This article is cited 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.

UDC: 517.98+517.55

Received: 10.11.2008
Received in revised form: 20.12.2008
Accepted: 29.01.2009

Language: English

Bibliographic databases:

•