The Cauchy problem in Sobolev spaces for Dirac operators

Abstract

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ⊂ ℝn with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one.