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July  2009, 8(4): 1291-1302. doi: 10.3934/cpaa.2009.8.1291

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

 1 Sez. di Matematica per I'Ingegneria, Dip. di Matematica Pura e Applicata, Università dell'Aquila, 67040 Roio Poggio (AQ) 2 Dipartimento Metodi e Modelli Matematici, per le Scienze Applicate, Università di Roma "La Sapienza", Via A. Scarpa 16, 00161 Roma 3 Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan

Received  May 2008 Revised  December 2008 Published  March 2009

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.
Citation: Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
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