Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak-strong uniqueness for solutions to mean-field games

Rita Ferreira, Diogo Gomes, Vardan Voskanyan

TL;DR

The paper studies weak-strong uniqueness for stationary first-order mean-field games on $\mathbb{T}^d$ within a monotone-operator framework. It introduces a linearization strategy to compare a weak solution $(\tilde{m},\tilde{u})$ with a strong solution $(m,u)$ and derives explicit structural conditions on the Hamiltonian that guarantee their equality. The main result is proved via two complementary routes: (a) a perturbation of the variational formulation under monotonicity assumptions, and (b) a linearized, adjoint-based approach yielding a linear elliptic problem for the difference in potentials. These contributions extend weak solution analysis by establishing weak-strong uniqueness under concrete regularity and monotonicity hypotheses, and they apply to common MFG Hamiltonians, aiding well-posedness and potential numerical analyses.

Abstract

This paper addresses the crucial question of solution uniqueness in stationary first-order Mean-Field Games (MFGs). Despite well-established existence results, establishing uniqueness, particularly for weaker solutions in the sense of monotone operators, remains an open challenge. Building upon the framework of monotonicity methods, we introduce a linearization method that enables us to prove a weak-strong uniqueness result for stationary MFG systems on the d-dimensional torus. In particular, we give explicit conditions under which this uniqueness holds.

Weak-strong uniqueness for solutions to mean-field games

TL;DR

The paper studies weak-strong uniqueness for stationary first-order mean-field games on within a monotone-operator framework. It introduces a linearization strategy to compare a weak solution with a strong solution and derives explicit structural conditions on the Hamiltonian that guarantee their equality. The main result is proved via two complementary routes: (a) a perturbation of the variational formulation under monotonicity assumptions, and (b) a linearized, adjoint-based approach yielding a linear elliptic problem for the difference in potentials. These contributions extend weak solution analysis by establishing weak-strong uniqueness under concrete regularity and monotonicity hypotheses, and they apply to common MFG Hamiltonians, aiding well-posedness and potential numerical analyses.

Abstract

This paper addresses the crucial question of solution uniqueness in stationary first-order Mean-Field Games (MFGs). Despite well-established existence results, establishing uniqueness, particularly for weaker solutions in the sense of monotone operators, remains an open challenge. Building upon the framework of monotonicity methods, we introduce a linearization method that enables us to prove a weak-strong uniqueness result for stationary MFG systems on the d-dimensional torus. In particular, we give explicit conditions under which this uniqueness holds.

Paper Structure

This paper contains 6 sections, 9 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Assumptions and preliminary results
  3. Proof of the main result
  4. Conditions $(a)$
  5. Conditions $(b)$
  6. Appendix: Solvability

Key Result

Theorem 1.3

Let $( \tilde{m}, \tilde{u}) \in L^{r}({\mathbb{T}^d})\times W^{1, \gamma}({\mathbb{T}^d})$ be a weak solution and $( {m}, {u}) \in L^{r_1}({\mathbb{T}^d})\times W^{1, \gamma_1}({\mathbb{T}^d})$ be a strong solution to Problem P1. Suppose that Assumptions a-4, asmp:monotonicity, a-2, a-3, a-1 hold Then, we have that

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 2.3
  • Remark 2.5
  • Theorem 2.6: Vitali--Lebesgue
  • Remark 2.7
  • Remark 2.8
  • Remark 2.11
  • Remark 2.15
  • ...and 20 more