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The Fischer-Riesz equations method in the ill-posed Cauchy problem for systems with injective symbols. (English) Zbl 0834.35133

Let \(S\) be a closed subset of \(\partial O\) of positive measure, and \(f_j\in L^q (G_j/ S)\), \(j= 0, 1, \dots, p-1\), be some known sections on \(S\). It is required to find \(f\in H^q_{P,B} (O)\) such that \(B_j f|_S= f_j\), \(j= 0, 1, \dots, p-1\). \(H^q_{P,B} (O)\) is the generalized Hardy space of solutions of the system \(Pf=0\) in \(O\) satisfying the condition \[ \sum_{j=1}^{p-1} \int _{\partial O} |B_j f|^q ds< \infty. \] Using the Fourier series analysis the authors investigate solvability conditions and the regularization of the above problem.

MSC:

35R25 Ill-posed problems for PDEs
35C15 Integral representations of solutions to PDEs
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References:

[1] Lavrent, ev Some ill - posed problems of mathematical physics Nauk SSSR Novosibirsk transl Springer - Verlag, English (1962)
[2] Tarkhanov, A criterion for solvability of the ill - posed Cauchy problem for elliptic systems Dokl Akad transl in Sov Math Dokl, Nauk SSSR English pp 308– (1989)
[3] Lavrent, ev and Ill - posed problems of mathematical physics and analysis Moscow Nauka transl Amer Math Providence, English (1980)
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