Abstract
In this paper 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 ⊂ ℝn 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 ∈ ℕ, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ∂D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.
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Original Russian Text © I.V. Shestakov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 7, pp. 51–64.
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Shestakov, I.V. The Cauchy problem in Sobolev spaces for Dirac operators. Russ Math. 53, 43–54 (2009). https://doi.org/10.3103/S1066369X09070056
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DOI: https://doi.org/10.3103/S1066369X09070056