Unique Localization of Unknown Boundaries in a Conducting Medium from Boundary Measurements

Bruno Canuto

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 7, page 1-22
  • ISSN: 1292-8119

Abstract

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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 ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T0,σ) measured at a given time T0 for σ ∈ Aout ⊂ 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.

How to cite

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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 7 (2010): 1-22. <http://eudml.org/doc/90591>.

@article{Canuto2010,
abstract = { 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 ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T0,σ) measured at a given time T0 for σ ∈ Aout ⊂ A, we prove that I and D are uniquely localized from knowledge of all possible pairs of input-output data $(g,u(T_0)_\{\mid A_\{\{\rm out\}\}\})$. The same result holds when a mean value of the temperature is measured over a small interval of time. },
author = {Canuto, Bruno},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Inverse boundary value problems; cavities; corrosion; uniqueness.; inverse boundary value problems; corrosion; uniqueness},
language = {eng},
month = {3},
pages = {1-22},
publisher = {EDP Sciences},
title = {Unique Localization of Unknown Boundaries in a Conducting Medium from Boundary Measurements},
url = {http://eudml.org/doc/90591},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Canuto, Bruno
TI - Unique Localization of Unknown Boundaries in a Conducting Medium from Boundary Measurements
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 1
EP - 22
AB - 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 ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T0,σ) measured at a given time T0 for σ ∈ Aout ⊂ A, we prove that I and D are uniquely localized from knowledge of all possible pairs of input-output data $(g,u(T_0)_{\mid A_{{\rm out}}})$. The same result holds when a mean value of the temperature is measured over a small interval of time.
LA - eng
KW - Inverse boundary value problems; cavities; corrosion; uniqueness.; inverse boundary value problems; corrosion; uniqueness
UR - http://eudml.org/doc/90591
ER -

References

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  2. B. Canuto and O. Kavian, Determining Coefficients in a Class of Heat Equations via Boundary Measurements. SIAM J. Math. Anal. (to appear).  Zbl0981.35096
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  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).  
  6. Rakesh and W.W. Symes, Uniqueness for an Inverse Problem for the Wave Equation. Comm. Partial Differential Equations13 (1988) 87-96.  Zbl0667.35071
  7. J.-C. Saut and B. Scheurer, Unique Continuation for Some Evolution Equations. J. Differential Equations66 (1987) 118-139.  Zbl0631.35044
  8. S. Vessella, Stability Estimates in an Inverse Problem for a Three-Dimensional Heat Equation. SIAM J. Math. Anal.28 (1997) 1354-1370.  Zbl0888.35130
  9. S. Vessella, Private Comunication.  

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