On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

Abstract

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic 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 [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].