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

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.