LIPSCHTTZ CLASSES OF SOLUTIONS TO CEFUIAIN ELLIPTIC EQUATIONS

Abstract. We consider weak solutions u in the Sobolev space!7'j,,o"(Q), O C R ' , to elliptic equations of the form divA(c,Vu) B(r,Vu). The Lipschitz continuity of u over O is characterized by the growth ofthe local-La-average of Vu. Previous results cover the case in which the structure exponent d> n. We give here a proof valid for all l. < a < oo. L. Introduction and main results Theorem 1.1 follows from results in [HL]. Throughout O will be a connected open subset of R". When /: O--+ R-, 0 < k < 1,, we write Theorem L.L. Let u beharmonicinthe unit disk D c R2 and0 < fr < 1