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

This article is cited in 3 scientific papers (total in 3 papers)

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.

Full text: PDF file (350 kB)
References: PDF file   HTML file
UDC: 517.98+517.55
Received: 10.11.2008
Received in revised form: 20.12.2008
Accepted: 29.01.2009
Language:

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  isi  elib  elib
    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  mathnet  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
  • Журнал Сибирского федерального университета. Серия "Математика и физика"
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