On elliptic extensions in the disk

Given two arbitrary functions f, g on the boundary of the unit disk D in R^2, it is shown that there exist a second order uniformly elliptic operator L and a function v in L^p, with L^p second derivatives, 1<p<2, satisfying Lv=0 a.e. in D and with v = f and the normal derivative dv/dn =g on the boundary. A similar extension property was proved by R. Cavazzoni (2003) for any pair of functions f, g that are analytic; our result is obtained under weaker regularity assumptions, e.g. with f'(theta) and g Holder continuous with exponent > 1/2.