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

Author O. E. Yaremko
ѕензенский государственный университет

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. Ц 2009. Ц V.428, є2. Ц P. 151Ц152. (in Russian)

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

3. LavrentТev M.A. Variational Methods for Boundary Value Problems: for Systems of Elliptic Equations. Ц Reprint. Ц USA: Dover Publications, 2006.

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

5. Gradstein I.S., Ryzhik I.M. Tables of integrals, sums, series of products, 4th eds. Ц 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. Ц New York: Dover, 1972. Ц P. 771Ц802.

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

8. Bavrin I.I., Yaremko O.E., Fourier integral on compact and their application to moments problem// Dokl. RAN. Ц 2000. Ц V.374, є2. Ц P. 177Ц179. (in Russsian)

9. Lavrentiev M.M., Romanov V.G., Shishatskiy S.P. Ill Posed problems of mathematical physics and analysis. Ц 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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