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| In this talk we investigate time dependent mean-field games (MFG) with superquadratic Hamiltonians and a power-like dependence on the measure. Existence and uniqueness of smooth solutions is established under a set of conditions depending on the dimension as well as on the growth of the Hamiltonian. In particular, our results recover the quadratic case. This is done by recurring to a delicate argument that combines the non-linear adjoint method with polynomial estimates for solutions of the Fokker-Planck in terms of $L^\infty L^\infty$-norms of $D_pH$. To the best of our knowledge, superquadratic MFG have not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of Hamilton-Jacobi equations with superquadratic Hamiltonians. This is a based on a joint work with D. Gomes and H. S\'anchez-Morgado. |
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