This article is cited in 2 papers

On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces

Ivan V. Shestakov, Alexander A. Shlapunov

Institute of Mathematics, Siberian Federal University

Abstract: Let $D$ be a bounded domain in $\mathbb R^n$ ($n\ge 2$) with a smooth boundary $\partial D$. We describe necessary and sufficient solvability conditions (in Sobolev spaces in $D$) of the ill-posed non-homogeneous Cauchy problem for a partial differential operator $A$ with injective symbol and of order $m\ge 1$. Moreover, using bases with the double orthogonality property we construct Carleman's formulae for (vector-) functions from the Sobolev space $H^s(D)$, $s\ge m$, by their Cauchy data on $\Gamma$ and the values of $Au$ in $D$ where $\Gamma$ is an open (in the topology of $\partial D$) connected part of the boundary.

Keywords: ill-posed Cauchy problem, Carleman's formula, bases with double orthogonality.

UDC: 517.955

Received: 11.10.2007
Received in revised form: 20.11.2007
Accepted: 05.12.2007

Language: English

Bibliographic databases:

•