Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform estimates for the planning problem with potential   In this paper, we study a priori estimates for a first-order mean-field
planning problem with a potential. In the theory of mean-field games (MFGs), a
priori estimates play a crucial role to prove the existence of classical
solutions. In particular, uniform bounds for the density of players'
distribution and its inverse are of utmost importance. Here, we investigate a
priori bounds for those quantities for a planning problem with a non-vanishing
potential. The presence of a potential raises non-trivial difficulties, which
we overcome by exploring a displacement-convexity property for the mean-field
planning problem with a potential together with Moser's iteration method. We
show that if the potential satisfies a certain smallness condition, then a
displacement-convexity property holds. This property enables $L^q$ bounds for
the density. In the one-dimensional case, the displacement-convexity property
also gives $L^q$ bounds for the inverse of the density. Finally, using these
$L^q$ estimates and Moser's iteration method, we obtain $L^\infty$ estimates
for the density of the distribution of the players and its inverse.

3
The categories of this paper are: physics
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph