2010 04 14 Fathi FCAM Hamilton-Jacobi

(These notes need to be proofread. Colliand 19:27, 14 April 2010 (UTC))

I am interested in studying mainly the dynamics associated with a Hamiltonian $$H: R_x^n \times R+p^n \rightarrow R.$$

Properties of the Hamiltonian

The first components index position and the second set index momentum. We will need some compactness w.r.t. $x$. We will assume that $J$ is $Z^n$ periodic in $x$: $$H(x+k, p) = H(x, p) ~\forall x \in R^n, k \in Z^n, p \in R^n.$$ Thus, we can collapse the domain of $H$ to be $T^n \times R^n$. We will collapse the discussion further and restrict attention to Tonelli Hamiltonian:

1. $H$ is $C^r, ~r \geq 2$
2. $\frac{\partial^2 H}{\partial p^2} (x,p)$ is positive definite $\forall (x,p) \in T^n \times R^n$.
3. $\frac{H(x,p)}{\|p\|} \rightarrow + \infty$ as $\| p \| \rightarrow + \infty$.

An example of such a Hamiltonian is $$H(x,p) - \frac{1}{2} \|p \|_{euc}^2 + V(x)$$ where $V: T^n \rightarrow R$ is $C^2$.

Hamilton-Jacobi Equation, Solutions and Subsolutions

I am interested in the (stationary) ~Hamilton-Jacobi equation $HJE_c$. $$H(x, d_x u) = c.$$ A solution (global, classical) is a $C^1$ function $u: T^n \rightarrow R$ which satisfies the equation. A subsolution is a global, classical $C^1$ function $v: T^n \rightarrow R$ which satisfies $$H(x,d_x v) \leq C ~\forall x \in T^n$$.

For $C^1$ functions, we can define $\mathbb{H} (u) = \sup_{x \in T^n} H(x, d_x u)$. If $c \geq {\mathbb{H}} (u)$ where $u$ is a subsolution of the equation $HJE_c$

Proposition: If $u: T^n \rightarrow R$ is a $C^1$ solution of $HJE_c$ then $c = \overline{H} (0).$

Why? $\mathbb{H} (u) = \sup_{x \in T^n} H(x, d_x u) = c.$ Therefore $\overline{H} (0) \leq C$. Let $v: T^n \rightarrow R, ~C^1$ such that $v=u$ has a maximum on $T^n$. Therefore there exists $x_0$ with $d|_{x = x_0} (v-u) = 0.$ Therefore we find that $d_{x_0} u = d_{x_0} v$. Hence, $${\mathbb{H}} (v) \geq H(x_0, d_{x_0} v) = H(x_0, d_{x_0} u) = c.$$ This implies that ${\overline{H}} (0) \geq c$. Note that $C^1$ solutions do not always exist. There is a theory of generalized solutions for $HJE$. These are the viscosity solutions of Crandall-Lions (Hopf, Kruzkhov, Lax, ....). In fact, in this theory has global solutions.

In what may be the most cited unpublished paper (now that Gromov's has been published.....): Theorem (Lions, Papanicolau, Varadhan 87): There always exist viscosity solutions of $HJE_c$ for some $c$. These solutions are Lipschitz and the $c$ for which existence holds true is $c = {\overline{H}} (0)$.

When does there exist a $C^1$ solution?

Let $u: T^n \rightarrow R, ~C^1$. We have $\mathbb{H}(u)$ and we introduce the Hamiltonian defect $${\mathbb{H}} D (u) - \max_{x \in T^n} H(x, d_x u) - \min_{x \in T^n} H(x, d_x u).$$ Let's define $D(H) = \inf_{u: T^n \rightarrow R, ~C^1} {\mathbb{H}} D u.$

Theorem (Fathi 2000): There exists a $C^1$ solution if and only if $D(H) = 0.$

Let $U \subset R^n$. Let $u_l : U \rightarrow R, ~C^1$ with $u_l \rightarrow u$ uniformly and $\| d_x u_l \|_{euc} \rightarrow 1$ uniformly. Then, $u$ is $C^1$.

Critical Subsolutions

Theorem (Fathi and Siconoffi 2002): There always exists $C^1$ critical subsolutions. Improved by P. Bernard in 2006 to conclude $C^{1,1}$ subsolutions and cannont be better because there are counterexamples.

Can we say something if we know there are more regular critical subsolutions? This is what I want to talk about today.....

To discuss this, I have to recall the function $\overline{H}: R^n \rightarrow R$. For every $P \in R^n$, I can define $H_P (x,p) = H(x, P + p).$ We have then that $\mathbb{H} (P) = \mathbb{H}_P (0).$

Lemma (L-P-V): $\overline{H}: R^n \rightarrow R$ is convex and superlinear.

Let's take the subgradient. $$\partial {\overline{H}} (P) = \{ v \in R^n: \langle v, p \rangle \leq {\overline{H}} (P+p) - {\overline{H}}(P).\}$$ Give a geometrical-dynamical description of $\partial {\overline{H}} (P)$. Let $u: T^n \rightarrow R$ be a $C^{1,1}$ subsolution. Introduce the Hamiltonian gradient of $u$: $$X_u (x) = \frac{\partial H}{\partial p} (x, d_x u).$$ We observe that $X_u$ is Lipschitz in $T^n$. It therefore has a flow which is complete because $T^n$ is compact. We also have that $H(x, d_x u) \leq {\overline{H}} (0)$. Therefore, the set $$\mathcal{T}(u) = \{ x: H(x, d)x u) = \overline{H} (0) \}$$ is non-empty. Let $\mathcal{J} (u) = \cap_{t\in R} \phi_t^u ( \mathcal{T}(u))$ be the maximal subset of $\mathcal{T} (u)$. This set is invariant under the flow $\phi_t^u$. It is the projection of the orbits of the Hamiltonian flow induced by $H$ contained in the graph of $du$ at the highest sublevel set $H^{-1} ( \overline{H} (0) ).$

Theorem (Mather): The Aubry set of u, introduced above as $\mathcal{J} (u)$, is not empty.

Next consider the set of invariant measures $\mathcal{P} (u)$ consisting of probability meausres which are supported in $\mathcal{J} (u)$ and invariant under the flow $\phi_t^u$. By Krylov-Bogoliubov theorem, $\mathcal{P} (u) \neq \phi$, $\mathcal{T} (u)$ compact, convex for the weak topology.

Theorem: $\partial {\overline{H}} (0) = \{ \int_{T^n} X_u (x) d\mu (x) : \mu \in \mathcal{P} (u) \}.$

Suppose $n=2$, so we are on $T^2$, and suppose that $H$ is $C^3$ so there exists a critical subsolution $u$ which is $C^3$. Then $X_u$ is $C^2$. But we can now apply Denjoy-Schwartz theory.

Theorem (Denjoy-Schwartz): If the flow $\phi_t$ is $C^2$ on $T^2$ and A is a compact nonempty subset of $T^2$ in v and $\phi_t$ then one of the following happens:

1. A contains a fixed point of $\phi_t.$
2. The recurrent orbits are closed orbits and they are all homologous on $T^2.$
3. $A= T^2$, and every orbit is dense and in that case there is only one probability measure invariant under $\phi_t .$

Theorem Assume that $H: T^2 \times R^2 \rightarrow R$ is $C^3$ and $H_P (x,p) = H(x, P + p)$ has a $C^3$ critical subsolution. Then one of the following things happens: 1. $0 \in \partial \overline{H} (P)$ so $\overline{H} (P) = \min {\overline{H}}.$

2. $\exists ~\alpha \in Z^2$ and $\partial {\overline{H}} (P) \subset R^*_+ \alpha.$

3. $\partial {\overline{H}} (P)$ is exactly one point in $R^2 \backslash R Z^2.$ In that case $D \overline{H} (P)$ exists.