| Abstract: |
| In this presentation, I will discuss recent joint work with Diogo Gomes and Yuri Ashrafyan, where we investigate a first-order mean-field game (MFG) model with a novel mixed boundary condition. The new boundary conditions split the boundary into two parts: an entrance with an inflow Neumann boundary condition and an exit with a relaxed Dirichlet condition and a relaxed outflow Neumann condition. We further impose an auxiliary contact-set condition on the exit portion of the boundary that links the other two conditions. This approach offers three advantages. It indicates the exit/entry regions of the boundary based on the general structure without the need to know the exact values of boundary data, addresses the lack of uniqueness issues associated with Neumann boundary conditions, and prevents the artificial inflow of virtual agents through the exit boundary caused by an excessively high Dirichlet condition. The interior behavior of our model adheres to the standard MFG structure, consisting of a coupled system of a first-order separable Hamilton-Jacobi equation and a stationary transport equation. We exploit the separability of the Hamiltonian to establish a corresponding variational formulation for the MFG, which we use to prove the existence of solutions and the uniqueness of the gradient of the value function. |
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