Local Mean-Field Games: Existence

In this last chapter, we address the existence problem for local mean-field games. First, we illustrate the bootstrapping technique and put together the previous estimates. Thanks to this technique, we show that solutions of stationary MFGs are bounded a priori in all Sobolev spaces. This is an essential step for the two existence methods developed next. The first method is a regularization procedure in which we perturb the original local MFG into a non-local problem. By the results of the preceding chapter, this non-local problem admits a solution. Then, a limiting procedure gives the existence of a solution. The second method we consider is the continuation method. The key idea is to deform the original MFG into a problem that can be solved explicitly. Then, a topological argument shows that it is possible to deform the solution of the latter MFG into the former. This argument uses both the earlier bounds and an infinite dimensional version of the implicit function theorem.