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 J. Sib. Fed. Univ. Math. Phys., 2009, Volume 2, Issue 1, Pages 17–30 (Mi jsfu48)

Negative Sobolev Spaces in the Cauchy Problem for the Cauchy–Riemann Operator

Ivan V. Shestakov, Alexander A. Shlapunov

Institute of Mathematics, Siberian Federal University

Abstract: Let $D$ be a bounded domain in $\mathbb C^n$ ($n\ge1$) with a smooth boundary $\partial D$. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for the Cauchy–Riemann operator $\overline\partial$ in $D$. In particular, we describe traces of the corresponding Sobolev functions on $\partial D$ and give an adequate formulation of the problem. Then we prove the uniqueness theorem for the problem, describe its necessary and sufficient solvability conditions and produce a formula for its exact solution.

Keywords: negative Sobolev spaces, ill-posed Cauchy problem.

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UDC: 517.98+517.55
Accepted: 29.01.2009
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Citation: Ivan V. Shestakov, Alexander A. Shlapunov, “Negative Sobolev Spaces in the Cauchy Problem for the Cauchy–Riemann Operator”, J. Sib. Fed. Univ. Math. Phys., 2:1 (2009), 17–30

Citation in format AMSBIB
\Bibitem{SheShl09} \by Ivan~V.~Shestakov, Alexander~A.~Shlapunov \paper Negative Sobolev Spaces in the Cauchy Problem for the Cauchy--Riemann Operator \jour J. Sib. Fed. Univ. Math. Phys. \yr 2009 \vol 2 \issue 1 \pages 17--30 \mathnet{http://mi.mathnet.ru/jsfu48} \elib{https://elibrary.ru/item.asp?id=11716607} 

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This publication is cited in the following articles:
1. I. V. Shestakov, A. A. Shlapunov, “The Cauchy problem for operators with injective symbol in the Lebesgue space $L^2$ in a domain”, Siberian Math. J., 50:3 (2009), 547–559
2. Alexander A. Shlapunov, “Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators”, Zhurn. SFU. Ser. Matem. i fiz., 4:2 (2011), 217–228
3. Fedchenko D., Shlapunov A., “On the Cauchy Problem for the Dolbeault Complex in Spaces of Distributions”, Complex Var. Elliptic Equ., 58:11, SI (2013), 1591–1614
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