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Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions

Author

Listed:
  • Petar Popivanov

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

  • Angela Slavova

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

Abstract

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B 1 a Green function is constructed in the cases c > 0 , c ∉ − N , where c is the coefficient in front of u in the boundary condition ∂ u ∂ n + c u = f . To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λ u + c u = f . Elliptic boundary value problems for Δ m u = 0 in B 1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ u = 0 , Δ 2 u = 0 and Δ 3 u = 0 in B 1 as well as some additional results from the theory of spherical functions are proposed.

Suggested Citation

  • Petar Popivanov & Angela Slavova, 2022. "Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions," Mathematics, MDPI, vol. 10(23), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4413-:d:981441
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