Convergence of the solutions of the discounted equation

Abstract

We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_{\lambda }:M\to \mathbb{R}$ is the viscosity solution of the discounted equation $$\lambda u_{\lambda }(x)+H(x,d_x{u}_{\lambda })=c(H),$$ where $c(H)$ is the critical value, we prove that $u_{\lambda }$ converges uniformly, as $\lambda \to 0$, to a specific solution $u_0:M\to \mathbb{R}$ of the critical equation $$H(x,d_{x}u)=c(H).$$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.

35 pages

References in corpus (3)

• Convergence of the solutions of the discounted equation
• Weak KAM theorem on non compact manifolds
• Convergence of the solutions of the discounted equation: the discrete case

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