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 $\gamma >2$, and right-hand side in a Lebesgue space $L^q$. When $q>N\frac{\gamma -1}{\gamma }$ ($N$ is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent ${\alpha }_q>\frac{\gamma -2}{\gamma -1}$. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of $u$ in $L^q$. Our methods are based on blow-up techniques and a Liouville theorem.