Extrapolation for the L _p Dirichlet Problem in Lipschitz Domains

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the L ^p Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in ℝ ^d is solvable for some 1<p=p0<2(d−1)d−2, then it is solvable for all p satisfying p0<p<2(d−1)d−2+ε. The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality.