In the present paper we consider Hamilton-Jacobi equations of the form H ( x , u , ∇ u ) = 0 , x ∈ Ω H(x,u,\nabla u) = 0,\;x \in \Omega , where Ω \Omega is a bounded open subset of R n , H {R^n},H is a given continuous real-valued function of ( x , s , p ) ∈ Ω × R × R n (x,s,p) \in \Omega \times R \times {R^n} and ∇ u \nabla u is the gradient of the unknown function u u . We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of Ω \Omega . This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.