Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m
-
Yury G. Smirnov
and
Aleksei A. Tsupak
Abstract
The boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes.
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 18-01-00219 A
Funding statement: This work was supported by the Russian Foundation for Basic Research [grant No. 18-01-00219 A].
A On the fundamental solution to the Helmholtz equation in
ℝ
m
Let
is a solution to the equation
Since
where
and
it follows that
as
Let us show that the functions
Statement A.1.
The function
Proof.
We prove this statement in two steps.
1. Taking into account the asymptotic behavior [21] of Hankel functions at infinity, i.e.,
we obtain
From (A.1) and (A.3) it follows that
2. Since
we obtain
Further,
Finally, we arrive at
as desired. ∎
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research biography of a distinguished expert in the field of inverse problems: Professor Michael Victor Klibanov
- Unique continuation property of solutions to general second order elliptic systems
- Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations
- Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory
- Alternating direction based method for optimal control problem constrained by Stokes equation
- Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m
- Mathematical modelling of plasmonic strain sensors
- Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations
- Stability and the inverse gravimetry problem with minimal data
Articles in the same Issue
- Frontmatter
- Research biography of a distinguished expert in the field of inverse problems: Professor Michael Victor Klibanov
- Unique continuation property of solutions to general second order elliptic systems
- Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations
- Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory
- Alternating direction based method for optimal control problem constrained by Stokes equation
- Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m
- Mathematical modelling of plasmonic strain sensors
- Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations
- Stability and the inverse gravimetry problem with minimal data
