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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.01681 |
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| _version_ | 1866915827761020928 |
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| author | Alharbi, AbdulRahman M. Gomes, Diogo |
| author_facet | Alharbi, AbdulRahman M. Gomes, Diogo |
| contents | We study a class of local, first-order, stationary mean-field games (MFGs) on bounded domains with nonstandard mixed boundary conditions: prescribed inflow on $Γ_N$ and a relaxed Signorini-type exit condition on $Γ_D$ (complementarity between exit flux and boundary value).
For separable Hamiltonians, we overcome the lack of coercivity and the boundary complementarity constraints by introducing a monotone operator on a convex domain, augmented with an auxiliary nonnegative boundary variable $h$ encoding exit flux.
To address a constant-shift degeneracy in the value function $u$ (the transport equation depends only on $Du$), we employ a quotient-space formulation that restores coercivity. Using the Browder--Minty theorem,
we prove existence for a penalized operator $A_ε$ on a convex domain and pass to the limit as $ ε\to 0^+$. We obtain weak solutions $(m,u,h)$ solving the associated variational inequality, with $m \in L^{β+1}(Ω)$, $u \in W^{1,γ}(Ω)$, and $h$ in the dual trace space on $Γ_D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Monotone Operator Approach to Separable Mean-Field Games with Mixed Boundary Conditions Alharbi, AbdulRahman M. Gomes, Diogo Analysis of PDEs We study a class of local, first-order, stationary mean-field games (MFGs) on bounded domains with nonstandard mixed boundary conditions: prescribed inflow on $Γ_N$ and a relaxed Signorini-type exit condition on $Γ_D$ (complementarity between exit flux and boundary value). For separable Hamiltonians, we overcome the lack of coercivity and the boundary complementarity constraints by introducing a monotone operator on a convex domain, augmented with an auxiliary nonnegative boundary variable $h$ encoding exit flux. To address a constant-shift degeneracy in the value function $u$ (the transport equation depends only on $Du$), we employ a quotient-space formulation that restores coercivity. Using the Browder--Minty theorem, we prove existence for a penalized operator $A_ε$ on a convex domain and pass to the limit as $ ε\to 0^+$. We obtain weak solutions $(m,u,h)$ solving the associated variational inequality, with $m \in L^{β+1}(Ω)$, $u \in W^{1,γ}(Ω)$, and $h$ in the dual trace space on $Γ_D$. |
| title | A Monotone Operator Approach to Separable Mean-Field Games with Mixed Boundary Conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.01681 |