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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
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