期刊
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
卷 42, 期 4, 页码 1949-1970出版社
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2021179
关键词
Hamilton-Jacobi equation; viscosity solutions; propagation of singularities
资金
- National Natural Science Foundation of China [11801223, 11871267]
This paper investigates the global propagation of singularities of the viscosity solution to the discounted Hamilton-Jacobi equation. By reducing the original equation to a time-dependent evolutionary Hamilton-Jacobi equation, it is shown that singularities propagate along specific characteristics and time can extend indefinitely. The use of a-compactness of the Euclidean space and the local Lipschitz issue are key technical difficulties in studying the global results.
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].
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