New regularity estimates for fully nonlinear elliptic equations

Abstract

We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability $\epsilon =\epsilon (\lambda ,\Lambda ,n)$ in the celebrated $W^{2,\epsilon }$-regularity estimate satisfies $$\frac{{\left(1+\frac{2}{3}\left(1-\frac{\lambda }{\Lambda }\right)\right)}^{n-1}}{\ln n^4}\cdot {\left(\frac{\lambda }{\Lambda }\right)}^{n-1}\le \epsilon \le \frac{n\lambda }{(n-1)\Lambda +\lambda },$$ where $n\ge 3$ is the dimension and $0<\lambda <\Lambda$ are the ellipticity constants. In particular, ${\left(\frac{\Lambda }{\lambda }\right)}^{n-1}\epsilon (\lambda ,\Lambda ,n)$ blows-up, as $n\to \infty$; previous results yielded fast decay of such a quantity. The upper estimate improves the one obtained by Armstrong, Silvestre, and Smart in arXiv:1103.3677

arXiv commentAccepted for publication in the Journal de Mathématiques Pures et Appliquées