The duality method for mean field games systems

Lucio Boccardo; Luigi Orsina

doi:10.3934/cpaa.2022021

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2022, Volume 21, Issue 4: 1343-1360. Doi: 10.3934/cpaa.2022021

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The duality method for mean field games systems

Lucio Boccardo^1, and
Luigi Orsina^2, ,

1.
Istituto Lombardo - Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy
2.
Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy

^*Corresponding author
^*Corresponding author

Received: July 31, 2021

Revised: November 30, 2021

Early access: January 2022

Published: April 2022

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Abstract

In this paper we prove, using a duality method, existence of solutions for the nonlinear elliptic mean field games type system

$ \left\{ \begin{array}{cl} -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla} u) + u - {\mathop{{{\rm{div}}}}}(u\,A(x)\,{\nabla}\psi) = f(x) & {\rm{in \; \Omega ,}}\\ -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla}\psi) + \psi + A(x)\,{\nabla}\psi \cdot {\nabla} \psi = u^{p-1} & {\rm{in \; \Omega ,}} \\ u = 0 = \psi & {\rm{on \; \partial\Omega ,}} \end{array} \right. $

under different assumptions on $ p > 1 $, and the function $ f(x) $ in Lebesgue spaces.

Keywords:

Mathematics Subject Classification: 35J47, 35J60, 35J62.

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