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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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