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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.

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
Full Text: DOI
[1] M. M. Lavrent’ev, On Some Ill-Posed Problems of Mathematical Physics (Izd. SOANSSSR, Novosibirsk, 1962) [in Russian].
[2] A. A. Shlyapunov, ”The Cauchy problem for Laplace’s equation,” Sibirsk. Mat. Zh. 33(3), 205–215 (1992) [Siberian Math. J. 33 (3), 534–542 (1992)].
[3] T. Carleman, Les fonctions quasi analitiques (Gauthier-Villars, Paris, 1926), pp. 3–6.
[4] M. M. Lavrent’ev, ”On the Cauchy problem for the Laplace equation,” Izv. Akad. Nauk SSSR Ser. Mat. 20(6), 819–842 (1956).
[5] Sh. Yrmukhamedov, ”A Carleman function and the Cauchy problem for the Laplace equation,” Sibirsk. Mat. Zh. 45(3), 702–719 (2004) [Siberian Math. J. 45 (3), 580–595 (2004)].
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