The study of a time-periodic solution of the multidimensional wave equation ∂2∂⁡t2⁢u~-Δx⁢u~=f~⁢(x,t){\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)}, u~⁢(x,t)=ei⁢k⁢t⁢u⁢(x){\widetilde{u}(x,t)=e^{ikt}u(x)}, over the whole space ℝ3{\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)=∫Ωε⁢(x-ξ,λ)⁢ρ⁢(ξ,λ)⁢𝑑ξu(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, -Δx⁢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 ℝn{\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.

中文翻译：

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亥姆霍兹方程的Sommerfeld问题和逆问题

多维波动方程∂2∂⁡t2⁢u〜-Δx⁢u〜= f〜⁢（x，t）{\ frac {\ partial ^ {2}} {\ partial t的时间周期解的研究^ {2}} \ widetilde {u}-\ Delta_ {x} \ widetilde {u} =％\ widetilde {f}（x，t）}，u〜⁢（x，t）=ei⁢k⁢t⁢ u⁢（x）{\ widetilde {u}（x，t）= e ^ {ikt} u（x）}在整个空间ℝ3{\ mathbb {R} ^ {3}}中导致Sommerfeld辐射处于无限远。这是一个问题，描述了从有界区域中的源散射固定波的运动。找到此源的反问题等效于将Sommerfeld问题简化为有限域中Helmholtz方程的边值问题。因此，Sommerfeld问题是一个特殊的反问题。应该注意的是，在贝兹梅诺夫[IV贝兹梅诺夫的工作中，根据变分原理Sb，将Sommerfeld辐射条件转移到该区域的人工边界。数学。[185 1995，3，3–24]找到了这种边界条件的近似形式。在[TS Kalmenov和D. Suragan中，将Sommerfeld辐射条件转移到有限区域的边界，J。Comput。数学。数学。物理 52 2012，6，1063–1068]，对于复数参数λ，通过由有限域Ω中的积分给出的亥姆霍兹势的边界条件找到了这些边界条件的显式形式：（$ * $）u⁢ （x，λ）=∫Ωε⁢（x-ξ，λ）⁢ρ⁢（ξ，λ）⁢𝑑ξu（x，\ lambda）= \ int _ {\ Omega} \ varepsilon（x- \ xi，\ lambda） \ rho（\ xi，\ lambda）\，d \ xi {}其中ε⁢（x-ξ，λ）{\ varepsilon（x- \ xi，\ lambda）}是Helmholtz方程的基本解-Δx⁢ ε⁢（x）-λ⁢ε=δ⁢（x），-\ Delta_ {x} \ varepsilon（x）-\ lambda \ varepsilon = \ delta（x），ρ⁢（ξ，λ）{\ rho（\ xi，\ lambda）}是电势的密度，λ是复数，并且δ是狄拉克δ函数。这些边界条件具有以下性质：来自区域Ω到Ω{\ partial \ Omega}的驻波在没有反射的情况下通过ΩΩ{\ partial \ Omega}，即透明的边界条件。在目前的工作中，在一般情况下，在ℝn{\ mathbb {R} ^ {n}}中，n≥3{n \ geq 3}，我们证明了将Sommerfeld问题简化为边值问题的问题。有限域。在亥姆霍兹势（**）的必要条件下，还发现了其密度ρ⁢（ξ，λ）{\ rho（\ xi，\ lambda）}。这些边界条件具有以下性质：来自区域Ω到Ω{\ partial \ Omega}的驻波在没有反射的情况下通过ΩΩ{\ partial \ Omega}，即透明的边界条件。在目前的工作中，在一般情况下，在ℝn{\ mathbb {R} ^ {n}}中，n≥3{n \ geq 3}，我们证明了将Sommerfeld问题简化为边值问题的问题。有限域。在亥姆霍兹势（**）的必要条件下，还发现了其密度ρ⁢（ξ，λ）{\ rho（\ xi，\ lambda）}。这些边界条件具有以下性质：来自区域Ω到Ω{\ partial \ Omega}的驻波在没有反射的情况下通过ΩΩ{\ partial \ Omega}，即透明的边界条件。在目前的工作中，在一般情况下，在ℝn{\ mathbb {R} ^ {n}}中，n≥3{n \ geq 3}，我们证明了将Sommerfeld问题简化为边值问题的问题。有限域。在亥姆霍兹势（**）的必要条件下，还发现了其密度ρ⁢（ξ，λ）{\ rho（\ xi，\ lambda）}。