Calculus of Variations and Geometric Measure Theory
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P. J. Graber - A. R. Mészáros

Sobolev regularity for first order Mean Field Games

created by mészáros on 19 Aug 2017
modified on 24 Jan 2018


Accepted Paper

Inserted: 19 aug 2017
Last Updated: 24 jan 2018

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Year: 2017


In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in a recent paper of Prosinski and Santambrogio. Our methods apply to a large class of Hamiltonians and coupling functions.


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