IJPAM: Volume 83, No. 1 (2013)
Institute of Mathematics
National Academy of Sciences of Ukraine
Tereshchenkivska Str. 3, 01601, Kiev, UKRAINE
Abstract. We consider the commutative algebra over the field of complex numbers with the bases satisfying the conditions , . This algebra is unique and it is associated with the biharmonic equation. We consider monogenic functions (having the classic derivative in domains of the biharmonic plane , where are real) with values in . For these functions, we consider a Schwarz-type boundary value problem (associated with the main biharmonic problem) for a half-plane and for a disk of the biharmonic plane.
We obtain solutions in explicit forms by means of Schwarz-type
integrals and prove that the mentioned problem is solvable
unconditionally for a half-plane but it is solvable for a disk if
and only if a certain natural condition is satisfied.
Received: December 12, 2012
AMS Subject Classification: 30G35, 31A30
Key Words and Phrases: biharmonic equation, biharmonic algebra, biharmonic plane, monogenic function, Schwarz-type boundary value problem, biharmonic Schwarz integral
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DOI: 10.12732/ijpam.v83i1.13 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395