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Representation of harmonic functions as potentials and the Cauchy problem. (English. Russian original) Zbl 1161.31004
Math. Notes 83, No. 5, 693-706 (2008); translation from Mat. Zametki 83, No. 5, 763-778 (2008).
Let \(D\) be a bounded, simply connected domain in \({\mathbb R}^m\), \(m\in\{2,3\}\), with boundary \(\partial D\) consisting of a compact, connected part \(T\) of the plane \({\mathbb R}^2\times\{0\}\) for \(m=3\), or of a segment \([a,b]\times\{0\}\) for \(m=2\), and a smooth part \(S\) of a Lyapunov surface for \(m=3\), or a smooth arc of a curve for \(m=2\), lying in the half-space \({\mathbb R}^2\times [0,\infty[\) or \({\mathbb R}\times [0,\infty[\), respectively. The author presents an explicit continuation and a regularization formula for the solution of the Cauchy problem for the Laplace equation in \(D\) if the Cauchy data are given on \(S\). (This is one of the ill-posed problems.) Both formulae make use of the so-called Carleman function. The article closes by presenting a necessary and sufficient condition for the solvability of this Cauchy problem.

MSC:
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
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[1] M. M. Lavrent’ev, On Some Ill-Posed Problems of Mathematical Physics (Izd. SOANSSSR, Novosibirsk, 1962) [in Russian].
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[3] T. Carleman, Les fonctions quasi analitiques (Gauthier-Villars, Paris, 1926), pp. 3–6.
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