| Abstract: |
| The theory of monotone operators is the basis of a standard method of proving existence of solutions to Dirichlet problems involving a second-order elliptic PDE of the divergence form. In this context, it generalizes the direct method in the calculus of variations as well as the Lax-Milgram theorem.
On the other hand, presence of a monotonicity structure in mean field games has been known since the early papers of Lions et al. Moreover, this monotonicity property was exploited by Ferreira et al for proving existence of solutions in a weak sense, to various MFG problems.
More recently, in ongoing joint work with Ferreira and Gomes, we discovered how to systematize the aforementioned existence proofs using the abstract machinery of the monotone operator theory. Consequently, we achieve simpler and unified proofs for a wider range of problems. Furthermore, with new a priori bounds based on a novel idea, we obtain solutions in a stronger sense. |
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