Green's functions of some boundary value problems for the biharmonic equation

Abstract

In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that, the Green's function of the Navier problem and the integral representation of the solution to the Navier problem are presented. To study the Neumann-2 problem, the Green's function of the Neumann problem for the Poisson equation is discussed and on its basis the Green's function of the Neumann-2 problem is constructed. To illustrate the results obtained, solutions of the three considered homogeneous problems for the polynomial right-hand side of the equation are found.

Keywords:

• Green's function
• biharmonic equation
• Dirichlet problem
• Navier problem
• Neumann-2 problem
• polynomial solutions

AMS Subject Classifications:

• 35J08
• 31B30
• 35J40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the Government of the Russian Federation (decree no. 211 16.03.2013, contract no. 02.A03.21.0011).