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A Schwartz-type boundary value problem in a biharmonic plane

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Abstract

A commutative algebra 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)2 ≠ 0, is considered. The algebra B is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type Φ(xe 1 + ye 2) = U 1(x, y)e 1 + U 2(x, y)ie 1 + U 3(x, y)e 2 + U 4(x, y)ie 2, (x, y) ∈ D, when values of two components U 1, U 4 are given on the boundary of a domain D lying in the Cartesian plane xOy. We develop a method of its solving which is based on expressions of monogenic functions via corresponding analytic functions of the complex variable. For a half-plane and for a disk, solutions are obtained in explicit forms by means of Schwartz-type integrals.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str. 3, 01601, Kiev-4, Ukraine

    S. V. Gryshchuk & S. A. Plaksa

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  1. S. V. Gryshchuk
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  2. S. A. Plaksa
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Correspondence to S. V. Gryshchuk.

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Submitted by V. V. Zharinov

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Gryshchuk, S.V., Plaksa, S.A. A Schwartz-type boundary value problem in a biharmonic plane. Lobachevskii J Math 38, 435–442 (2017). https://doi.org/10.1134/S199508021703012X

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  • DOI: https://doi.org/10.1134/S199508021703012X

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