Solving Mean-Field Games with Monotonicity Methods in Banach Spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908886571679744 |
|---|---|
| author | Ferreira, Rita Gomes, Diogo Ucer, Melih |
| author_facet | Ferreira, Rita Gomes, Diogo Ucer, Melih |
| contents | This paper develops a unified framework for proving the existence of solutions to stationary first-order mean-field games (MFGs) based on the theory of monotone operators in Banach spaces. We cast the coupled MFG system as a variational inequality, overcoming the limitations of prior Hilbert-space approaches that relied on high-order regularization and typically yielded only weak solutions in the monotone operator sense. In contrast, with our low-order regularization, we obtain strong solutions.
Our approach addresses the non-coercivity of the underlying MFG operator through two key regularization strategies. First, by adding a low-order $p$-Laplacian term, we restore coercivity, derive uniform a priori estimates, and pass to the limit via Minty's method. This establishes, for the first time via monotonicity methods, the existence of strong solutions for models with both standard power-growth and singular congestion, with the latter requiring a careful restriction of the operator's domain. Second, for Hamiltonians with only minimal growth hypotheses, we regularize the Hamiltonian itself via infimal convolution to prove the existence of weak solutions.
Our Banach-space framework unifies and extends earlier existence results. By avoiding high-order smoothing, it not only provides a more direct theoretical path but is also ideally suited for modern numerical algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solving Mean-Field Games with Monotonicity Methods in Banach Spaces Ferreira, Rita Gomes, Diogo Ucer, Melih Analysis of PDEs 35J47, 35A01 This paper develops a unified framework for proving the existence of solutions to stationary first-order mean-field games (MFGs) based on the theory of monotone operators in Banach spaces. We cast the coupled MFG system as a variational inequality, overcoming the limitations of prior Hilbert-space approaches that relied on high-order regularization and typically yielded only weak solutions in the monotone operator sense. In contrast, with our low-order regularization, we obtain strong solutions. Our approach addresses the non-coercivity of the underlying MFG operator through two key regularization strategies. First, by adding a low-order $p$-Laplacian term, we restore coercivity, derive uniform a priori estimates, and pass to the limit via Minty's method. This establishes, for the first time via monotonicity methods, the existence of strong solutions for models with both standard power-growth and singular congestion, with the latter requiring a careful restriction of the operator's domain. Second, for Hamiltonians with only minimal growth hypotheses, we regularize the Hamiltonian itself via infimal convolution to prove the existence of weak solutions. Our Banach-space framework unifies and extends earlier existence results. By avoiding high-order smoothing, it not only provides a more direct theoretical path but is also ideally suited for modern numerical algorithms. |
| title | Solving Mean-Field Games with Monotonicity Methods in Banach Spaces |
| topic | Analysis of PDEs 35J47, 35A01 |
| url | https://arxiv.org/abs/2506.21212 |