Abstract:
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.
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