LetFbe a relatively closed subset of an open setGof the plane. Alice Roth in [10] proved that if a functionfis a uniform limit onFof holomorphic or meromorphic functions onG, it is possible to select the approximating functionsmin such a way that the difference functionfmcan be extended continuously into the boundary ofF. In this paper we consider the more general situation arising when one replaces ∂ƀ by an elliptic operatorp(D) with constant coefficients. We prove that the natural analog of Roth's theorem for the operatorP(D) holds, at least for bounded open setsG. In those cases where one can find an inversion of the domain, preserving the uniform approximation by solutions of the operator, unbounded open sets are allowed too. This hannens whenP(D)=δ andn=2. In this case we improve the main result of Goldstein andOwin [7], removing most oi the unnecessary conditions assumed in their work.