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Stable determination of a boundary coefficient in an elliptic equation. (English) Zbl 1155.35108

The paper is motivated by a practical problem of corrosion detection, where the corroded part is inaccessible to direct measurements. The mathematical problem reduces to a Cauchy problem for a second order elliptic problem. In this paper a logarithmic stability estimate for the function representing the amount of corrosion in dependence of a single indirect boundary measurement is derived. The stability result is based on Carleman estimates as used by A. L. Bukhgeim.

MSC:

35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B35 Stability in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
82D35 Statistical mechanics of metals
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References:

[1] DOI: 10.1088/0266-5611/19/4/312 · Zbl 1050.35134 · doi:10.1088/0266-5611/19/4/312
[2] Bukhgeim A. L., J. Inv. Ill-Posed Probl. 1 pp 17–
[3] DOI: 10.1088/0266-5611/20/1/003 · Zbl 1055.35135 · doi:10.1088/0266-5611/20/1/003
[4] DOI: 10.1088/0266-5611/15/6/303 · Zbl 0943.35100 · doi:10.1088/0266-5611/15/6/303
[5] Cheng J., Numer. Math. J. Chin. Univ. 14 pp 252–
[6] DOI: 10.1088/0266-5611/16/4/101 · Zbl 0957.65052 · doi:10.1088/0266-5611/16/4/101
[7] DOI: 10.1515/1569394042248247 · Zbl 1081.78006 · doi:10.1515/1569394042248247
[8] Choulli M., J. Inv. Ill-Posed Probl. 10 pp 601–
[9] DOI: 10.1088/0266-5611/15/1/008 · Zbl 0922.35188 · doi:10.1088/0266-5611/15/1/008
[10] DOI: 10.1007/978-3-642-61798-0 · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[11] DOI: 10.1088/0266-5611/13/4/006 · Zbl 0882.35133 · doi:10.1088/0266-5611/13/4/006
[12] DOI: 10.1007/s000000050114 · Zbl 0908.35137 · doi:10.1007/s000000050114
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