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

Abstract

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.

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

“…The applications of boundary value problems for elliptic equations with fractional order boundary operators were considered in works [11]- [13]. The analogues of the Neumann problems for the biharmonic and polyharmonic equations in the case of the integer order boundary operators were studied in works [15]- [19], while boundary operators fractional in Riemann-Liouville, Caputo and Hadamard-Marchaud sense were addressed in [20]- [23].…”

On solvability of some boundary value problems for a biharmonic equation with periodic conditions

“…Recently other types of boundary value problems for the biharmonic equation such as the Neumann problem [3-5, 9, 13, 16, 17, 23? , 24], the spectral Steklov problem [6], the Robin problem [7], generalized Robin boundary value problem [15], as well as fractional analogous of Neumann problem [1,2,21,22] are begun to investigate actively. In the paper, a new class of boundary value problems with periodic conditions is studied in the unit ball for an inhomogeneous biharmonic equation.…”

About solvability of some boundary value problems for poisson equation in the ball

“…In this paper, we introduce fractional analogues of the boundary operators D α ν , and for the equation (1) we study the boundary value problem with the boundary condition (2) for all values of the parameter α ∈ (0, ∞). Moreover, we investigate solvability of some analogues of periodic boundary value problems and an analog of Samarskii-Ionkin type boundary value problems for circular domains.…”