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

Abstract

This paper studies a maximal L q -regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having sub-quadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a L p duality method, which exploits the regularity properties of solutions to stationary Fokker-Planck equations, for which we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic problems are the treatment of equations with a zero-th order term and the analysis of the end-point summability threshold q = d(γ − 1)/γ, d being the dimension of the ambient space and γ the growth of the first-order term.

A priori Lipschitz estimates for nonlinear equations with mixed local and nonlocal diffusion via the adjoint-Bernstein method

“…see e.g. [27]. Moreover, the technique of the present paper applies with few modifications to problems driven by the more general local-nonlocal operator once more that, though one expects the same results of the case of a local diffusion, being the dominating part of the diffusive term, obtaining a gradient estimate through the standard methods, such as the Bernstein one, is by no means immediate even for the stationary problem involving the mixed operator.…”

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

“…In such limiting case, we also enlight the role of zero-th order terms. This was first observed in [23] for parabolic problems and then in [38] for elliptic equations equipped with Dirichlet boundary conditions in the case p = 2, when the first-order term has subquadratic growth. The superquadratic case with linear diffusion has been recently addressed in [21], up to the endpoint threshold.…”

“…For such value of q the maximal regularity result for linear diffusions is new when γ ≥ 2. The case (ii) in the regime γ < 2 is covered by the results in [38].…”

“…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.…”

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

“…[8]), it should be possible to control u in W 2,q 0 whenever f lies in a set of L q 0 uniformly integrable functions. This has been shown to hold for subquadratic (γ < 2) problems [12].…”