# Integral representations for harmonic and biharmonic functions in the unit ball (in Russian)

Author O. E. Yaremko
yaremki@mail.ru
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Abstract In the paper an analytical solution of the analytic continuation problem in the unit N-dimensional ball by its values on the inner sphere is found. A generalization of the classical Poisson formula is obtained. A solution of the similar analytic continuation problem for certain known operator expressions of a given potential on the inner sphere is obtained. A new result of the moment problem making a connection between the moment problem on a segment and on the half-axis is found.
Keywords extension problem; Poisson formula; moment problems; biharmonic function; harmonic function; unit ball
Reference 1. Bavrin I.I. Cauchy integral formula in the ring inverse problem// Dokl. RAN. – 2009. – V.428, ¹2. – P. 151–152. (in Russian)

2. Bavrin I.I. Operational methods in complex analysis. – M.:Prometey, 1991. (in Russian)

3. Lavrent’ev M.A. Variational Methods for Boundary Value Problems: for Systems of Elliptic Equations. – Reprint. – USA: Dover Publications, 2006.

4. Belov Yu.Ya. Inverse Problems for Partial Differential Equations. – Utrecht: VSP, 2002.

5. Gradstein I.S., Ryzhik I.M. Tables of integrals, sums, series of products, 4th eds. – M.Fizmatgiz, 1963. (in Russian)

6. Abramowitz M., Stegun I.A., Orthogonal Polynomials Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. – New York: Dover, 1972. – P. 771–802.

7. Ditkin V.A., Prudnikov A.P., Integral transforms and operational calculus. – M: Nauka, 1974. (in Russian)

8. Bavrin I.I., Yaremko O.E., Fourier integral on compact and their application to moments problem// Dokl. RAN. – 2000. – V.374, ¹2. – P. 177–179. (in Russsian)

9. Lavrentiev M.M., Romanov V.G., Shishatskiy S.P. Ill Posed problems of mathematical physics and analysis. – M. Nauka, 1980. (in Russsian)

10. Weisstein, Eric W., Biharmonic Equation// Wolfram MathWorld.

Pages 67-73
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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