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Schwartz-type boundary-value problems for canonical domains in a biharmonic plane

Abstract

A commutative algebra \( \mathbbm{B} \) over the complex field with a basis {e1, e2} satisfying the conditions \( {\left({e}_1^2+{e}_2^2\right)}^2=0,{e}_1^2+{e}_2^2\ne 0 \) is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form.

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Correspondence to Serhii V. Gryshchuk.

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Dedicated to V.Ya. Gutlyanskii on the occasion of his 80th birthday

Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 18, No. 3, pp. 338–358, July–September, 2021.

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Gryshchuk, S.V., Plaksa, S.A. Schwartz-type boundary-value problems for canonical domains in a biharmonic plane. J Math Sci 259, 37–52 (2021). https://doi.org/10.1007/s10958-021-05599-6

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Keywords

  • Biharmonic equation
  • biharmonic algebra
  • biharmonic plane
  • monogenic function
  • Schwartz-type boundary-value problem
  • biharmonic Schwartz-type integral