$W^{2,p}$- and $W^{1,p}$-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations

Abstract

In this paper we extend Caffarelli's result on interior $W^{2,p}$-estimates for viscosity solutions of uniformly elliptic equations and prove $W^{2,p}$-estimates at a flat boundary. Moreover we extend a result of A. Świech and prove $W^{1,p}$-estimates at the boundary. Thereafter we combine these results and prove global $W^{2,p}$-estimates for equations with dependence on $Du$ and $u$. Finally, we show that the previous estimates lead to an existence result for $W^{2,p}$-strong solutions.