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The Fischer-Riesz equations method in the ill-posed Cauchy problem for systems with injective symbols

The Fischer-Riesz equations method in the ill-posed Cauchy problem for systems with injective... -- 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 [11] 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inverse and Ill-Posed Problems de Gruyter

The Fischer-Riesz equations method in the ill-posed Cauchy problem for systems with injective symbols

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0928-0219
eISSN
1569-3945
DOI
10.1515/jiip.1993.1.4.307
Publisher site
See Article on Publisher Site

Abstract

-- 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 [11] 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

Journal of Inverse and Ill-Posed Problems – de Gruyter

Published: Jan 1, 1993

References

  • ev Some ill - posed problems of mathematical physics Nauk SSSR Novosibirsk transl Springer - Verlag
    Lavrent
  • A criterion for solvability of the ill - posed Cauchy problem for elliptic systems Dokl Akad transl in Sov Math Dokl
    Tarkhanov
  • ev and Ill - posed problems of mathematical physics and analysis Moscow Nauka transl Amer Math Providence
    Lavrent