-- An original way of applying the Fourier series theory to investigation of solvability conditions and regularization of solutions of the Cauchy problem with data on a piece of the domain boundary for systems with injective symbols is presented. In his early paper  M. M. Lavrenfev brilliantly grasped some aspects of approximate solution of the Cauchy problem for the Laplace equation. This article got the theory of conditionally stable problems off the ground. In our paper a constructive approach to the study of one important class of conditionally stable problems, namely, the Cauchy problems with data on a boundary subset for solutions of elliptic and more general systems, is worked out. Let P G dop(£ -» F) be a differential operator (DO) of type E --> F and order p with injective symbol on an open set X c K n . Here E = Xx<Ck and F = Xxfc1 are (trivial) vector bundles over X whose sections of the class 0 are interpreted as columns of functions from 0(X), that is 0(E) = [0(X)]*, and for F by analogy. Thereby, DO P is given by a (/xfc)-matrix of scalar DO's of order < p on X.
Journal of Inverse and Ill-Posed Problems – de Gruyter
Published: Jan 1, 1993
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