Abstract

Let Ω Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = nΣi,j=1 aij∂iju + nΣj=1 bj∂ju + cu where aji=aij ϵC, and bj, c ϵ L ∞, C ≤ ). We show that the realization A0 of A in C0: = {u ϵ C: u|∂Ω=0} given by D:= {u ϵ C0 ∩ W2,n loc } a0u:= Au generates a bounded holomorphic C0-semigroup on C0. The result is in particular true if Ω is a Lipschitz domain. so far the best known result seems to be the case where Ω has C2-boundary [12, Secton 3.1.5]. We also study the elliptic problem -Au = f u|∂Ω = g