Unique localization of unknown boundaries in a conducting medium from boundary measurements

Canuto, Bruno

doi:10.1051/cocv:2002001

Canuto, Bruno

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 1-22

Résumé

We consider the problem of localizing an inaccessible piece $I$ of the boundary of a conducting medium $Ω$ , and a cavity $D$ contained in $Ω$ , from boundary measurements on the accessible part $A$ of $\partial Ω$ . Assuming that $g (t, σ)$ is the given thermal flux for $(t, σ) \in (0, T) \times A$ , and that the corresponding output datum is the temperature $u (T_{0}, σ)$ measured at a given time $T_{0}$ for $σ \in A_{out} \subset A$ , we prove that $I$ and $D$ are uniquely localized from knowledge of all possible pairs of input-output data $(g, u {(T_{0})}_{∣ A_{out}})$ . The same result holds when a mean value of the temperature is measured over a small interval of time.

MR Zbl

DOI : 10.1051/cocv:2002001

Classification : 35R30
Keywords: inverse boundary value problems, cavities, corrosion, uniqueness

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     author = {Canuto, Bruno},
     title = {Unique localization of unknown boundaries in a conducting medium from boundary measurements},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--22},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {7},
     doi = {10.1051/cocv:2002001},
     mrnumber = {1925019},
     zbl = {1053.35128},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2002001/}
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Canuto, Bruno. Unique localization of unknown boundaries in a conducting medium from boundary measurements. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 1-22. doi: 10.1051/cocv:2002001

Bibliographie
Cité par

[1] K. Bryan and L.F. Caudill, An Inverse Problem in Thermal Imaging. SIAM J. Appl. Math. 56 (1996) 715-735. | Zbl | MR

[2] B. Canuto and O. Kavian, Determining Coefficients in a Class of Heat Equations via Boundary Measurements. SIAM J. Math. Anal. (to appear). | Zbl | MR

[3] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1. Wiley, New York (1953). | Zbl

[4] N. Garofalo and F.H. Lin, Monotonicity Properties of Variational Integrals, $A_{p}$ Weights and Unique Continuation. Indiana Univ. Math. J. 35 (1986) 245-268. | Zbl | MR

[5] O.A. Ladyzhenskaja, V.A. Solonnikov and N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. AMS, Providence, RI, Trans. Math. Monographs 23 (1968). | Zbl

[6] Rakesh and W.W. Symes, Uniqueness for an Inverse Problem for the Wave Equation. Comm. Partial Differential Equations 13 (1988) 87-96. | Zbl | MR

[7] J.-C. Saut and B. Scheurer, Unique Continuation for Some Evolution Equations. J. Differential Equations 66 (1987) 118-139. | Zbl | MR

[8] S. Vessella, Stability Estimates in an Inverse Problem for a Three-Dimensional Heat Equation. SIAM J. Math. Anal. 28 (1997) 1354-1370. | Zbl | MR

[9] S. Vessella, Private Comunication.

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