Representation of harmonic functions as potentials and the Cauchy problem. (English. Russian original) Zbl 1161.31004

Yarmukhamedov, Sh.

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.

Reviewer: Eleutherius Symeonidis (Eichstätt)

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

Keywords:

harmonic function; potential; Cauchy problem; Laplace equation; Carleman function; Lyapunov condition; ill-posed problem

References:

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[3]	T. Carleman, Les fonctions quasi analitiques (Gauthier-Villars, Paris, 1926), pp. 3–6.
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