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.
Diogo A. GOMES. Hiroyoshi MITAKE. Hung V. TRAN. "The selection problem for discounted Hamilton–Jacobi equations: some non-convex cases." J. Math. Soc. Japan 70 (1) 345 - 364, January, 2018. https://doi.org/10.2969/jmsj/07017534