Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations · arXivDesk2004.12269Apr 26, 2020viscosity solution, contact Hamiltonian, action minimizer, Aubry-Mather theory, weak KAM solution, Peierls barrier
Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations
Yanan Wang, Jun Yan, Jianlu Zhang
Abstract
For any compact connected manifold M, we consider the generalized contact Hamiltonian H(x,p,u) defined on T∗M×R which is conex in p
and monotonically increasing in
. Let
uε−:M→R be the viscosity solution of the parametrized contact Hamilton-Jacobi equation
H(x,∂xuε−(x),εuε−(x))=c(H) with
being the Mañé Critical Value. We prove that
converges uniformly, as
ε→0+ , to a specfic viscosity solution
of the critical equation
H(x,∂xu0−(x),0)=c(H) which can be characterized as a minimal combination of associated Peierls barrier functions.