Over-determined boundary value problems for PDE and their application in the wave propagation theory

Abstract

The solvability conditions of the over-determined boundary value problems for PDE with constant coefficients in the form of connection between Fourier transforms of boundary functions are obtained. The jump problem on the hyperplane for PDE is formulated and investigated. Two-dimensional Helmholtz equation is considered as an example. It is proved that the mixed type conjugation problem for Helmholtz equation, to which the diffraction problem of the electromagnetic wave on thin conducting ribbons is reduced, is equivalent to one-sided mixed type boundary value problem. The uniqueness theorem is proved for this problem.

Keywords:

• over-determined problems
• Fourier transform
• Helmholtz equation
• diffraction problem

AMS Subject Classifications:

• 35G15
• 35A22
• 78A25

Notes

Dedicated to Professor Lin Wei on the occasion of his 80th birthday.

Supported by RFBR 12-01-97012-r-povolzh’e-a and the Ministry of education and science of the Russian Federation.