Weak Kam Theory for General Hamilton-jacobi Equations I: the Solution Semigroup under Proper Conditions
نویسندگان
چکیده
We consider the following evolutionary Hamilton-Jacobi equation with initial condition: { ∂tu(x, t) +H(x, u(x, t), ∂xu(x, t)) = 0, u(x, 0) = φ(x). Under some assumptions on H(x, u, p) with respect to p and u, we provide a variational principle on the evolutionary Hamilton-Jacobi equation. By introducing an implicitly defined solution semigroup, we extend Fathi’s weak KAM theory to certain more general cases, in which H explicitly depends on the unknown function u. As an application, we show the viscosity solution of the evolutionary Hamilton-Jacobi equation with initial condition tends asymptotically to the weak KAM solution of the following stationary Hamilton-Jacobi equation: H(x, u(x), ∂xu(x)) = 0. .
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