A necessary and sufficient condition for convergence of the Lax-Oleinik semigroup for reversible Hamiltonians on R-n

作者：Liu, Qihuai^*; Wang, Kaizhi^*; Wang, Lin^*; Yan, Jun^*

来源：Journal of Differential Equations, 2016, 261(10): 5289-5305.

DOI：10.1016/j.jde.2016.08.001

• Lax-Oleinik semigroup
• Convergence
• Hamilton-Jacobi equations
• Non-compact manifold
• Reversible Hamiltonians

This paper is devoted to the study of the convergence of the Lax-Oleinik semigroup associated with reversible Hamiltonians H(x, p) on R-n. We provide a necessary and sufficient condition for the convergence of the semigroup. We also give an example to show that for irreversible Hamiltonians on R-n, even if the Hamiltonian is integrable and the initial data is Lipschitz continuous and bounded, the corresponding Lax-Oleinik semigroup may not converge.

• 出版日期2016-11-15
• 单位复旦大学; 上海交通大学; 桂林电子科技大学; 清华大学