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A Monotone Operator Approach to Separable Mean-Field Games with Mixed Boundary Conditions

AbdulRahman M. Alharbi, Diogo Gomes

Abstract

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 $Γ_D$.

A Monotone Operator Approach to Separable Mean-Field Games with Mixed Boundary Conditions

Abstract

We study a class of local, first-order, stationary mean-field games (MFGs) on bounded domains with nonstandard mixed boundary conditions: prescribed inflow on and a relaxed Signorini-type exit condition on (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 encoding exit flux. To address a constant-shift degeneracy in the value function (the transport equation depends only on ), we employ a quotient-space formulation that restores coercivity. Using the Browder--Minty theorem, we prove existence for a penalized operator on a convex domain and pass to the limit as . We obtain weak solutions solving the associated variational inequality, with , , and in the dual trace space on .
Paper Structure (44 sections, 33 theorems, 194 equations)

This paper contains 44 sections, 33 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. The Model Problem
  3. Methodology and Operators
  4. Main Results
  5. Acknowledgments:
  6. Notation, Assumptions, and Preliminary Results
  7. Notation
  8. Assumptions
  9. The problem data
  10. The Hamiltonian
  11. Example.
  12. Monotone Operators and the Browder--Minty Theorem
  13. Auxiliary Bounds on the Hamiltonian
  14. Basic auxiliary bounds
  15. Convex combination bounds
  16. ...and 29 more sections

Key Result

Theorem 1.1

Suppose that Assumptions assumset:D and assumset:S hold, and let $\epsilon > 0$. Then, there exists a triplet $(m_{\epsilon} ,u _{\epsilon},h_{\epsilon}) \in \mathcal{X}^+$ such that for all $(\mu,v,k) \in \mathcal{X}^+$.

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6: KiSt00, Theorem 1.7
  • Definition 2.7
  • ...and 66 more