### Abstract

We consider the inverse problem to the refraction problem div((1 + (k-1)χD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω. The inverse problem is to determine the size and the location of an unknown object D from the boundary measurement Λ_{D}(g) = u|_{∂Ω}. The results of this paper are twofold: stability and estimation of size of D. We first obtain upper and lower bounds of the size of D by comparing Λ_{D}(g) with the Dirichlet data corresponding to the harmonic equation with the same Neumann data g. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of D) of the gradient of the solution u to the refraction problem with the Neumann data g satisfying some mild conditions.

Original language | English |
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Pages (from-to) | 1389-1405 |

Number of pages | 17 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 28 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1997 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*28*(6), 1389-1405. https://doi.org/10.1137/S0036141096299375

}

*SIAM Journal on Mathematical Analysis*, vol. 28, no. 6, pp. 1389-1405. https://doi.org/10.1137/S0036141096299375

**The inverse conductivity problem with one measurement : Stability and estimation of size.** / Kang, Hyeonbae; Seo, Jin Keun; Sheen, Dongwoo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The inverse conductivity problem with one measurement

T2 - Stability and estimation of size

AU - Kang, Hyeonbae

AU - Seo, Jin Keun

AU - Sheen, Dongwoo

PY - 1997/1/1

Y1 - 1997/1/1

N2 - We consider the inverse problem to the refraction problem div((1 + (k-1)χD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω. The inverse problem is to determine the size and the location of an unknown object D from the boundary measurement ΛD(g) = u|∂Ω. The results of this paper are twofold: stability and estimation of size of D. We first obtain upper and lower bounds of the size of D by comparing ΛD(g) with the Dirichlet data corresponding to the harmonic equation with the same Neumann data g. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of D) of the gradient of the solution u to the refraction problem with the Neumann data g satisfying some mild conditions.

AB - We consider the inverse problem to the refraction problem div((1 + (k-1)χD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω. The inverse problem is to determine the size and the location of an unknown object D from the boundary measurement ΛD(g) = u|∂Ω. The results of this paper are twofold: stability and estimation of size of D. We first obtain upper and lower bounds of the size of D by comparing ΛD(g) with the Dirichlet data corresponding to the harmonic equation with the same Neumann data g. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of D) of the gradient of the solution u to the refraction problem with the Neumann data g satisfying some mild conditions.

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U2 - 10.1137/S0036141096299375

DO - 10.1137/S0036141096299375

M3 - Article

AN - SCOPUS:0001607563

VL - 28

SP - 1389

EP - 1405

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -