Some properties and applications of the integrodifferential operators of hadamard-marchaud type in the class of harmonic functions

Modified Riemann - Liouville integro-differential operators in the class of harmonic functions and their applications

“…Substituting in this case ( ) ≡ into the formulation of problem (15), we have = 0, or identity (16). Thus, under conditions (13) and (14) we get ( ) ≡ 0. If (16) holds true, then an arbitrary constant also solves homogeneous problem (15).…”

“…Problem L. Find a function ( ) harmonic in ball Ω such that function [ ]( ) is continuous in Ω and satisfies the identity [ ]( ) = ( ), ∈ Ω, on sphere Ω. We note that similar problems with boundary operators of integer high order were considered in works [4], [6]- [9], and fractional order operators were studied in works [2], [10]- [14].…”

“…3) if the solution exists, then under condition (14) it is unique and under condition (16) it is unique up an additive constant.…”

On solvability of a boundary value problem for an inhomogeneous polyharmonic equation with a fractional order boundary operator

“…We note that the boundary value problems for elliptic second order equations with fractional order boundary operators were studied in works [2]- [10]. The applications of boundary value problems for elliptic equations with fractional order boundary operators were considered in works [11]- [13].…”

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

“…In [4] the author proves that operators J α μ and D α μ for any μ > 0, α ∈ (0, ∞) are mutually inverse in the class of harmonic functions in the ball, and applies these operators for study of solvability of certain boundary-value problems for the Laplace equation. Note also that properties and applications of operators J α μ and D α μ in the case μ ≥ 0, α ∈ (0, 1) are studied in [5]. Analogous questions on operators of differentiation of fractional order in the Riemann-Liouville and Caputo sense are studied in [6].…”