Mixed Problems with a Parameter

Abstract

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