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
Issue Date:
23-Nov-2022
Publisher:
MDPI
Citation:
Popivanov, P.; Slavova, A. Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions. Mathematics 2022, 10, 4413. https://doi.org/10.3390/math10234413
Series/Report no.:
Mathematics;10, 4413
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 𝐵1 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+𝑐𝑢=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 Λ𝑢+𝑐𝑢=𝑓
. Elliptic boundary value problems for Δ𝑚𝑢=0
in 𝐵1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ𝑢=0, Δ2𝑢=0 and Δ3𝑢=0 in 𝐵1 as well as some additional results from the theory of spherical functions are proposed.
