The Sommerfeld problem and inverse problem for the Helmholtz equation

Abstract

Abstract The study of a time-periodic solution of the multidimensional wave equation ∂ 2 ∂ ⁡ t 2 ⁢ u ~ - Δ x ⁢ u ~ = f ~ ⁢ ( x , t ) {\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)} , u ~ ⁢ ( x , t ) = e i ⁢ k ⁢ t ⁢ u ⁢ ( x ) {\widetilde{u}(x,t)=e^{ikt}u(x)} , over the whole space ℝ 3 {\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 , λ ) = ∫ Ω ε ⁢ ( x - ξ , λ ) ⁢ ρ ⁢ ( ξ , λ ) ⁢ 𝑑 ξ u(x,\lambda)=\int_{\Omega}\varepsilon(x-\xi,\lambda)\rho(\xi,\lambda)\,d\xi{} where ε ⁢ ( x - ξ , λ ) {\varepsilon(x-\xi,\lambda)} are fundamental solutions of the Helmholtz equation, - Δ x ⁢ ε ⁢ ( x ) - λ ⁢ ε = δ ⁢ ( x ) , -\Delta_{x}\varepsilon(x)-\lambda\varepsilon=\delta(x), ρ ⁢ ( ξ , λ ) {\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 ∂ ⁡ Ω {\partial\Omega} pass ∂ ⁡ Ω {\partial\Omega} without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in ℝ n {\mathbb{R}^{n}} , n ≥ 3 {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 ρ ⁢ ( ξ , λ ) {\rho(\xi,\lambda)} has also been found.