The Over-Determined Boundary Value Problems and Convolution Type Integral Transformations

The over-determined boundary value problems for elliptic equations in the half-plane and in the half-space are investigated. A connection is found between traces of solutions and their normal derivatives on the boundary of the domain. In the case of the Laplace equation and the Helmholtz equation, the method of Green functions and the method of the Fourier integral transformation on tangent variables are used. It is established that the solvability conditions of over-determined problems generate pairs of mutually inverse integral transformations of the convolution type. It is shown how the problem of diffraction of an electromagnetic wave by conductive thin screens is reduced to integral equations, using the solvability conditions of an auxiliary over-determined problem for Maxwell equations.