New regularity estimates for fully nonlinear elliptic equations

TL;DR

This paper derives new quantitative estimates for the integrability of the Hessian of solutions to fully nonlinear elliptic equations, revealing how the optimal integrability exponent depends on ellipticity constants and dimension.

Contribution

It provides the first explicit bounds on the optimal Hessian integrability exponent for fully nonlinear elliptic equations, improving previous decay estimates and revealing dimension-dependent blow-up behavior.

Findings

Optimal Hessian integrability exponent bounds derived

Shows blow-up of integrability exponent as dimension increases

Improves previous estimates by Armstrong, Silvestre, and Smart

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