In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of timedependent mean-field games.D.
We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as the key techniques used in the subquadratic case do not extend to the superquadratic setting. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton-Jacobi equation are obtained in the present paper through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton-Jacobi equation with polynomial estimates for solutions of the Fokker-Planck equation. Existence of classical solutions can is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to the current understanding of superquadratic Hamilton-Jacobi equations.
We establish sharp W 2,p regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator F. By means of geometric tangential methods, we show that if the recession of the operator F -formally given bywith appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they are new even under convexity of the frozen coefficient operator, M → F(x 0 , M), as oscillation is measured only at the recession level. The methods further yield BMO regularity of the hessian, provided the source lies in that space. As a final application, we establish the density of W 2,p solutions within the class of all continuous viscosity solutions, for generic fully nonlinear operators F. This result gives an alternative tool for treating common issues often faced in the theory of viscosity solutions.
In this paper, we prove the existence of classical solutions for time dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument, which combines Lipschitz regularity for the HamiltonJacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
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