On Elliptic Extensions in the Disk

Given two arbitrary functions f ⁽⁰⁾, f ⁽¹⁾ on the boundary of the unit disk D in ({mathbb R}^2), it is shown that there exists a second order uniformly elliptic operator L and a function v in L ^p , with L ^p second derivatives (1?p?Lv?=?0 a.e. in D and with v?=?f ⁽⁰⁾ and (frac{ partial v}{partial n} = f^{(1)}) on (partial{D}). A similar extension property was proved in Cavazzoni (2003) for any pair of functions f ⁽⁰⁾, f ⁽¹⁾ that are analytic; a result is obtained under weaker regularity assumptions, e.g. with (frac{partial f^{(0)}}{partial theta}) and f ⁽¹⁾ Hölder continuous with exponent (eta > frac{1}{2}).