Some estimates for the planning problem with potential

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