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. We conclude with an application of our estimates to prove existence and uniqueness of solutions for a particular first-order mean-field planning problem with a potential.

在本文中，我们研究了具有潜力的一阶均值规划问题的先验估计。在均值场博弈（MFG）理论中，先验估计对证明经典解的存在起着至关重要的作用。尤其重要的是，对于球员分布密度及其倒数的统一界线至关重要。在这里，我们调查那些数量不存在的计划问题的先验界限。势能的存在会带来非平凡的困难，我们通过探索具有势能的均值场规划问题的位移-凸性质以及Moser的迭代方法来克服这些困难。我们表明，如果势能满足某个小的条件，那么位移-凸性将成立。此属性启用\（L ^ q \）密度的界线。在一维情况下，位移-凸性还给出了密度倒数的\（L ^ q \）界。最后，使用这些（L ^ q \）估计和Moser的迭代方法，我们获得了（L ^ \ infty \）估计值，用于估计玩家分布的密度及其倒数。我们以估计值的应用来结束，以证明具有潜力的特定一阶平均场规划问题的解的存在性和唯一性。