Generalized convergence of solutions for nonlinear Hamilton–Jacobi equations with state-constraint.

For a continuous Hamiltonian H : (x , p , u) ∈ T ⁎ R n × R → R , we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint { H (x , D u , λ u) ≤ C λ , x ∈ Ω λ ⊂ R n , H (x , D u , λ u) ≥ C λ , x ∈ Ω ‾ λ ⊂ R n , as λ → 0 +. When H satisfies certain convex, coercive and monotone conditions, the domain Ω λ : = (1 + r (λ)) Ω keeps bounded, star-shaped for all λ > 0 with lim λ → 0 + ⁡ r (λ) = 0 , and lim λ → 0 + ⁡ C λ = c (H) equals the ergodic constant of H (⋅ , ⋅ , 0) , we prove the convergence of solutions u λ to a specific solution of the critical equation { H (x , D u , 0) ≤ c (H) , x ∈ Ω ⊂ R n , H (x , D u , 0) ≥ c (H) , x ∈ Ω ‾ ⊂ R n. We also discuss the generalization of such a convergence for equations with more general C λ and Ω λ. [ABSTRACT FROM AUTHOR]