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

We study the local Holder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth gamma > 2, and right-hand side in a Lebesgue space Lq. When q > N gamma-1 gamma (N is the dimension of the Euclidean space), we obtain the optimal Holder continuity exponent alpha q > gamma-2 gamma-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.