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The selection problem for discounted Hamilton-Jacobi equations: some non-convex cases. (English) Zbl 1403.35082
Summary: Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

##### MSC:
 35F21 Hamilton-Jacobi equations 35B40 Asymptotic behavior of solutions to PDEs 37J50 Action-minimizing orbits and measures (MSC2010) 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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