$W^1,p$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations

We prove W 1,p -interior estimates for viscosity solutions of fully nonlinear, uniformly elliptic equations.The estimates complement C 1,↵ -regularity results of L. Ca↵arelli.We apply the estimates to obtain existence and uniqueness of viscosity and strong solutions of Dirichlet boundary value problems.0. Introduction.The paper deals with fully nonlinear partial di↵erential equations of the formwhere ⌦ is a bounded domain in R n , f 2 L p (⌦), andAbove S(n) denotes the set of n ⇥ n real symmetric matrices and is equipped with its usual order; that is, 0  P means 0  hP ⇣, ⇣i for ⇣ 2 R n , where h•, •i is the Euclidean inner product, and X  Y means 0  Y X.The equation (0.1) will be assumed to be uniformly elliptic (with constants , ⇤,  ⇤), by which we mean that if 0  P 2 S(n), then