$ such that for all $(t,y)\in S_{\lambda}(s,x,t_{0}(T,R,\lambda))$ , $h\in[0,\frac{1}{2}(t-s))$ and $z\in B(0,\lambda(t-s))$ we have that

\displaystyle A_{s,t+h}(x,y+z)+A_{s,t-h}(x,y-z)-2A_{s,t}(x,y)\leqslant\frac{C_{0}(T,R,\lambda)}{t-s}(h^{2}+|z|^{2}).

(2)

The function $(t,y)\mapsto A_{s,t}(x,y)$ is semiconvex on the cone $S_{\lambda}(s,x,t_{0}(T,R,\lambda))$ and there exists $C_{1}(T,R,\lambda)>0$ such that for all $(t,y)\in S_{\lambda}(s,x,t_{0}(T,R,\lambda))$ , $h\in[0,\frac{1}{2}(t-s))$ and $z\in B(0,\lambda(t-s))$ we have that

\displaystyle A_{s,t+h}(x,y+z)+A_{s,t-h}(x,y-z)-2A_{s,t}(x,y)\geqslant-\frac{C_{1}(T,R,\lambda)}{t-s}(h^{2}+|z|^{2}).

(3)

For all $t\in(s,s+\tau]$ , the function $A_{s,t}(x,\cdot)$ is uniformly convex on $B(x,\lambda(t-s))$ , and there exists $C_{2}(T,R,\lambda)>0$ such that for all $y\in B(x,\lambda(t-s))$ and $z\in B(0,\lambda(t-s))$ we have that

$\displaystyle A_{s,t}(x,y+z)+A_{s,t}(x,y-z)-2A_{s,t}(x,y)\geqslant\frac{C_{2}(T,R,\lambda)}{t-s}|z|^{2}.$

Moreover, $C(T,R,\lambda)$ is continuous with respect to $R$ .

Remark A.3.

In this paper, for any fixed $T>0$ , we choose $\lambda:=\lambda_{2}(T)$ and $R:=\lambda_{2}(T)T$ , where $\lambda_{2}(T)$ is defined in Lemma 2.6. Assume that $x\in S_{\lambda}(0,0,s)$ , then by Lemma 2.6 and the definition of $S_{\lambda}(s,x,\tau)$ , we can let $t_{0}(s,x,T,R)=T-s$ and $C_{i}(T,R,\lambda)$ only depends on $T$ for $i=1,2,3$ .

Moreover, if we consider the domain $[0,T]\times\bar{B}(x_{0},R)$ for any $x_{0}\in\mathbb{R}^{n}$ , then $C_{i}(T,R,\lambda)$ only depends on initial point $x_{0}$ and $T$ for $i=1,2,3$ .

Proposition A.4.

Suppose L satisfies condition (L1)-(L3). Then for any fixed $T>0$ , $R>0$ , $\lambda>0$ and $(s,x)\in[0,T]\times\bar{B}(0,R)$ , the function $(t,y)\mapsto A_{s,t}(x,y)$ is of class $C^{1,1}_{loc}$ on the cone $S_{\lambda}(s,x,t_{0}(T,R,\lambda))$ , where $t_{0}(T,R,\lambda)$ is that in Proposition A.2. Moreover, we have

	$\displaystyle D_{y}A_{s,t}(x,y)=L_{v}(t,\xi(t),\dot{\xi}(t)),$
	$\displaystyle D_{x}A_{s,t}(x,y)=-L_{v}(s,\xi(s),\dot{\xi}(s)),$
	$\displaystyle D_{t}A_{s,t}(x,y)=-E_{s,t,x,y},$

where $\xi\in\Gamma_{x,y}^{s,t}$ is the unique minimizer for $A_{s,t}(x,y)$ and

\displaystyle E_{s,t,x,y}:=H(t,\xi(t),p(t))

is the energy of the Hamilton trajectory $(\xi,p)$ with

p(\tau):=L_{v}(\tau,\xi(\tau),\dot{\xi}(\tau)),\quad\tau\in[s,t].

Appendix B Proof of Lemma 2.2 and Lemma 2.6

The convex conjugate of a superlinear function $\theta_{T}$ is defined as

\theta_{T}^{*}(s)=\sup_{r\geqslant 0}\{rs-\theta_{T}(r)\},\quad s\geqslant 0.

In view of the superlinear growth of $\theta_{T}$ , it is clear that $\theta_{T}^{*}$ is well defined and satisfies

\theta_{T}(r)+\theta_{T}^{*}(s)\geqslant rs,\quad r,s\geqslant 0,

which in turn can be used to show that $\theta_{T}^{*}(s)/s\rightarrow+\infty$ as $s\rightarrow+\infty$ .

Proof of Lemma 2.2.

For item (1), let $k=\mbox{\rm{Lip}}[f]+1$ . Then for any $x\in\mathbb{R}^{n}$ , $0\leqslant t_{1}<t_{2}\leqslant T$ and $z\in\mathbb{R}^{n}$ , we have

	$\displaystyle A_{t_{1},t_{2}}(z,x)=$	$\displaystyle\,\inf_{\xi\in\Gamma_{z,x}^{t_{1},t_{2}}}\int_{t_{1}}^{t_{2}}L(s,\xi,\dot{\xi})\ ds$
	$\displaystyle\geqslant$	$\displaystyle\,\inf_{\xi\in\Gamma_{z,x}^{t_{1},t_{2}}}\int_{t_{1}}^{t_{2}}\theta_{T}(\|\dot{\xi}\|)\ ds-c_{T}(t_{2}-t_{1})$
	$\displaystyle\geqslant$	$\displaystyle\,\inf_{\xi\in\Gamma_{z,x}^{t_{1},t_{2}}}k\int_{t_{1}}^{t_{2}}\|\dot{\xi}\|\ ds-(\theta_{T}^{*}(k)+c_{T})(t_{2}-t_{1})$
	$\displaystyle\geqslant$	$\displaystyle\,k\|z-x\|-(\theta_{T}^{*}(k)+c_{T})(t_{2}-t_{1}).$

Therefore,

		$\displaystyle f(x)+A_{t_{1},t_{2}}(x,x)-f(z)-A_{t_{1},t_{2}}(z,x)$
	$\displaystyle\leqslant$	$\displaystyle\,\mbox{\rm Lip}[f]\cdot\|z-x\|-k\|z-x\|+(\theta_{T}^{*}(k)+c_{T})\ (t_{2}-t_{1})+\int_{t_{1}}^{t_{2}}L(s,x,0)\ ds$
	$\displaystyle\leqslant$	$\displaystyle\,-\|z-x\|+[\theta_{T}^{*}(k)+c_{T}+\overline{\theta}_{T}(0)]\ (t_{2}-t_{1})\ .$

Now, taking $\lambda_{1}=\theta_{T}^{*}(k)+c_{T}+\overline{\theta}_{T}(0)$ , it follows that

\Lambda_{t_{1},t_{2}}^{x}:=\{z:f(z)+A_{t_{1},t_{2}}(z,x)\leqslant f(x)+A_{t_{1},t_{2}}(x,x)\}\subset\overline{B}\big{(}x,\lambda_{1}(t_{2}-t_{1})\big{)}.

(B.1)

Therefore $\Lambda_{t_{1},t_{2}}^{x}$ is compact and the infimum in (2.3) is attained, i.e., $Z(f,t_{1},t_{2},x)\neq\emptyset$ . Moreover, due to (B.1), for any $z_{t_{1},t_{2},x}\in Z(f,t_{1},t_{2},x)$ , we have

|z_{t_{1},t_{2},x}-x|\leqslant\lambda_{1}(t_{2}-t_{1}).

For item (2), A similar result holds for the sup-convolution defined in (2.4). ∎

Proof of Lemma 2.6.

Set $(t,x)\in(0,T]\times\mathbb{R}^{n}$ is a differentiable point of $u$ . Due to Proposition 2.1 and Proposition 2.5, the solution of (2.2) with terminal condition

\begin{cases}\xi(t)=x\\ p(t)=\nabla u(t,x)\end{cases}

is the unique minimizer for $u(t,x)$ . Lemma 2.2 (1) implies $|\xi(0)-x|\leqslant\lambda_{1}(T,\mbox{\rm Lip}[u_{0}])t$ . Now, denote that

\displaystyle E(s):=H(s,\xi(s),p(s)),\qquad s\in[0,t].

By Lemma 2.3, we know that $p(0)\in D^{-}u_{0}(\xi(0))$ . This implies $|p(0)|\leqslant\mbox{\rm Lip}\,[u_{0}]$ and

\displaystyle E(0)=H(0,\xi(0),p(0))\leqslant\theta^{*}_{T}(|p(0)|)+c_{T}\leqslant\theta^{*}_{T}(\mbox{\rm Lip}\,(u_{0}))+c_{T}.

Notice that

		$\displaystyle\frac{d}{ds}E(s)=\frac{d}{ds}H(s,\xi(s),p(s))=H_{t}+H_{x}\cdot\dot{\xi}(s)+H_{p}\cdot\dot{p}(s)$
	$\displaystyle=$	$\displaystyle H_{t}+H_{x}\cdot H_{p}+H_{p}\cdot(-H_{x})=H_{t}(s,\xi(s),p(s))=-L_{t}(s,\xi(s),\dot{\xi}(s)).$

Thus, we have

	$\displaystyle E(t)$	$\displaystyle=E(0)+\int_{0}^{t}\frac{d}{ds}E(s)\ ds$
		$\displaystyle=E(0)+\int_{0}^{t}L_{t}(s,\xi(s),\dot{\xi}(s))\ ds$
		$\displaystyle\leqslant E(0)+\int_{0}^{t}\Big{(}\widetilde{C}_{1}(T)+\widetilde{C}_{2}(T)L(s,\xi(s),\dot{\xi}(s))\Big{)}\ ds$
		$\displaystyle\leqslant E(0)+t\ \widetilde{C}_{1}(T)+\widetilde{C}_{2}(T)\int_{0}^{t}L(s,\xi(0)+s(x-\xi(0)),\frac{x-\xi(0)}{t})\ ds$
		$\displaystyle\leqslant E(0)+\widetilde{C}_{1}(T)T+\widetilde{C}_{2}(T)\int_{0}^{t}\overline{\theta}_{T}(\|\frac{x-\xi(0)}{t}\|)\ ds$
		$\displaystyle\leqslant\theta^{*}_{T}(\mbox{\rm Lip}\,(u_{0}))+c_{T}+\widetilde{C}_{1}(T)T+\widetilde{C}_{2}(T)T\cdot\overline{\theta}_{T}(\lambda_{1}(T,\mbox{\rm Lip}[u_{0}])t).$

Since

\displaystyle|\nabla u(t,x)|\leqslant\overline{\theta}_{T}^{*}(|\nabla u(t,x)|)+\overline{\theta}_{T}(1)\leqslant H(t,x,\nabla u(t,x))+\overline{\theta}_{T}(1)=E(t)+\overline{\theta}_{T}(1),

it follows that

	$\displaystyle\|\nabla u(t,x)\|\leqslant$	$\displaystyle\theta^{*}_{T}(\mbox{\rm Lip}\,(u_{0}))+c_{T}+\widetilde{C}_{1}(T)T+\widetilde{C}_{2}(T)T\cdot\overline{\theta}_{T}(\lambda_{1}(T,\mbox{\rm Lip}[u_{0}])t)+\overline{\theta}_{T}(1)$
	$\displaystyle:=$	$\displaystyle F_{1}(T).$

By proposition 2.4 (1), we obtain

	$\displaystyle\|u_{t}(t,x)\|$	$\displaystyle=\|-H(t,x,\nabla u(t,x))\|\leqslant\theta_{T}^{}(\|\nabla u(t,x)\|)+c_{0}+\|\bar{\theta}_{T}^{}(\|\nabla u(t,x)\|)\|$
		$\displaystyle\leqslant\theta_{T}^{}(F_{1}(T))+c_{0}+\|\bar{\theta}_{T}^{}(F_{1}(T))\|:=F_{2}(T).$

Therefore,

\displaystyle|Du(t,x)|=(|\nabla u(t,x)|^{2}+|u_{t}(t,x)|^{2})^{\frac{1}{2}}\leqslant\big{(}F_{1}^{2}(T)+F_{2}^{2}(T)\big{)}^{\frac{1}{2}}:=F_{0}(T).

Combing this with Proposition 2.4 (2), we conclude that $u$ is a Lipschitz function on $(0,T]\times\mathbb{R}^{n}$ and $\mbox{\rm Lip}\,[u]\leqslant F_{0}(T)$ . ∎

Appendix C Proof of Lemma 3.8

We define $F:\mathbb{R}^{n}\times[0,t]\to\mathbb{R}^{n}$ as:

F(x,s)=\mathbf{x}_{x}(s),\quad s\in[0,+\infty),

where $\mathbf{x}_{x}:[0,+\infty)\to\mathbb{R}^{n}$ is that in Theorem 2.9.

Similar to [8, Lemma 2.1], $F(x,s)$ has the properties (a), (b) and (c) stated in Lemma 3.8. Due to Theorem 3.3, $\mathbf{x}_{x}(s)$ is a locally Lipschitz curve. Thus, to prove that $F(x,s)=\mathbf{x}_{x}(s)$ is continuous, it remains to show $\mathbf{x}_{x}(s)$ is continuous with respect to $x$ . We prove it in Lemma C.2.

Lemma C.1.

For any fixed $x\in\mathbb{R}^{n}$ and $t>0$ , let $t_{x,T}>0$ be defined in Lemma 2.8. Then for any $0<s<t<T$ with $t-s\leqslant t_{x,T}$ , the map $z\mapsto y_{s,t,z}$ is Lipschitz on $B(x,\lambda_{2}(T)t_{x,T})$ with Lipschitz constant $K_{1}(x,T)$ which only depends on $x,T$ .

Proof.

For any $x_{1},x_{2}$ with $|x_{1}-x_{2}|<\lambda_{2}(T)t_{x,T}$ and $|x_{1}-x_{2}|<r$ , we denote by $y_{s,t,x_{1}}$ and $y_{s,t,x_{2}}$ the unique maximizers of $T^{+}_{s,t}u(t,x_{1})$ and $T^{+}_{s,t}u(t,x_{2})$ respectively. Notice that $A_{s,t}(x_{1},\cdot)$ is uniformly convex in the ball $B(x_{1},2\lambda_{2}(T)t_{x,T})$ with convexity constant $C_{2}(x,T)/t$ for $t\in(0,t_{T})$ .Then we have

\frac{C_{2}(x,T)}{t}|y_{s,t,x_{1}}-y_{s,t,x_{2}}|^{2}\leqslant A_{s,t}(x_{1},y_{s,t,x_{1}})-A_{s,t}(x_{1},y_{s,t,x_{2}})-D_{y}A_{s,t}(x_{1},y_{s,t,x_{2}})(y_{s,t,x_{1}}-y_{s,t,x_{2}}).

On the other hand, since $C(T)$ is the semiconcave constant of $u(t,\cdot)$ for $t\in[0,T]$ , we have

		$\displaystyle\,A_{s,t}(x_{1},y_{s,t,x_{1}})-A_{s,t}(x_{1},y_{s,t,x_{2}})\leqslant u(t,y_{s,t,x_{1}})-u(t,y_{s,t,x_{2}})$
	$\displaystyle\leqslant$	$\displaystyle\,D_{y}A_{s,t}(x_{2},y_{s,t,x_{2}})(y_{s,t,x_{1}}-y_{s,t,x_{2}})+C(x,T)\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}.$

Notice that $D_{y}A_{t}(x_{2},y_{s,t,x_{2}})\in\nabla^{+}u(t,y_{s,t,x_{2}})$ . Thus,

		$\displaystyle\,\frac{C_{2}(x,T)}{t-s}\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}$
	$\displaystyle\leqslant$	$\displaystyle\,\Big{(}D_{y}A_{s,t}(x_{2},y_{s,t,x_{2}})-D_{y}A_{s,t}(x_{1},y_{s,t,x_{2}})\Big{)}(y_{s,t,x_{1}}-y_{s,t,x_{2}})+C(x,T)\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{C_{0}(x,T)}{t-s}\|x_{1}-x_{2}\|\cdot\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|+C(x,T)\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}.$

Therefore, due to $t_{x,T}:=\frac{C_{2}(x,T)}{2C(x,T)}$ , for any $0<t-s\leqslant t_{x,T}$ , we have

|y_{s,t,x_{1}}-y_{s,t,x_{2}}|\leqslant\frac{C_{0}(x,T)}{C_{2}(x,T)-C(x,T)\cdot(t-s)}|x_{1}-x_{2}|\leqslant\frac{2C_{0}(x,T)}{C_{2}(x,T)}|x_{1}-x_{2}|:=K_{1}(x,T)|x_{1}-x_{2}|.

∎

Lemma C.2.

For any $t>0$ , $\mathbf{x}_{x}(t)$ is continuous with respect to $x$ .

Proof.

Let $\Omega_{n}(x):=\{B(x,\lambda_{2}(nT)nT)\}$ and assume $t_{x_{1},T}\geqslant t_{x_{2},T}$ . By Theorem 2.9, we construct an new curve $\widetilde{\mathbf{x}}_{x_{1}}(t)$ defined by $\Omega_{n}(x_{1})$ and $\widetilde{t}_{x_{1},T}:=t_{x_{2},T}$ . Then, for $(n-1)T\leqslant t<nT$ with $n\in\mathbb{N}$ , by Lemma C.1, we have

|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{2}}(t)|\leqslant K_{0}(nT)\cdot|x_{1}-x_{2}|,\quad\forall|x_{1}-x_{2}|<K_{1}(T)t_{x_{2},T}.

(C.1)

where $K_{0}(T)=(K_{1}(T))^{k_{1}}$ and $k_{1}$ depends on $\widetilde{t}_{x_{1},T}=t_{x_{2},T}$ .

On the other hand, due to Theorem 2.8, Proposition A.2 and Proposition 2.4(2),

t_{x,T}=\frac{C_{2}(x,T)}{2C(x,T)}

is continuous with respect to $x$ . Therefore, for $0\leqslant t<T$ , we have

|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{1}}(t)|\leqslant\widetilde{K}_{0}(T)\cdot|t_{x_{1},T}-t_{x_{2},T}|,

where $\widetilde{K}_{0}(T):=(K_{1}(T))^{k_{1}}$ and $k_{1}$ depends on $t_{x_{1},T}$ . Combining this with (C.1), one obtain that for any $t\in[0,T)$ ,

	$\displaystyle\|\mathbf{x}_{x_{2}}(t)-\mathbf{x}_{x_{1}}(t)\|\leqslant$	$\displaystyle\,\|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{2}}(t)\|+\|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{1}}(t)\|$
	$\displaystyle\leqslant$	$\displaystyle\,K_{0}(T)\cdot\|x_{1}-x_{2}\|+\widetilde{K}_{0}(T)\cdot\|t_{x_{1},T}-t_{x_{2},T}\|.$

Similarly, for $t\in[(n-1)T,nT)$ , there exists constant $K_{0}(n,T)$ and $\widetilde{K}_{0}(n,T)$ such that

\displaystyle|\mathbf{x}_{x_{2}}(t)-\mathbf{x}_{x_{1}}(t)|\leqslant K_{0}(n,T)\cdot|x_{1}-x_{2}|+\widetilde{K}_{0}(n,T)\cdot|t_{x_{1},nT}-t_{x_{2},nT}|.

Since $t_{x,nT}$ is continuous with respect to $x$ for each $n\in\mathbb{N}$ , we conclude that for any $t>0$ , $\mathbf{x}_{x}(t)$ is continuous with respect to $x$ . ∎

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		$\displaystyle\,\frac{C_{2}(x,T)}{t-s}\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}$
	$\displaystyle\leqslant$	$\displaystyle\,\Big{(}D_{y}A_{s,t}(x_{2},y_{s,t,x_{2}})-D_{y}A_{s,t}(x_{1},y_{s,t,x_{2}})\Big{)}(y_{s,t,x_{1}}-y_{s,t,x_{2}})+C(x,T)\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}$
	$\displaystyle\leqslant$	$\displaystyle\,\frac{C_{0}(x,T)}{t-s}\|x_{1}-x_{2}\|\cdot\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|+C(x,T)\|y_{s,t,x_{1}}-y_{s,t,x_{2}}\|^{2}.$

	$\displaystyle\|\mathbf{x}_{x_{2}}(t)-\mathbf{x}_{x_{1}}(t)\|\leqslant$	$\displaystyle\,\|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{2}}(t)\|+\|\widetilde{\mathbf{x}}_{x_{1}}(t)-\mathbf{x}_{x_{1}}(t)\|$
	$\displaystyle\leqslant$	$\displaystyle\,K_{0}(T)\cdot\|x_{1}-x_{2}\|+\widetilde{K}_{0}(T)\cdot\|t_{x_{1},T}-t_{x_{2},T}\|.$

Global propagation of singularities for discounted Hamilton-Jacobi equations

Abstract.

Key words and phrases:

1. introduction

2. Global propagation of singularities for time-dependent Hamiltonian

Proposition 2.1.

2.1. Semiconcave functions

2.2. Lax-Oleinik operator in time-dependent case and a priori estimate

Lemma 2.2.

Lemma 2.3.

Proposition 2.4.

Proposition 2.5.

Lemma 2.6.

Corollary 2.7.

2.3. Propagation of singularities

Proposition 2.8.

Proof.

Theorem 2.9.

Proof.

Remark 2.10.

2.4. Local Lipschitz of singular curve

Theorem 2.11.

Lemma 2.12.

Proof.

3. Global propagation of singularities for discounted Hamiltonian

3.1. Global propagation of singularities for discounted Hamiltonian

Lemma 3.1.

Remark 3.2.

Theorem 3.3.

Proof.

3.2. Homotopy equivalence

Definition 3.4.

Theorem 3.5.

Corollary 3.6.

Theorem 3.7.

Lemma 3.8.

Lemma 3.9.

Lemma 3.10.

Proof.

Proof of Theorem 3.5.

3.3. genuine propagation of singularities

Definition 3.11.

Corollary 3.12.

3.4. Existence of global Lipschitz viscosity solution of (HJλ)

Theorem 3.13.

Remark 3.14.

Lemma 3.15.

Lemma 3.16.

Proof.

Lemma 3.17.

Proof.

Lemma 3.18.

Proof of Theorem 3.13.

Appendix A Regularity properties of fundamental solutions

Proposition A.1.

Proposition A.2.

Remark A.3.

Proposition A.4.

Appendix B Proof of Lemma 2.2 and Lemma 2.6

Proof of Lemma 2.2.

Proof of Lemma 2.6.

Appendix C Proof of Lemma 3.8

Lemma C.1.

Proof.

Lemma C.2.

Proof.

References

3.4. Existence of global Lipschitz viscosity solution of (HJ_λ)