GREEN'S FUNCTIONS OF SOME BOUNDARY VALUE PROBLEMS FOR BYHARMONIC OPERATORS AND THEIR CORRECT CONSTRICTIONS
Ключевые слова:
biharmonic equations, Dirichlet problem, biharmonic operator, domain of operator definition, correct problems, correct operator constrictions.Аннотация
In this paper, a constructive method is given for constructing the Green function of the Dirichlet
problem for a biharmonic equation in a multidimensional ball.
The need to study boundary value problems for elliptic equations is dictated by numerous practical applications
in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity
theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the
Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise.
There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For
many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the
construction of the Green function is an open problem. For the ball, the Green function of the internal and external
Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases.
Finding general correct boundary value problems for differential equations is always an urgent problem. The
abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a
method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of
symmetric operators with finite defect indices was developed in detail. Many problems for partial differential
equations lead to operators with infinite defect indices.
In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to
describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain
minimum operator, independently of each other, in terms of the inverse operator.
In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's
function.
