Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity

Abstract In the paper we prove the convergence of viscosity solutions u λ {u_{\lambda}} as λ → 0 + {\lambda\rightarrow 0_{+}} for the parametrized degenerate viscous Hamilton–Jacobi equation H ⁢ ( x , d x ⁢ u , λ ⁢ u ) = α ⁢ ( x ) ⁢ Δ ⁢ u , α ⁢ ( x ) ≥ 0 , x ∈ 𝕋 n H(x,d_{x}u,\lambda u)=\alpha(x)\Delta u,\quad\alpha(x)\geq 0,\quad x\in\mathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * ⁢ M × ℝ → ℝ {H:T^{*}M\times\mathbb{R}\rightarrow\mathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ⁢ ( x , d x ⁢ u , 0 ) = α ⁢ ( x ) ⁢ Δ ⁢ u . H(x,d_{x}u,0)=\alpha(x)\Delta u.