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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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##### References:
 [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)]. [6] M. Ikehata, ”Inverse conductivity problem in the infinite slab,” Inverse Problems 17(3), 437–454 (2001). · Zbl 0980.35174 · doi:10.1088/0266-5611/17/3/305
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