2011/11/11 by Teixeira, Eduardo V. · 3 citations
#35B65 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1111.2728
This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D2u) = 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 Ln norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C1,α regularity estimates are delivered when f∈ Ln+ε. The limiting upper borderline case, f∈ L^∞, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on F, that u ∈ C1,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 a priori estimates on the C0,(n-2ε)/(n-ε) norm of u based on the Ln-ε norm of f, where ε is the Escauriaza universal constant. The exponent (n-2ε)/(n-ε) 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.