On the Cauchy problem for the elliptic complexes in spaces of distributions

Abstract

Let be a bounded domain in , , with a smooth boundary . We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex of first order operators. In particular, we describe traces on of the tangential part and the normal part of a (vector)-function from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue space , we construct the approximate and exact solutions to the Cauchy problem with the maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman’s formulae for a (vector-) function from the Sobolev space by its Cauchy data on a subset and the values of in modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.