Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D2u) =  f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the Ln norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C1,α regularity estimates are also delivered when f∈Ln+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f\in L^{n+\varepsilon}}$$\end{document} . The limiting upper borderline case, f∈L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \in L^\infty}$$\end{document} , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that u∈C1,Log-Lip\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u \in C^{1,{\rm Log-Lip}}}$$\end{document} , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the C0,n-2εn-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}$$\end{document} norm of u based on the Ln-ε norm of f, where ɛ is the Escauriaza universal constant. The exponent n-2εn-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{n-2\varepsilon}{n-\varepsilon}}$$\end{document} is optimal. When the source function f lies in Lq, n > q > n−ε, we also obtain the exact, improved sharp Hölder exponent of continuity.