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

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 $L^q$. When $q > N\frac{γ-1}γ$ ($N$ is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent $α_q > \frac{γ-2}{γ-1}$. 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.

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