Abstract
The time domain enclosure method is one of the analytical methods for inverse obstacle problems governed by partial differential equations in the time domain. This paper considers the case when the governing equation is given by the Maxwell system and consists of two parts. The first part establishes the base of the time domain enclosure method for the Maxwell system using a single set of the solutions over a finite time interval for a general (isotropic) inhomogeneous medium in the whole space. It is a system of asymptotic inequalities for the indicator function which may enable us to apply the time domain enclosure method to the problem of finding unknown penetrable obstacles embedded in various background media. As a first step of its expected applications, the case when the background medium is homogeneous and isotropic is considered and the time domain enclosure method is realized. This is the second part.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 17K05331
Award Identifier / Grant number: 18H01126
Funding statement: The author was partially supported by Grant-in-Aid for Scientific Research (C)(No. 17K05331) and (B)(No. 18H01126) of Japan Society for the Promotion of Science.
References
[1] H. Ammari, G. Bao and J. L. Fleming, An inverse source problem for Maxwell’s equations in magnetoencephalography, SIAM J. Appl. Math. 62 (2002), no. 4, 1369–1382. 10.1137/S0036139900373927Suche in Google Scholar
[2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd ed., Dover Publications, New York, 1986. Suche in Google Scholar
[3] D. E. Carlson, Linear thermoelasticity, Mechanics of Solids Vol. II, Springer, Berlin (1984), 297–345. 10.1007/978-3-642-69567-4_2Suche in Google Scholar
[4] R. Courant and D. Hilbert, Methoden der Mathematischen Physik. Vol. II, Interscience, New York, 1937. 10.1007/978-3-642-47434-7Suche in Google Scholar
[5] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution problems. I, Vol. 5, Springer, Berlin, 1992. Suche in Google Scholar
[6] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems 15 (1999), no. 5, 1231–1241. 10.1088/0266-5611/15/5/308Suche in Google Scholar
[7] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 8 (2000), no. 4, 367–378. 10.1515/jiip.2000.8.4.367Suche in Google Scholar
[8] M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Appl. Anal. 86 (2007), no. 8, 963–1005. 10.1080/00036810701460834Suche in Google Scholar
[9] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems 26 (2010), no. 5, Article ID 055010. 10.1088/0266-5611/26/5/055010Suche in Google Scholar
[10] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data, Inverse Problems 28 (2012), no. 4, Article ID 045010. 10.1088/0266-5611/28/4/045010Suche in Google Scholar
[11] M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems 31 (2015), no. 8, Article ID 085011. 10.1088/0266-5611/31/8/085011Suche in Google Scholar
[12] M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Probl. Imaging 10 (2016), no. 1, 131–163. 10.3934/ipi.2016.10.131Suche in Google Scholar
[13] M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain, Math. Methods Appl. Sci. 40 (2017), no. 4, 915–927. 10.1002/mma.4021Suche in Google Scholar
[14] M. Ikehata, On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method, Inverse Probl. Imaging 11 (2017), no. 1, 99–123. 10.3934/ipi.2017006Suche in Google Scholar
[15] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator, J. Inverse Ill-Posed Probl. 25 (2017), no. 6, 747–761. 10.1515/jiip-2016-0023Suche in Google Scholar
[16] M. Ikehata, On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body, J. Inverse Ill-Posed Probl. 26 (2018), no. 3, 369–394. 10.1515/jiip-2017-0066Suche in Google Scholar
[17] M. Ikehata, On finding the surface admittance of an obstacle via the time domain enclosure method, Inverse Probl. Imaging 13 (2019), no. 2, 263–284. 10.3934/ipi.2019014Suche in Google Scholar
[18] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: V. Using time-reversal invariance, J. Inverse Ill-Posed Probl. 27 (2019), no. 1, 133–149. 10.1515/jiip-2018-0046Suche in Google Scholar
[19] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data, J. Inverse Ill-Posed Probl. 28 (2020), no. 3, 349–366. 10.1515/jiip-2019-0039Suche in Google Scholar
[20] M. Ikehata and M. Kawashita, On finding a buried obstacle in a layered medium via the time domain enclosure method, Inverse Probl. Imaging 12 (2018), no. 5, 1173–1198. 10.3934/ipi.2018049Suche in Google Scholar
[21] M. Ikehata, M. Kawashita and W. Kawashita, On finding a buried obstacle in a layered medium via the time domain enclosure method in the case of possible total reflection phenomena, Inverse Probl. Imaging 13 (2019), no. 5, 959–981. 10.3934/ipi.2019043Suche in Google Scholar
[22] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl. 36, Oxford University, Oxford, 2008. 10.1093/acprof:oso/9780199213535.001.0001Suche in Google Scholar
[23] H.-M. Nguyen and V. Vinoles, Electromagnetic wave propagation in media consisting of dispersive metamaterials, C. R. Math. Acad. Sci. Paris 356 (2018), no. 7, 757–775. 10.1016/j.crma.2018.05.012Suche in Google Scholar
[24] H.-M. Nguyen and M. S. Vogelius, Approximate cloaking using transformation optics for acoustic and electromagnetic waves, Acta Math. Vietnam. 45 (2020), no. 1, 261–280. 10.1007/s40306-019-00334-5Suche in Google Scholar
[25] B. O’Neill, Elementary Differential Geometry, 2nd ed., Elsevier/Academic Press, Amsterdam, 2006. Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Lipschitz stability in inverse source problems for degenerate/singular parabolic equations by the Carleman estimate
- The asymptotic behavior of the solution of an inverse problem for the pseudoparabolic equation
- On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement II. The non-radial case
- Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation
- On finding a penetrable obstacle using a single electromagnetic wave in the time domain
- Direct and inverse problems for time-fractional heat equation generated by Dunkl operator
- Agent-based mathematical model of COVID-19 spread in Novosibirsk region: Identifiability, optimization and forecasting
- Certain inverse uniqueness from the quotients of scattering coefficients
- On recovery of an unbounded bi-periodic interface for the inverse fluid-solid interaction scattering problem
- Approximate Lipschitz stability for phaseless inverse scattering with background information
- The mean field games system: Carleman estimates, Lipschitz stability and uniqueness
Artikel in diesem Heft
- Frontmatter
- Lipschitz stability in inverse source problems for degenerate/singular parabolic equations by the Carleman estimate
- The asymptotic behavior of the solution of an inverse problem for the pseudoparabolic equation
- On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement II. The non-radial case
- Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation
- On finding a penetrable obstacle using a single electromagnetic wave in the time domain
- Direct and inverse problems for time-fractional heat equation generated by Dunkl operator
- Agent-based mathematical model of COVID-19 spread in Novosibirsk region: Identifiability, optimization and forecasting
- Certain inverse uniqueness from the quotients of scattering coefficients
- On recovery of an unbounded bi-periodic interface for the inverse fluid-solid interaction scattering problem
- Approximate Lipschitz stability for phaseless inverse scattering with background information
- The mean field games system: Carleman estimates, Lipschitz stability and uniqueness