The Sommerfeld problem and inverse problem for the Helmholtz equation

The study of a time-periodic solution of the multidimensional wave equation {\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)}, {\widetilde{u}(x,t)=e^{ikt}u(x)}, over the whole space {\mathbb{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, 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, J. Comput. Math. Math. Phys. 52 2012, 6, 1063–1068], for a complex parameter λ, 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 Ω:($*$)u(x,\lambda)=\int_{\Omega}\varepsilon(x-\xi,\lambda)\rho(\xi,\lambda)\,d\xi{}where {\varepsilon(x-\xi,\lambda)} are fundamental solutions of the Helmholtz equation,-\Delta_{x}\varepsilon(x)-\lambda\varepsilon=\delta(x),{\rho(\xi,\lambda)} is a density of the potential, λ is a complex number, and δ is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Ω to {\partial\Omega} pass {\partial\Omega} without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in {\mathbb{R}^{n}}, {n\geq 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.