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Periodic homogenization of Hamilton-Jacobi equations: Additive eigenvalues and variational formula. (English) Zbl 0871.49025
Summary: The classical first-order Hamilton-Jacobi equation \(u^\varepsilon_t+ H(D u^\varepsilon,x/ \varepsilon)=0\) has a unique viscosity solution satisfying an initial condition \(u^\varepsilon=u_0\) when \(t=0\). When \(H\) is periodic in the second variable, it is known that \(u^\varepsilon\) converges as \(\varepsilon\to 0\) to the unique viscosity solution of the “averaged equation” \(u_t+ \bar H(Du)=0\), with the same initial condition. In this paper, we assume that \(H\) is convex and we use optimal control theory to establish a relationship between the effective Hamiltonian \(\bar H\) and a nonlinear “additive” eigenvalue equation. For any convex Hamiltonian \(H\) we prove existence and uniqueness of this additive eigenvalue, and we derive a variational formula for \(\bar H\).

MSC:
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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