Stationary fully nonlinear mean-field games

TL;DR

This paper studies fully nonlinear mean-field games derived from a variational problem involving a Hessian-dependent functional, establishing regularity, existence, and explicit solutions, with potential for broader generalizations.

Contribution

It introduces a novel variational framework for fully nonlinear mean-field games and proves regularity, existence, and explicit solutions within this setting.

Findings

Established sharp regularity of solutions in Sobolev spaces

Proved existence of minimizers and solutions to the MFG system

Provided explicit solutions in a unidimensional example

Abstract

In this paper we examine fully nonlinear mean-field games associated with a minimization problem. The variational setting is driven by a functional depending on its argument through its Hessian matrix. We work under fairly natural conditions and establish improved (sharp) regularity for the solutions in Sobolev spaces. Then, we prove the existence of minimizers for the variational problem and the existence of solutions to the mean-field games system. We also investigate a unidimensional example and unveil new information on the explicit solutions. Our findings can be generalized to a larger class of operators, yielding information on a broader range of examples.