# Unique Determination of the Shape of a Scattering Screen from a Passive Measurement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Antennas

#### 1.2. Mathematical Background

#### 1.3. Definitions and Theorems

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- $x,y,\dots $ represent variables in ${\mathbb{R}}^{3}$, and we associate to them various projections described below.
- ${x}^{\prime},{y}^{\prime},\dots $ mean variables in ${\mathbb{R}}^{2}$ or projections to ${\mathbb{R}}^{2}$. For example if $x=(1,2,3)\in {\mathbb{R}}^{3}$ then in that context ${x}^{\prime}=(1,2)\in {\mathbb{R}}^{2}$, but we could have $d{y}^{\prime}$ in an integral over a subset of ${\mathbb{R}}^{2}$ without having to define the variable y separately.
- ${x}^{0},{y}^{0},\dots $ denote lifts to ${\mathbb{R}}^{3}$, meaning ${x}^{0}=({x}^{\prime},0)$. For example if ${x}^{\prime}=(-1,-2)$ then ${x}^{0}=(-1,-2,0)$. This notation can also be used as a projection ${\mathbb{R}}^{3}\to {\mathbb{R}}^{2}\times \left\{0\right\}$. So, if $x=(1,2,3)$ then ${x}^{0}=(1,2,0)$. Essentially ${{x}^{\prime}}^{0}={\left({x}^{\prime}\right)}^{0}={x}^{0}$ and ${{x}^{0}}^{\prime}={\left({x}^{0}\right)}^{\prime}={x}^{\prime}$ but we do not use this combined notation explicitly.
- $\Phi $ is reserved for the fundamental solution to $(\Delta +{k}^{2})$, defined in Lemma 2.
- ${u}^{+},{u}^{-}$ mean the function u restricted to ${\mathbb{R}}^{2}\times {\mathbb{R}}_{+}$ and ${\mathbb{R}}^{2}\times {\mathbb{R}}_{-}$, respectively. If their variable is in ${\mathbb{R}}^{2}\times \left\{0\right\}$ then they are the two-sided limits (traces) as ${x}_{3}\to 0$. We often use ${\partial}_{3}{u}^{+}$ and ${\partial}_{3}{u}^{-}$. These are simply the derivatives in the ${x}_{3}$-direction of ${u}^{+}$ and ${u}^{-}$, respectively. Often this is evaluated on ${\mathbb{R}}^{2}\times \left\{0\right\}$ where it then denotes the one-sided derivative, i.e., the trace of ${\partial}_{3}{u}^{\pm}$.
- ${\tilde{H}}^{-1/2}\left({\Omega}_{0}\right)$: this is the set of ${H}^{-1/2}\left({\mathbb{R}}^{2}\right)$ distributions whose support is contained in $\overline{{\Omega}_{0}}$, where we recall that ${\Omega}_{0}$ signifies the shape of a screen $\Omega $.

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Representation Theorems

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

- ${\nabla}_{y}{\left|x-y\right|}^{s}=-s\frac{x-y}{\left|x-y\right|}{\left|x-y\right|}^{s-1}$ for all $s\in \mathbb{R}$,
- ${\nabla}_{y}{e}^{ik\left|x-y\right|}=-ik\frac{x-y}{\left|x-y\right|}{e}^{ik\left|x-y\right|}$, and
- ${\nabla}_{y}{e}^{-ik\widehat{x}\xb7y}=-ik\widehat{x}{e}^{-ik\widehat{x}\xb7y}$.

**Proof**

**of**

**Theorem**

**1.**

## 3. Solving the Inverse Problem

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Proof**

**of**

**Theorem**

**2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Maxwell, J.C. On physical lines of force. Philos. Mag.
**1861**, 90, 11–23. [Google Scholar] [CrossRef] - Sylvester, J.; Uhlmann, G. A global uniqueness theorem for an inverse boundary value problem. Ann. Math.
**1987**, 125, 153–169. [Google Scholar] [CrossRef] - Uhlmann, G. Inside Out: Inverse Problems and Applications; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Bukhgeim, A.L. Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl.
**2008**, 16, 19–33. [Google Scholar] [CrossRef] - Guillarmou, C.; Tzou, L. Calderón inverse problem with partial data on Riemann surfaces. Duke Math. J.
**2011**, 158, 83–120. [Google Scholar] [CrossRef] [Green Version] - Imanuvilov, O.Y.; Uhlmann, G.; Yamamoto, M. The Calderón problem with partial data in two dimensions. J. Am. Math. Soc.
**2010**, 23, 655–691. [Google Scholar] [CrossRef] [Green Version] - Dos Santos Ferreira, D.; Kenig, C.E.; Salo, M. Determining an unbounded potential from Cauchy data in admissible geometries. Comm. Part. Differ. Eq.
**2013**, 38, 50–68. [Google Scholar] [CrossRef] [Green Version] - Blåsten, E.; Imanuvilov, O.Y.; Yamamoto, M. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Probl. Imaging
**2015**, 9, 709–723. [Google Scholar] - Blåsten, E.; Tzou, L.; Wang, J. Uniqueness for the inverse boundary value problem with singular potentials in 2D. Math. Z.
**2019**. [Google Scholar] [CrossRef] [Green Version] - Colton, D.; Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory; Applied Mathematical Sciences; Springer: Berlin/Heidelberg, Germany, 1992; Volume 93. [Google Scholar]
- Lax, P.; Phillips, R. Scattering Theory; Academic Press: New York, NY, USA; London, UK, 1967. [Google Scholar]
- Colton, D.; Kirsch, A. A simple method for solving inverse scattering problems in the resonance region. Inverse Probl.
**1996**, 12, 383–393. [Google Scholar] [CrossRef] - Kirsch, A.; Grinberg, N. The Factorization Method for Inverse Problems; Oxford Lecture Series in Mathematics and Its Applications; Oxford University Press: Oxford, UK, 2008; Volume 36. [Google Scholar]
- Alves, C.J.S.; Ha-Duong, T. On inverse scattering by screens. Inverse Probl.
**1997**, 13, 1161–1176. [Google Scholar] [CrossRef] - Cakoni, F.; Colton, D.; Darrigrand, E. The inverse electromagnetic scattering problem for screens. Inverse Probl.
**2003**, 19, 627–642. [Google Scholar] [CrossRef] [Green Version] - Colton, D.; Sleeman, B. Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math.
**1983**, 31, 253–259. [Google Scholar] [CrossRef] - Isakov, V. Inverse Problems for Partial Differential Equations, 2nd ed.; Springer: New York, NY, USA, 2006. [Google Scholar]
- Alessandrini, G.; Rondi, L. Determining a sound-soft polyhedral scatterer by a single far-field measurement. Proc. Am. Math. Soc.
**2005**, 35, 1685–1691. [Google Scholar] [CrossRef] - Cheng, J.; Yamamoto, M. Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Probl.
**2003**, 19, 1361–1384. [Google Scholar] [CrossRef] - Elschner, J.; Yamamoto, M. Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave. Inverse Probl.
**2008**, 24, 035004. [Google Scholar] [CrossRef] [Green Version] - Liu, H.; Petrini, M.; Rondi, L.; Xiao, J. Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements. J. Differ. Eq.
**2017**, 262, 1631–1670. [Google Scholar] [CrossRef] [Green Version] - Liu, H.; Rondi, L.; Xiao, J. Mosco convergence for H(curl) spaces, higher integrability for Maxwell’s equations, and stability in direct and inverse EM scattering problems. J. Eur. Math. Soc. (JEMS)
**2019**, 21, 2945–2993. [Google Scholar] [CrossRef] [Green Version] - Liu, H.; Zou, J. Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Probl.
**2006**, 22, 515–524. [Google Scholar] [CrossRef] - Rondi, L. Stable determination of sound-soft polyhedral scatterers by a single measurement. Indiana Univ. Math. J.
**2008**, 57, 1377–1408. [Google Scholar] [CrossRef] [Green Version] - Honda, N.; Nakamura, G.; Sini, M. Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators. Math. Ann.
**2013**, 355, 401–427. [Google Scholar] [CrossRef] - Blåsten, E.; Päivärinta, L.; Sylvester, J. Corners always scatter. Commun. Math. Phys.
**2014**, 331, 725–753. [Google Scholar] [CrossRef] [Green Version] - Päivärinta, L.; Salo, M.; Vesalainen, E.V. Strictly convex corners scatter. Rev. Mat. Iberoam.
**2017**, 33, 1369–1396. [Google Scholar] [CrossRef] [Green Version] - Blåsten, E. Nonradiating sources and transmission eigenfunctions vanish at corners and edges. SIAM J. Math. Anal.
**2018**, 50, 6255–6270. [Google Scholar] - Blåsten, E.; Liu, H. On corners scattering stably, nearly non-scattering interrogating waves, and stable shape determination by a single far-field pattern. Indiana Univ. Math. J.
**2019**, in press. [Google Scholar] - Blåsten, E.; Liu, H. Recovering piecewise constant refractive indices by a single far-field pattern. Inverse Probl.
**2020**. [Google Scholar] [CrossRef] - Hu, G.; Salo, M.; Vesalainen, E. Shape identification in inverse medium scattering problems with a single far-field pattern. SIAM J. Math. Anal.
**2016**, 48, 152–165. [Google Scholar] [CrossRef] [Green Version] - Ikehata, M. Reconstruction of a source domain from the Cauchy data. Inverse Probl.
**1999**, 15, 637–645. [Google Scholar] [CrossRef] - Kusiak, S.; Sylvester, J. The scattering support. Comm. Pure Appl. Math.
**2003**, 56, 1525–1548. [Google Scholar] [CrossRef] - Kusiak, S.; Sylvester, J. The convex scattering support in a background medium. SIAM J. Math. Anal.
**2005**, 36, 1142–1158. [Google Scholar] [CrossRef] [Green Version] - Päivärinta, L.; Rempel, S. A deconvolution problem with the kernel 1/|x| on the plane. Appl. Anal.
**1987**, 26, 105–128. [Google Scholar] [CrossRef] - Päivärinta, L.; Rempel, S. Corner singularities of solutions to Δ
^{±1/2}u=f in two dimensions. Asymptotic Anal.**1992**, 5, 429–460. [Google Scholar] [CrossRef] - Stephan, E.P. Boundary integral equations for screen problems in
**R**^{3}. Integr. Eq. Oper. Theory**1987**, 10, 236–257. [Google Scholar] [CrossRef] - Evans, L.C. Partial Differential Equations, 2nd ed.; Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2010; Volume 19. [Google Scholar]
- Deng, Y.; Liu, H.; Uhlmann, G. On regularized full- and partial-cloaks in acoustic scattering. Comm. Part. Differ. Eq.
**2017**, 42, 821–851. [Google Scholar] [CrossRef] [Green Version]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Blåsten, E.; Päivärinta, L.; Sadique, S.
Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. *Mathematics* **2020**, *8*, 1156.
https://doi.org/10.3390/math8071156

**AMA Style**

Blåsten E, Päivärinta L, Sadique S.
Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. *Mathematics*. 2020; 8(7):1156.
https://doi.org/10.3390/math8071156

**Chicago/Turabian Style**

Blåsten, Emilia, Lassi Päivärinta, and Sadia Sadique.
2020. "Unique Determination of the Shape of a Scattering Screen from a Passive Measurement" *Mathematics* 8, no. 7: 1156.
https://doi.org/10.3390/math8071156