论文标题
潜在的计划问题的统一估计
Uniform estimates for the planning problem with potential
论文作者
论文摘要
在本文中，我们研究了具有潜力的一阶平均计划问题的先验估计。在平均游戏理论（MFG）的理论中，先验估计在证明经典解决方案的存在方面起着至关重要的作用。特别是，统一的范围对于玩家的分布及其逆密度至关重要。在这里，我们研究了这些数量的先验界限，以解决具有无危险潜力的计划问题。潜力的存在会带来非平凡的困难，我们通过探索平均场计划问题的位移 - 凸度特性以及Moser的迭代方法来克服的困难。我们表明，如果潜力满足一定的较小条件，则位移凸度属性将保持。该属性启用$ l^q $的密度边界。在一维情况下，位移凸度属性还为密度倒数提供了$ l^q $界限。最后，使用这些$ l^q $估计和Moser的迭代方法，我们获得了$ l^\ infty $估计，用于播放器的分布及其逆密度。
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.
