Elliptic extensions in the disk with operators in divergence form

This paper proves that, given the unit disk D on the the plane, there exist a linear second order uniformly elliptic operator L in divergence form with bounded measurable coefficients and a function u in W^1,p(D), 1<p<2, such that Lu=0 in D and with u and its conormal derivative assigned on the boundary. The data on the boundary are regular functions, satisfying suitable mean properties.