## 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
Full Text:

### 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.