Geodesic
The Cauchy problem in Sobolev spaces for Dirac operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 51-64

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In this paper we consider the Cauchy problem as a typical example of ill-posed boundary value problems. We describe the necessary and sufficient solvability conditions for the Cauchy problem for a Dirac operator $A$ in Sobolev spaces in a bounded domain $D\subset\mathbb R^n$ with piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of the harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function $u$ from the Sobolev space $H^s(D)$, $s\in\mathbb N$, by its values on $\Gamma$ and values $Au$ in $D$, where $\Gamma$ is an open connected subset of the boundary $\partial D$. It is worth pointing out that we impose no assumptions about geometric properties of the domain $D$, except for its connectedness.
Keywords: Cauchy problem, Dirac operators
Mots-clés : Carleman formula. 
@article{IVM_2009_7_a4,
     author = {I. V. Shestakov},
     title = {The {Cauchy} problem in {Sobolev} spaces for {Dirac} operators},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {51--64},
     publisher = {mathdoc},
     number = {7},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/}
}
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I. V. Shestakov. The Cauchy problem in Sobolev spaces for Dirac operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 51-64. http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/