Résumé en anglais

Under a nondegeneracy condition on the boundary, we prove a comparison principle for discontinuous viscosity sub- and supersolutions of the generalized Dirichlet boundary-value problem for a first-order Hamilton-Jacobi equation ⎧⎩⎨⎪⎪H(x,u,Du)=0inΩ,Max(H(x,u,Du);u−φ)⩾0on∂Ω,Min(H(x,u,Du);u−φ)⩽0on∂Ω. For optimal control problems, we interpret this nondegeneracy as a condition on the controlled vector fields. Finally, we use this to extend classical singular perturbation results to degenerated elliptic equations.