Averaging of Hamilton-Jacobi equations over Hamiltonian flows

Hitoshi Ishii; Taiga Kumagai

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arXiv · 1912.09153

Averaging of Hamilton-Jacobi equations over Hamiltonian flows

Hitoshi Ishii·Taiga Kumagai

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Abstract

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense, and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian $G$.

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Hitoshi Ishii, Taiga Kumagai. 2019-12-19. Averaging of Hamilton-Jacobi equations over Hamiltonian flows. https://arxiv.org/abs/1912.09153

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