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Averaging of Hamilton-Jacobi equations along divergence-free vector fields

  • * Corresponding author: Hitoshi Ishii

    * Corresponding author: Hitoshi Ishii 
The first author is partially supported by the JSPS grants: KAKENHI #16H03948, #18H00833, #20K03688, #20H01817. The second author is partially supported by the JSPS grants: KAKENHI #20K03688
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  • 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 divergence-free vector fields. 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 $.

    Mathematics Subject Classification: Primary: 35B25, 35F21; Secondary: 35F30, 35D40, 49L25.


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  • Figure 1.  $ N = 6 $

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