Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.15174 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913954401353728 |
|---|---|
| author | Alharbi, Abdulrahman Gomes, Diogo Di Fazio, Giuseppe Ucer, Melih |
| author_facet | Alharbi, Abdulrahman Gomes, Diogo Di Fazio, Giuseppe Ucer, Melih |
| contents | We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x, Du, m) = 0$ and a transport equation $-\operatorname{div}(m D_pH(x, Du, m)) = 0$ in a domain $Ω\subset \mathbb{R}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally Hölder continuous in $Ω$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity for Weak Solutions to First-Order Local Mean Field Games Alharbi, Abdulrahman Gomes, Diogo Di Fazio, Giuseppe Ucer, Melih Analysis of PDEs 35Q89 (Primary) 35B65 (Secondary) We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x, Du, m) = 0$ and a transport equation $-\operatorname{div}(m D_pH(x, Du, m)) = 0$ in a domain $Ω\subset \mathbb{R}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally Hölder continuous in $Ω$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems. |
| title | Regularity for Weak Solutions to First-Order Local Mean Field Games |
| topic | Analysis of PDEs 35Q89 (Primary) 35B65 (Secondary) |
| url | https://arxiv.org/abs/2411.15174 |