Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: First order Mean Field Games with density constraints: Pressure equals
  Price   In this paper we study Mean Field Game systems under density constraints as
optimality conditions of two optimization problems in duality. A weak solution
of the system contains an extra term, an additional price imposed on the
saturated zones. We show that this price corresponds to the pressure field from
the models of incompressible Euler's equations {\`a} la Brenier. By this
observation we manage to obtain a minimal regularity, which allows to write
optimality conditions at the level of single agent trajectories and to define a
weak notion of Nash equilibrium for our model.

11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ