On the problem of maximal LqL^q-regularity for viscous Hamilton-Jacobi equations

For $q>2, \gamma > 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-\Delta u + |Du|^\gamma = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q > d \frac{\gamma-1}\gamma$.Comment: 11 page

On the existence and uniqueness of solutions to time-dependent fractional MFG

We establish existence and uniqueness of solutions to evolutive fractional Mean Field Game systems with regularizing coupling, for any order of the fractional Laplacian $s\in(0,1)$. The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime $s>1/2$ the solution of the system is classical, while if $s\leq 1/2$ we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons.Comment: 42 page

Lipschitz regularity for viscous Hamilton-Jacobi equations with LpL^p terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.Comment: 31 page

High-order estimates for fully nonlinear equations under weak concavity assumptions

This paper studies a priori and regularity estimates of Evans-Krylov type in H\"older spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems

On the strong maximum principle for fully nonlinear parabolic equations of second order

We provide a proof of strong maximum and minimum principles for fully nonlinear uniformly parabolic equations of second order. The approach is of parabolic nature, slightly differs from the earlier one proposed by L. Nirenberg and does not exploit the parabolic Harnack inequality

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

We prove new (sharp) Liouville-type properties via degenerate Hadamard three-sphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci's extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework.Comment: 17 page

Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds

In this paper, we study Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue scale, under various integrability regimes. Our method is based on an integral refinement of the Bernstein method, and leads to ``semilinear Calder\'on-Zygmund'' type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed.Comment: 23 page