Consider the Hamilton-Jacobi equation H(x, P+Du) =λ, there exists one and only one real numberλ∈R such that the equation has a global viscosity solution. It is called the effective Hamiltonian and denoted by H(P).
The physical meaning of the effective Hamiltonian is that it represents the eigenvalue of an eigenstate. It is important in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. It is also closely related to the Weak KAM theory and Aubry-Mather theory...
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