Mixed Problems with a Parameter

Let $X$ be a smooth $n$-dimensional manifold and $D$ be an open connected set in $X$ with smooth boundary $\mathrm{\partial }D$. Perturbing the Cauchy problem for an elliptic system $Au=f$ in $D$ with data on a closed set $\text{iG}\subset \mathrm{\partial }D$ we obtain a family of mixed problems depending on a small parameter $\epsilon >0$. Although the mixed problems are subject to a non-coercive boundary condition on $\mathrm{\partial }D\setminus \text{iG}$ in general, each of them is uniquely solvable in an appropriate Hilbert space ${\text{cD}}_T$ and the corresponding family $\{u_{\epsilon }\}$ of solutions approximates the solution of the Cauchy problem in ${\text{cD}}_T$ whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in ${\text{cD}}_T$ is equivalent to the boundedness of the family $\{u_{\epsilon }\}$. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing its solution. Examples for Dirac operators in the Euclidean space ${\mathbb{R}}^n$ are considered. In the latter case we obtain a family of mixed boundary problems for the Helmholtz equation.