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The Cauchy problem in Sobolev spaces for Dirac operators. (English. Russian original) Zbl 1181.35042
Russ. Math. 53, No. 7, 43-54 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 7, 51-64 (2009).
Summary: We consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator $$A$$ in Sobolev spaces in a bounded domain $$D \subset \mathbb R^n$$ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of 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 $$u$$ in a Sobolev space $$H^s(D),s\in \mathbb N$$, from 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.
##### MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 58J05 Elliptic equations on manifolds, general theory 58J32 Boundary value problems on manifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
Cauchy problem; Dirac operators; Carleman formula
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##### References:
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