IJPAM: Volume 83, No. 1 (2013)

SCHWARZ-TYPE INTEGRALS IN A BIHARMONIC PLANE

S.V. Gryshchuk$^1$, S.A. Plaksa$^2$
$^{1,2}$Institute of Mathematics
National Academy of Sciences of Ukraine
Tereshchenkivska Str. 3, 01601, Kiev, UKRAINE


Abstract. We consider the commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. 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 $\{xe_1+ye_2\}$, where $x,y$ are real) with values in $\mathbb{B}$. 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
Year: 2013
Volume: 83
Issue: 1