CAUCHY PROBLEM FOR FIRST-ORDER LINEAR ELLIPTIC SYSTEMS WITH CONSTANT COEFFICIENTS IN A THREE-DIMENSIONAL BOUNDED DOMAIN

Abstract

The paper studies the problem of continuing the solution of linear systems of elliptic type of the first order with constant coefficients in a domain  by its known values ​​on the smooth part of the boundary, i.e. the Cauchy problem is studied for solving linear systems of equations of the elliptic type of the first order with constant coefficients. The problem under consideration belongs to the ill-posed problems of mathematical physics, since there is no continuous dependence of the solution on the initial data. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain, and the Cauchy data on a part of the boundary of the domain are given exact. For this ill-posed problem, an explicit continuation formula is obtained. An estimate of the stability of the solution of the Cauchy problem in the classical sense is obtained.