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 $\varepsilon = \varepsilon(\lambda, \Lambda, n)$ in the celebrated $W^{2, \varepsilon}$-regularity estimate satisfies $$\frac{ \left (1+ \frac{2}{3}\left(1- \frac{\lambda}{\Lambda} \right )\right )^{n-1}}{\ln n^4} \cdot \left( \frac{\lambda}{\Lambda} \right) ^{n-1} \le \varepsilon \le \frac{n\lambda}{(n-1)\Lambda +\lambda}, $$ where $n\ge 3$ is the dimension and $0< \lambda < \Lambda$ are the ellipticity constants. In particular, $\left( \frac{\Lambda}{\lambda} \right) ^{n-1} \varepsilon(\lambda, \Lambda, n)$ 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