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

Abstract

We study the local Hölder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When q>Nγ−1γ (N is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent αq>γ−2γ−1. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.