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

Abstract

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 Calderón-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.

Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms

“…Remark 4.1. The regularity of the domain can be considerably weakened, as soon as the Sobolev inequality and the divergence theorem hold, and the results continue to be valid for problems posed on more general manifolds, as in [40], with bounds depending on the underlying geometry.…”

“…Remark 3.2. The linear case ã(t) ≡ ã > 0 has been already covered in [24] in the periodic setting and later in [40] for Neumann problems when λ = 0. Here, we are also able to deal with the borderline integrability exponent q = N (γ−1) γ .…”

“…In the case p = 2, estimate (1.1) was proved in [53], whilst an estimate of the form (1.2) was conjectured by P.-L. Lions [55,54] and has been the focus of a recent intensive research, especially in connection with the analysis of Mean Field Games systems: the recent work [24] addressed the conjecture of maximal regularity for the viscous problem in the periodic case, the later developments in [40,38] treated global regularity for boundary-value problems with Neumann and Dirichlet boundary conditions respectively, while interior estimates in the superquadratic regime has been the subject of the paper [25] through a different approach based on a blow-up argument.…”

“…The approach used in our main results (Theorem 2.1 and Theorem 3.1) is based on the socalled integral Bernstein method, see [53] and the later papers [47,5] together with the recent developments [24,40,22].…”

On the optimal $L^q$-regularity for viscous Hamilton-Jacobi equations with sub-quadratic growth in the gradient

“…We conclude by saying that our underlying motive for this analysis relies on the application of the maximal regularity estimates to the existence problem of classical solutions to the systems of PDEs arising in the theory of Mean Field Games introduced by J.-M. Lasry-P.-L. Lions [36], when the coupling term of the Hamilton-Jacobi equation has power-like growth, cf [20] and [22,Theorems 1.4 and 1.5]. Some recent results in this direction through maximal regularity in the case of defocusing systems [20] posed on convex domains of the Euclidean space and Neumann boundary conditions have been recently discussed in [30].…”