The Sommerfeld problem and inverse problem for the Helmholtz equation

The study of a time-periodic solution of the multidimensional wave equation partial derivative(2)/partial derivative t(2)(u) over tilde - Delta(x)(u) over tilde = (f) over tilde (x, t), (u) over tilde (x, t) = e(ikt)u(x), over the whole space R-3 leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 (1995), no. 3, 3-24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, I. Comput. Math. Math. Phys. 52 (2012), no. 6, 1063-1068], for a complex parameter lambda, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain Omega: u(x, lambda) = integral(Omega) epsilon(x - xi, lambda)rho(xi, lambda) d xi (*) where epsilon(x - xi, lambda) are fundamental solutions of the Helmholtz equation, -Delta(x)epsilon(x) - lambda epsilon = delta(x), rho(xi, lambda) is a density of the potential, lambda is a complex number, and delta is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Omega to partial derivative Omega pass partial derivative Omega without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in R-n, n >= 3, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (*), its density rho(xi, lambda) has also been found.