Homoclinic orbits and critical points of barrier functions

Abstract

We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on ${\mathbb{T}}^n$. We also prove a critical point theorem for barrier functions, and the existence of such homoclinic orbits on ${\mathbb{T}}^2$ as an application.

References in corpus (2)

• Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds
• Propagation of singularities for weak KAM solutions and barrier functions

Cited by in corpus (7)

• Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus
• Lasry-Lions, Lax-Oleinik and Generalized characteristics
• A way to cross double resonance
• Uniform hyperbolicity of invariant cylinder
• Generalized characteristics and Lax-Oleinik operators: global theory
• Lasry-Lions approximations for discounted Hamilton-Jacobi equations
• Singularities of solutions of Hamilton-Jacobi equations