GLOBAL PROPAGATION OF SINGULARITIES FOR DISCOUNTED HAMILTON-JACOBI EQUATIONS

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation lambda v(x) + H(x, Dv(x)) = 0, x is an element of R-n. (HJ(lambda)) with fixed constant lambda is an element of R+. We reduce the problem for equation (HJ(lambda)) into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of (HJ(lambda)) propagate along locally Lipschitz singular characteristics x(s) : [0, t] -> R-n and time t can extend to +infinity. Essentially, we use a-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of u and the complement of Aubry set using the basic idea from [9].