Lipschitz Continuity Results for Minimax Solutions of Path-Dependent Hamilton–Jacobi Equations

The paper proves three Lipschitz properties for the minimax solution ϕ—local/global uniform-norm Lipschitz in the history, a special-norm Lipschitz result (with discrete lags and an L2-history term), and time-Lipschitz on compact Lipschitz-bounded classes—using a careful doubling-of-variables scheme built on ci-smooth Lyapunov–Krasovskii penalties with a time-dependent weight a(t) and the minimax representation; see Theorem 3.2, Corollary 3.5, and Theorem 4.2 with Assumptions 2.1, 3.1, and 4.1, respectively . By contrast, the candidate solution asserts a key derivative estimate for a penalty functional of the form ∂tΨε + ⟨s,∇x1Ψε − ∇x2Ψε⟩ ≤ C(1+‖s‖)Ψε and then differentiates along arcs using s, not ẏ. This mixes the ci-chain rule (which involves the actual velocities ẏ1,ẏ2) with an arbitrary parameter s and omits the time-weight a(t) that the paper crucially uses (cf. the use of γ and ν=a(t)√(ε+γ), together with inequalities like (3.3) and the derivative bounds that lead to µ̇≤0) . The model’s compactness argument for the reachable arcs also bypasses the paper’s precise construction of K(D) and its compactness via established results . The high-level outline mirrors the paper’s themes, but the proof hinges on an unjustified penalty derivative inequality and a ẏ-versus-s mismatch, so it does not constitute a correct proof.