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arXiv · 2203.06092

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

Marco Cirant·Gianmaria Verzini
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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 $L^q$. 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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Marco Cirant, Gianmaria Verzini. 2022-03-11. Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms. https://arxiv.org/abs/2203.06092

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