Universal moduli of continuity for solutions to fully nonlinear elliptic equations

Details Universal moduli of continuity for solutions to fully nonlinear elliptic equations Universal moduli of continuity for solutions to fully nonlinear elliptic equations

2011-11-11

arxiv.org/abs/1111.2728v2

Mathematics

Analysis of PDEs

16 pages - improvement of C^{1,Log-Lip} regularity theory, comments on BMO estimates on Du and D^2u

Scientific paper

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,\alpha}$ regularity estimates are delivered when $f\in L^{n+\epsilon}$. The limiting upper borderline case, $f\in L^\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \in C^{1,\mathrm{Log-Lip}}$, 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 \textit{a priori} estimates on the $C^{0,\frac{n-2\epsilon}{n-\epsilon}}$ norm of $u$ based on the $L^{n-\epsilon}$ norm of $f$, where $\epsilon$ is the Escauriaza universal constant. The exponent $\frac{n-2\epsilon}{n-\epsilon}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-\epsilon$, we also obtain the exact, improved sharp H\"older exponent of continuity.

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