Lyapunov stability and uniqueness problems for Hamilton-Jacobi equations without monotonicity

The paper establishes Theorem 1.1, namely the decomposition of the Mañé set Ñ into the union of the Φ^H_t-invariant subsets Ñ_{u−} associated with all backward weak KAM (viscosity) solutions u−, under assumptions (H1)–(H3). It proves (i) Ñ_{u−} ⊂ Ñ via invariance and calibration of orbits starting on the 1-graph of u−, and (ii) Ñ ⊂ ⋃_{u−∈S−} Ñ_{u−} by constructing a backward solution from a semi-static orbit and showing the orbit lies in Λ_{u−} (Steps 1–5 of the proof of Theorem 1.1). These steps are explicitly given in Lemma 3.5 and equation (3.14) for the first inclusion and in the five-step proof of Theorem 1.1 for the second inclusion, culminating with p0 = Du±(x0) and u0 = u±(x0) (Proposition 3.9 and subsequent steps) . The candidate solution reproduces the same two-inclusion strategy with standard weak KAM/variational arguments and minor stylistic differences (e.g., using U(y)=inf_{s>0} h_{x0,u0}(y,s) rather than the paper’s u(x,t)=inf_{s∈ℝ} h_{x(s),u(s)}(x,t)). Both are correct and essentially the same proof style. Definitions of semi-static curves, the Mañé set, weak KAM solutions, and the semigroup/implicit action machinery invoked match the paper’s framework (Definitions 3.1–3.4 and the semigroup DPP) . Proposition 3.9 (Ñ_{u−}=⋂_{t≥0}Φ^H_{−t}(Λ_{u−})) also aligns with the candidate’s use of the closure/invariance of Λ_{u−} (Corollary 3.4) .