Calculus of Variations and Geometric Measure Theory

A. Davini - A. Fathi - R. Iturriaga - M. Zavidovique

Convergence of the solutions of the discounted equation

created by davini on 12 Jul 2023


Published Paper

Inserted: 12 jul 2023
Last Updated: 12 jul 2023

Journal: Invent. Math.
Volume: 206
Number: 1
Pages: 29–55
Year: 2016

ArXiv: 1408.6712 PDF


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 \[ \lambda u_{\lambda}(x)+H(x,d_x u_{\lambda})=c(H)\qquad\hbox{in $M$}, \] 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 \[ H(x,d_xu)=c(H)\qquad\hbox{in $M$}. \] We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.