On uniqueness in the inverse transmission scattering problem. (English) Zbl 0728.35148
In this paper uniqueness results are proved for the inverse scattering problems where the unknown scatterer D is a bounded open set and some coefficients of an elliptic equation are unknown as well. Let \(D^ i\) be the bounded open set in \(R^ n\), \(D^ e=R^ n\setminus D^ i\), \(u=u^ e\) in \(D^ e\), \(u=u^ i\) in \(D^ i\), \(\chi\) (D) the characteristic function of D, \(a=1+(\mu -1)\chi (D)\), \(c=1+(\rho -1)\chi (D)\). Further, let u be a solution of \(div(a\nabla u)+k^ 2cu=0\), satisfying \(u^ i=u^ e\), \(\partial u^ e/\partial N=\mu \partial u^ i/\partial N\) on \(\partial D\), \(u^ e(x)=\exp (ix\cdot \xi +u^{e_ 0}(x)\), \(| \xi | =k\), \(r^{(n-1)/2}(\partial u^{e_ 0}/\partial r-iku^{e_ 0})\to 0\) for \(r\to \infty\). The author applies ideas of Nachman, Sylvester, Uhlmann and own results for this special problem under consideration.
Reviewer: G.Anger (Berlin)
MSC:
| 35R30 | Inverse problems for PDEs |
| 35P25 | Scattering theory for PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
References:
| [1] | Angell T. S., Inverse Problems 3 pp 149– (1987) · Zbl 0658.35093 · doi:10.1088/0266-5611/3/2/004 |
| [2] | Colton D., Integral Equations Methods is Scattering Theory (1983) |
| [3] | Isakov V., Soviet Math. Dokl 17 pp 1338– (1976) |
| [4] | Isakov V., Comm. Pure Appl. Math. 41 pp 865– (1988) · Zbl 0676.35082 · doi:10.1002/cpa.3160410702 |
| [5] | V. Isakov Completeness of products and inverse problems for PDE] J. of Diff. Equations (in Print). |
| [6] | Isakov V., AMS math. |
| [7] | Lax P., Scattering Theory (1967) |
| [8] | Lax P., Comm. Pure Appl. Math. 30 pp 195– (1977) · Zbl 0335.35075 · doi:10.1002/cpa.3160300204 |
| [9] | Majda A., Comm. Pure Appl. Math. 30 pp 169– (1977) |
| [10] | Taylor M., Comm. in part. Diff. Equat. 2 (4) pp 395– (1977) · Zbl 0357.35010 · doi:10.1080/03605307708820033 |
| [11] | Melrose R., Comm. Pure Appl. Math. 33 (4) pp 461– (1980) · Zbl 0435.35066 · doi:10.1002/cpa.3160330402 |
| [12] | Miranda C., Ergebn. der Math. (1970) |
| [13] | Morrey C. B., Die Grundlehren der Math. 130 (1966) |
| [14] | Morrey C. B., Ann. Math. 128 pp 531– (1988) · Zbl 0675.35084 · doi:10.2307/1971435 |
| [15] | Nachman A., Comm. Math. Physics 115 pp 595– (1973) · Zbl 0644.35095 · doi:10.1007/BF01224129 |
| [16] | Phillips R., Proc. Symp. in Pure Math. 130 pp 153– (1966) |
| [17] | Stampacchia G., :Les Presses de l’Univ. de Montreal 23 (1966) |
| [18] | Sylvester J., Ann. Math. 125 pp 153– (1987) · Zbl 0625.35078 · doi:10.2307/1971291 |
| [19] | Sylvester J., SIAM Proc. Series 125 pp 99– (1990) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
