#### Vol. 15, No. 5, 2022

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Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations

### Maxime Zavidovique

Vol. 15 (2022), No. 5, 1287–1311
##### Abstract

We study solutions of Hamilton–Jacobi equations of the form

 $\lambda \alpha \left(x\right){u}_{\lambda }\left(x\right)+H\left(x,{D}_{x}{u}_{\lambda }\right)=c,$

where $\alpha$ is a nonnegative function, $\lambda$ a positive constant, $c$ a constant and $H$ a convex coercive Hamiltonian. Under suitable conditions on $\alpha$ we prove that the functions ${u}_{\lambda }$ converge as $\lambda \to 0$ to a function ${u}_{0}$ that is a solution of the critical equation $H\left(x,{D}_{x}{u}_{0}\right)=c$.

##### Keywords
discounted Hamilton–Jacobi equations, viscosity solutions, weak KAM theory, Mather measures
##### Mathematical Subject Classification
Primary: 35D40, 35F21, 49L25