Convergence of Viscosity Solutions of Generalized Contact Hamilton–Jacobi Equations

Abstract

For any compact connected manifold M, we consider the generalized contact Hamiltonian H(x, p, u) defined on $T^*M\times \mathbb {R}$ which is convex in p and monotonically increasing in u. Let $u_\varepsilon ^-:M\rightarrow \mathbb {R}$ be the viscosity solution of the parametrized contact Hamilton–Jacobi equation

$$\begin{aligned} H(x,d_x u_\varepsilon ^-(x),\varepsilon u_\varepsilon ^-(x))=c(H), \end{aligned}$$

with c(H) being the Mañé Critical Value. We prove that $u_\varepsilon ^-$ converges uniformly, as $\varepsilon \rightarrow 0_+$, to a specific viscosity solution $u_0^-$ of the critical equation

$$\begin{aligned} H(x,d_x u_0^-(x),0)=c(H), \end{aligned}$$

which can be characterized as a minimal combination of the associated Peierls barrier functions.

Appendix A: The Proof of Lemma 2.3

For any $\varepsilon \in (0,1]$ and $\phi \in C^0(M,\mathbb {R})$, we have

$$\begin{aligned} \mathcal {T}^{\varepsilon -}_t\phi (x)=\inf _{\gamma (t)=x}\bigg \{\phi (\gamma (0))+\int ^t_0L(\gamma (s),\dot{\gamma }(s),\varepsilon \mathcal {T}^{\varepsilon -}_s\phi (\gamma (s)))+c(H)\,\mathrm{d}s\bigg \}, \end{aligned}$$

implicitly defined, with the infimum taken among all piecewise $C^1$ curve. Furthermore, for each $\phi \in C(M,\mathbb {R})$,

$$\begin{aligned} \lim _{t\rightarrow +\infty }\mathcal {T}^{\varepsilon -}_t\phi (x)=u^-_{\varepsilon }(x), \end{aligned}$$

where $u_\varepsilon ^-$ is the unique weak KAM solution of $H(x,\partial _xu_{\varepsilon }^-,\varepsilon u_{\varepsilon }^-)=c(H)$, see Theorem 1.4 in [13] and Appendix B in [14]. Now we are ready to give a proof of Lemma 2.3.

Lemma A.1

For any $\phi \in C(M,\mathbb {R})$ with $\Vert \phi \Vert \leqq 1$, the family $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$ is uniformly bounded.

Proof

We claim that $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$ is uniformly bounded from below. Without loss of generality, we assume $\mathcal {T}^{\varepsilon -}_t\phi (x)<0$ for some $\varepsilon \in (0,1]$ and $(x,t)\in M\times [1,+\infty )$, otherwise 0 would be a lower bound of the family. Let $\gamma _{x,\varepsilon }:[0,t]\rightarrow M$ be the associated minimizer of $\mathcal {T}^{\varepsilon -}_t\phi (x)$. Then, there are two probabilities:

Case I There exists $s_0\in [0,t)$ such that $\mathcal {T}^{\varepsilon -}_{s_0}\phi (\gamma _{x,\varepsilon }(s_0))=0$ and $\mathcal {T}^{\varepsilon -}_{s}\phi (\gamma _{x,\varepsilon }(s))<0, s\in (s_0,t]$.

Case II $\mathcal {T}^{\varepsilon -}_s\phi (\gamma _{x,\varepsilon }(s))<0$ for all $s\in [0,t]$.

      For Case I, due to (H3), L is strictly decreasing of u. Therefore,

$$\begin{aligned} \mathcal {T}^{\varepsilon -}_t\phi (x)&=\mathcal {T}^{\varepsilon -}_{s_0}\phi (\gamma _{x,\varepsilon }(s_0))+\int ^t_{s_0}L(\gamma _{x,\varepsilon }(s),\dot{\gamma }_{x,\varepsilon }(s),\varepsilon \mathcal {T}^{\varepsilon -}_s\phi (\gamma _{x,\varepsilon }(s)))+c(H)\,\mathrm{d}s\\&\geqq \int ^t_{s_0}L(\gamma _{x,\varepsilon }(s),\dot{\gamma }_{x,\varepsilon }(s),0)+c(H)\,\mathrm{d}s\\&\geqq h^{t-s_0}(\gamma _{x,\varepsilon }(s_0),x)+c(H)(t-s_0). \end{aligned}$$

It is well known that $h^{t-s_0}(\gamma _{x,\varepsilon }(s_0),x)+c(H)(t-s_0)$ is uniformly bounded from below (see Lemma 5.3.2 in [8] for instance). Moreover, the lower bound can be made independent of the selection of $\varepsilon \in (0,1]$ and $(x,t-s_0)\in M\times (0,+\infty )$.

For Case II,

$$\begin{aligned} \mathcal {T}^{\varepsilon -}_t\phi (x)&=\phi (\gamma _{x,\varepsilon }(0))+\int ^t_{0}L(\gamma _{x,\varepsilon }(s),\dot{\gamma }_{x,\varepsilon }(s),\varepsilon \mathcal {T}^{\varepsilon -}_s\phi (\gamma _{x,\varepsilon }(s)))+c(H)\,\mathrm{d}s\\&\geqq \min _{x\in M}\phi (x)+\int ^t_{0}L(\gamma _{x,\varepsilon }(s),\dot{\gamma }_{x,\varepsilon }(s),0)+c(H)\,\mathrm{d}s\\&\geqq -1+h^t(\gamma _{x,\varepsilon }(0),x)+c(H)t. \end{aligned}$$

Hence, $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1]\}$ is bounded from below. Still the lower bound is independent of the selection of $\varepsilon \in (0,1]$ and $(x,t)\in M\times [1,+\infty )$.

      As a summary, the family $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$ is uniformly bounded from below.

      We claim $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$ is uniformly bounded from above. Without loss of generality, we assume $\mathcal {T}^{\varepsilon -}_t\phi (x)>0$ for some $\varepsilon \in (0,1]$ and $(x,t)\in M\times [1,+\infty )$, otherwise 0 is a upper bound of $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$. Let $\beta :[0,t]\rightarrow M$ be the associated minimizer of $h^t(\gamma _{x,\varepsilon }(0),x)$, i.e.,

$$\begin{aligned} h^t(\gamma _{x,\varepsilon }(0),x)=\int ^t_0L(\beta (s),\dot{\beta }(s),0)\,\mathrm{d}s. \end{aligned}$$

There are also two probabilities:

Case I’ $\mathcal {T}^{\varepsilon -}_s\phi (\beta (s))>0$ for each $s\in [0,t]$. Hence,

$$\begin{aligned} \mathcal {T}^{\varepsilon -}_t\phi (x)&\leqq \phi (\beta (0))+\int ^t_0L(\beta (s),\dot{\beta }(s),\varepsilon \mathcal {T}^{\varepsilon -}_s\phi (\beta (s)))+c(H)\,\mathrm{d}s\\&\leqq \max _{x\in M}\phi (x)+\int ^t_0L(\beta (s),\dot{\beta }(s),0)+c(H)\,\mathrm{d}s\\&\leqq 1+h^t(\gamma _{x,\varepsilon }(0),x)+c(H)t. \end{aligned}$$

Since $t\geqq 1$, $h^t(\gamma _{x,\varepsilon }(0),x)+c(H)t$ is bounded from above. Hence, $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,t],t\geqq 1\}$ is uniformly bounded from above.

Case II’ There exists $s_1\in [0,t)$ such that $\mathcal {T}^{\varepsilon -}_{s_1}\phi (\beta (s_1))=0$ and $\mathcal {T}^{\varepsilon -}_s\phi (\beta (s))>0,s\in (s_1,t]$. Then,

$$\begin{aligned} \mathcal {T}^{\varepsilon -}_t\phi (x)&\leqq \mathcal {T}^{\varepsilon -}_{s_1}\phi (\beta (s_1))+\int ^t_{s_1}L(\beta (s),\dot{\beta }(s),\varepsilon \mathcal {T}^{\varepsilon -}_s\phi (\beta (s)))+c(H)\,\mathrm{d}s\\&\leqq \int ^t_{s_1}L(\beta (s),\dot{\beta }(s),0)+c(H)\,\mathrm{d}s\\&=h^{t-s_1}(\beta (s_1),x)+c(H)(t-s_1). \end{aligned}$$

      If $t-s_1\geqq \frac{1}{2}$, then $h^{t-s_1}(\beta (s_1),x)+c(H)(t-s_1)$ is bounded from above. If not, then $s_1>\frac{1}{2}$. Note that

$$\begin{aligned} h^t(\gamma _{x,\varepsilon }(0),x)=h^{s_1}(\gamma _{x,\varepsilon }(0),\beta (s_1))+h^{t-s_1}(\beta (s_1),x). \end{aligned}$$

We derive that

$$\begin{aligned}&h^{t-s_1}(\beta (s_1),x)+c(H)(t-s_1)=\bigg (h^t(\gamma _{x,\varepsilon }(0),x)+c(H)t\bigg )\\&\quad -\bigg (h^{s_1}(\gamma _{x,\varepsilon }(0),\beta (s_1))+c(H)s_1\bigg ), \end{aligned}$$

      Note that the first term is bounded from above ($t\geqq 1$) and the second term is bounded from below ($s_1> 1/2$), see Lemma 5.3.2 in [8]. Hence, $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$ is uniformly bounded from above. $\square$

Lemma A.2

The family $\{u_\varepsilon ^-\}$ is uniformly bounded for all $\varepsilon \in (0,1]$.

Proof

Due to the boundedness of $\{\mathcal {T}^{\varepsilon -}_t\phi (\cdot )|\varepsilon \in (0,1],t\geqq 1\}$, there exists a $K>0$ such that for $\phi \in C(M,\mathbb {R})$ satisfying $\Vert \phi \Vert \leqq 1$,

$$\begin{aligned} |\mathcal {T}^{\varepsilon -}_t\phi (x)|\leqq K, (x,t)\in M\times [1,+\infty ) \text{ and } \varepsilon \in (0,1]. \end{aligned}$$

Since $\lim _{t\rightarrow +\infty }\mathcal {T}^{\varepsilon -}_t\phi (x)=u_{\varepsilon }^{-}(x)$ is uniquely established, we get $|u^-_\varepsilon (x)|\leqq K$ for all $\varepsilon \in (0,1]$, immediately. $\square$

Lemma A.3

The map $x \rightarrow u_{\varepsilon }^-(x)$ is equi-Lipschitz for $\varepsilon \in (0,1]$.

Proof

Let $x,y\in M$ and $\alpha :[0,d_R(x,y)]\rightarrow M$ be a geodesic connecting x and y. Note that $\sqrt{\langle \dot{\alpha },\dot{\alpha }\rangle }_R\leqq 1$, where $\langle \cdot ,\cdot \rangle _R$ is the Riemannian metric on M. We derive that

$$\begin{aligned} u^-_{\varepsilon }(y)-u^-_{\varepsilon }(x)&\leqq \int ^{d_R(x,y)}_0L(\alpha (s),\dot{\alpha }(s),\varepsilon u^-_{\varepsilon }(\alpha (s)))+c(H)\,\mathrm{d}s\\&\leqq \int ^{d_R(x,y)}_0L(\alpha (s),\dot{\alpha }(s),0)+\Delta \cdot K+c(H)\,\mathrm{d}s\\&\leqq (C_1+\Delta \cdot K+c(H))d_R(x,y), \end{aligned}$$

where $C_1$ is a uniform constant such that

$$\begin{aligned} L(x,v,0)\leqq C_1,\quad \forall \Vert v\Vert _R\leqq 1. \end{aligned}$$

By switching the role of x and y, we derive

$$\begin{aligned} u^-_{\varepsilon }(x)-u^-_{\varepsilon }(y)\leqq (C_1+\Delta \cdot K+c(H))d_R(x,y), \end{aligned}$$

and then

$$\begin{aligned} |u^-_{\varepsilon }(x)-u^-_{\varepsilon }(y)|\leqq (C_1+\Delta \cdot K+c(H))d_R(x,y), \end{aligned}$$

This finishes the proof. $\square$