On Selection of Solutions to Vectorial Hamilton-jacobi System

In this paper, we discuss some principles on the selection of the solutions of a special vectorial Hamilton-Jacobi system defined by (1.1) below. We first classify the equivalent solutions based on a wellknown construction and then discuss some selection principles using intuitive descriptions of coarseness or maximality of solutions. We also discuss a selection principle based on comparison of the projections of a solution with certain smooth functions and introduce a notion of partial viscosity solutions for the vectorial Hamilton-Jacobi equation, which generalizes the well-known notion of viscosity solutions in the scalar case. Since we mainly work in the framework of Sobolev solutions, we show that, as in the scalar case, for certain domains the partial viscosity principle may rule out many interesting Lipschitz solutions.