Elliptic extensions in the disk with operators in divergence form

Let phi_0 and phi_1 be regular functions on the boundary bD of the unit disk D in R^2, such that the integrals over bD of phi_1(t) e of sin(t)* (phi_1(t) - phi_0(t)) are both vanishing. It is proved that 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|_{bD}= phi_0 and the conormal derivative \partial u/\partial N|_{bD}=phi_1.