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
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.
negative Sobolev spaces, ill-posed Cauchy problem.
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Received in revised form: 20.12.2008
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
\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.
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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
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
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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