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# Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem

• Aim of this paper is to show that some of the results in the weak KAM theory for $1^{s t}$ order convex Hamilton-Jacobi equations (see [11], [13]) can be extended to systems of convex Hamilton-Jacobi equations with implicit obstacles and to the obstacle problem. We obtain two results: a comparison theorem for systems lacking strict monotonicity; a representation formula for the obstacle problem involving the distance function associated to the Hamiltonian of the equation.
Mathematics Subject Classification: Primary: 49L25; Secondary: 35F30, 35B25.

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