Technical Summary: Weak KAM Theorems for Subriemannian Lagrangians Depending on the Unknown Function

Problem Statement

The paper addresses the extension of Weak KAM (Kolmogorov-Arnold-Moser) theory to a specific class of variational problems defined on subriemannian manifolds. The core challenge lies in two simultaneous generalizations of the classical setting:

1. Geometric Constraint: The Lagrangian is defined only on the horizontal distribution $D$ of a subriemannian manifold $(M, D, \langle \cdot, \cdot \rangle)$, rather than the full tangent bundle $TM$. This restricts admissible curves to be "horizontal" (tangent to $D$ almost everywhere).
2. Functional Dependence: The Lagrangian $L(v, u)$ depends explicitly on the unknown function $u$ (the solution to the associated Hamilton-Jacobi equation), in addition to the velocity $v$. This introduces a nonlinearity that prevents the direct application of standard autonomous Lagrangian techniques.

The authors aim to establish the existence and uniqueness of viscosity solutions to the associated Cauchy problem and to prove the convergence of the Lax-Oleinik semigroup to a stationary solution (the Weak KAM theorem) under these constraints.

Methodology

1. Variational Framework and Definitions

The authors define the space of horizontal absolutely continuous curves, denoted $W^{1,1}_D([a, b])$, and the subriemannian length. They introduce the subriemannian contact Hamiltonian $H: T^*M \times \mathbb{R} \to \mathbb{R}$ via the Legendre-Fenchel transform of the Lagrangian:
$$H(p, u) = \max \{ p(v) - L(v, u) : v \in D, \pi(v) = \pi^*(p) \}$$
The Lagrangian $L \in C^2(D \times \mathbb{R})$ is assumed to satisfy three critical conditions:

• Uniform Superlinearity: $L(v, 0) \geq k\|v\| + C(k)$ for all $k \geq 0$.
• Bounded Monotonicity: $-\lambda \leq \partial_u L(v, u) < 0$, ensuring the dependence on $u$ is strictly decreasing but bounded.
• Strict Convexity: $\partial^2_{vv} L(v, u)(w, w) \geq \kappa \|w\|^2$.

2. Tonelli's Theorem for Non-Autonomous Lagrangians

A significant portion of the methodology is dedicated to proving a Tonelli-type theorem for the action functional $A_L(\gamma) = \int L(t, \dot{\gamma}(t)) dt$ in the subriemannian setting.

• Uniform Integrability: The authors prove that families of curves with bounded action have uniformly integrable velocities (Lemma 1). This is crucial for applying compactness arguments in $W^{1,1}$.
• Lower Semicontinuity: They establish that the action functional is lower semicontinuous with respect to uniform convergence (Theorem 4). The proof involves reducing the problem to local charts where the horizontal distribution is trivial and utilizing the strict convexity of $L$.
• Existence of Minimizers: Using the Arzelà-Ascoli theorem and the lower semicontinuity of the action, they prove the existence of minimizers for the action between any two points (Corollary 3).

3. The Lax Semigroup and Fixed Point Argument

The authors define a Lax-Oleinik semigroup $T_t$ acting on continuous functions $\phi \in C(M)$:
$$T_t[\phi](x) = \inf \left\{ \phi(\gamma(0)) + \int_0^t L(\dot{\gamma}(s), u(s, \gamma(s))) ds : \gamma \in W^{1,1}_D([0, t]), \gamma(t) = x \right\}$$
where $u$ is the solution being sought.

• Fixed Point Construction: They define an operator $A_\phi$ on the space $C([0, T] \times M)$ and prove it is a contraction in a generalized sense (Lemma 3). By iterating this operator, they establish the existence of a unique fixed point $u$, which corresponds to the viscosity solution of the Cauchy problem.
• Semigroup Properties: They prove the semigroup property $T_{s+t} = T_s \circ T_t$ (Corollary 4) and establish monotonicity and non-expansiveness properties (Propositions 3 and 4).

4. Convergence to Stationary Solutions

To prove the Weak KAM theorem, the authors analyze the asymptotic behavior of $T_t[\phi]$.

• Uniform Bounds and Equi-Lipschitz Continuity: They show that for Lipschitz initial data $\phi$, the family $T_t[\phi] + c_0 t$ is uniformly bounded and equi-Lipschitz (Lemma 4 and Lemma 5), where $c_0$ is a critical constant related to the Mañé critical value.
• Limit Existence: Under an additional assumption on the growth of the Hamiltonian's spatial derivative (Assumption 1), they prove that the sequence $T_t[\phi] + c_0 t$ converges uniformly to a function $\psi$ as $t \to \infty$. This limit $\psi$ is a viscosity solution to the stationary Hamilton-Jacobi equation $H(x, D_x \psi, \psi) = c_0$.

Key Results

1. Existence and Uniqueness (Theorem 1): For any continuous initial condition $\phi$, there exists a unique viscosity solution $u \in C([0, \infty) \times M)$ to the Cauchy problem:
   $$\begin{cases} u_t + H(x, D_x u, u) = 0 \\ u(0, x) = \phi(x) \end{cases}$$
   This solution is represented variationally by the Lax-Oleinik formula.

2. Weak KAM Theorem (Theorem 2):

   • There exists a critical constant $c_0 \in \mathbb{R}$ such that the family $T_t[\phi] + c_0 t$ is uniformly bounded and equi-Lipschitz for Lipschitz $\phi$.
   • There exists a stationary solution $\psi \in \text{Lip}(M, d)$ satisfying $H(x, D_x \psi, \psi) = c_0$.
   • Under Assumption 1, the limit $\lim_{t \to \infty} (T_t[\phi] + c_0 t)$ exists uniformly and yields a stationary solution.

3. Tonelli Minimizers: The paper establishes that for the subriemannian Lagrangian depending on $u$, action-minimizing curves exist between any two points, extending classical Tonelli results to this constrained, non-autonomous setting.

Significance and Claims

The paper claims to successfully extend the foundational results of Weak KAM theory to the context of subriemannian geometry where the Lagrangian depends on the unknown function.

• Geometric Extension: The authors demonstrate that the bracket-generating condition on the distribution $D$ is sufficient to ensure that the induced Carnot-Caratheodory distance generates the manifold topology, allowing for the definition of viscosity solutions and the application of compactness arguments (Arzelà-Ascoli) in the subriemannian context.
• Functional Dependence: By handling the dependence on $u$ (specifically the monotonicity condition $-\lambda \leq \partial_u L < 0$), the authors overcome the difficulty that the Lagrangian is not autonomous. This allows the treatment of Hamilton-Jacobi equations where the Hamiltonian depends on the solution itself, a scenario common in control theory and mean field games but previously less explored in subriemannian settings.
• Robustness of Variational Methods: The work reinforces that the variational approach (Lax-Oleinik semigroups) remains a powerful tool for proving existence, uniqueness, and convergence of solutions even when the underlying geometry is non-integrable (subriemannian) and the dynamics are non-autonomous.

The authors do not claim to solve open problems regarding the regularity of solutions beyond Lipschitz continuity, nor do they propose new experimental applications. The contribution is strictly theoretical, providing the necessary analytical framework to treat this specific class of Hamilton-Jacobi equations.