ON THE SOLVABILITY OF BOUNDARY VALUE PROBLEMS WITH GENERAL CONDITIONS FOR THE TRIHARMONIC EQUATION IN A BALL

Abstract

 The need to study boundary value problems for elliptic and parabolic equations is dictated by numerous practical applications in the theoretical study of processes in hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, and quantum physics. This paper
investigates the solvability of a boundary value problem with general conditions for the triharmonic equation in a unit ball.The validity of the analogue of the Almansi representation is proved. For completeness of presentation, a representation of the Green’s functions of the Dirichlet-2 problem is given. This article indicates the difference between the Green’s function of the real Dirichlet
problem and the Green’s function of the Dirichlet-2 problem. It is known that the results of  differential equations with partial derivatives in the entire space or differential equations without boundary conditions are in a sense final. The theory of boundary value problems for general differential operators is currently a relevant and rapidly developing part of the theory of differential
equations. However, there is a shortage of explicitly solvable problems on the path of further development of the theory of boundary value problems of differential equations. Over the past decades, sufficient material has been accumulated on the constructive construction of solutions to boundary value problems for model equations with partial derivatives. This article relates to this topical issue.