Geodesic
On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 1, pp. 52-62

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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, bases with double orthogonality.
Mots-clés : Carleman's formula 
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Ivan V. Shestakov; Alexander A. Shlapunov. On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 1, pp. 52-62. http://geodesic.mathdoc.fr/item/JSFU_2008_1_1_a5/