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

Petar Popivanov and Angela Slavova ()
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Petar Popivanov: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Angela Slavova: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Mathematics, 2022, vol. 10, issue 23, 1-13

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.

Keywords: Laplace operator; biharmonic and polyharmonic operators; Dirichlet, Neumann and Robin boundary value problems; Green function for elliptic second order operator; solutions into explicit form of boundary value problems; Lopatinskii (elliptic) boundary conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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