Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance


Authors: V. Bacchelli, M. Di Cristo, E. Sincich and S. Vessella
Journal: Trans. Amer. Math. Soc. 366 (2014), 3965-3995
MSC (2010): Primary 35R30, 35R25, 35R35
DOI: https://doi.org/10.1090/S0002-9947-2014-05807-8
Published electronically: April 7, 2014
MathSciNet review: 3206449
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of determining an unaccessible part of the boundary of a conductor by means of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.


References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
  • [2] Giovanni Alessandrini, Elena Beretta, Edi Rosset, and Sergio Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 4, 755-806. MR 1822407 (2002b:35196)
  • [3] Valeria Bacchelli, Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition, Inverse Problems 25 (2009), no. 1, 015004, 4. MR 2465336 (2009m:35531), https://doi.org/10.1088/0266-5611/25/1/015004
  • [4] E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data, SIAM J. Math. Anal. 30 (1999), no. 1, 220-232. MR 1656995 (99j:35229), https://doi.org/10.1137/S0036141097325733
  • [5] P. Bison, M. Ceseri, D. Fasino, and G. Inglese, Active infrared thermography in non-destructive evaluation of surface corrosion. II. Heat exchange between specimen and environment, Applied and industrial mathematics in Italy II, Ser. Adv. Math. Appl. Sci., vol. 75, World Sci. Publ., Hackensack, NJ, 2007, pp. 161-171. MR 2367569, https://doi.org/10.1142/9789812709394_0015
  • [6] P. Bison, M. Ceseri, D. Fasino, and G. Inglese, Domain derivative approach to active infrared thermography, Inverse Probl. Sci. Eng. 18 (2010), no. 7, 873-889. MR 2743228 (2011i:35252), https://doi.org/10.1080/17415977.2010.492511
  • [7] Fioralba Cakoni and Rainer Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging 1 (2007), no. 2, 229-245. MR 2282267 (2008j:35181), https://doi.org/10.3934/ipi.2007.1.229
  • [8] B. Canuto, E. Rosset, and S. Vessella, Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries, Trans. Amer. Math. Soc. 354 (2002), no. 2, 491-535. MR 1862557 (2002k:35311), https://doi.org/10.1090/S0002-9947-01-02860-4
  • [9] Michele Di Cristo and Luca Rondi, Examples of exponential instability for inverse inclusion and scattering problems, Inverse Problems 19 (2003), no. 3, 685-701. MR 1984884 (2004d:35247), https://doi.org/10.1088/0266-5611/19/3/313
  • [10] Michele Di Cristo, Luca Rondi, and Sergio Vessella, Stability properties of an inverse parabolic problem with unknown boundaries, Ann. Mat. Pura Appl. (4) 185 (2006), no. 2, 223-255. MR 2214134 (2006m:35378), https://doi.org/10.1007/s10231-005-0152-x
  • [11] Luis Escauriaza and Francisco Javier Fernández, Unique continuation for parabolic operators, Ark. Mat. 41 (2003), no. 1, 35-60. MR 1971939 (2004b:35136), https://doi.org/10.1007/BF02384566
  • [12] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [13] Gabriele Inglese, An inverse problem in corrosion detection, Inverse Problems 13 (1997), no. 4, 977-994. MR 1463588, https://doi.org/10.1088/0266-5611/13/4/006
  • [14] Victor Isakov, On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data, Inverse Probl. Imaging 2 (2008), no. 1, 151-165. MR 2375327 (2009h:35444), https://doi.org/10.3934/ipi.2008.2.151
  • [15] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822 (39 #3159b)
  • [16] Gary M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), no. 2, 329-352. MR 779924 (87j:35101)
  • [17] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. MR 1465184 (98k:35003)
  • [18] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 0350178 (50 #2671)
  • [19] Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. MR 0159139 (28 #2357)
  • [20] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740. MR 0288405 (44 #5603)
  • [21] Carlo Domenico Pagani and Dario Pierotti, Identifiability problems of defects with the Robin condition, Inverse Problems 25 (2009), no. 5, 055007, 12. MR 2501025 (2010k:35539), https://doi.org/10.1088/0266-5611/25/5/055007
  • [22] Eva Sincich, Stability for the determination of unknown boundary and impedance with a Robin boundary condition, SIAM J. Math. Anal. 42 (2010), no. 6, 2922-2943. MR 2745797 (2011m:35425), https://doi.org/10.1137/100788343
  • [23] Sergio Vessella, Stability estimates in an inverse problem for a three-dimensional heat equation, SIAM J. Math. Anal. 28 (1997), no. 6, 1354-1370. MR 1474218 (99d:35178), https://doi.org/10.1137/S0036141095294262
  • [24] Sergio Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse Problems 24 (2008), no. 2, 023001, 81. MR 2408537 (2009c:35192), https://doi.org/10.1088/0266-5611/24/2/023001
  • [25] M. Vollmer, K. P. Möllmann, Infrared thermal imaging: fundamental, research and applications, Wiley-VCH Verlag GmbH & Co KGaA, Weinheim, 2010.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35R30, 35R25, 35R35

Retrieve articles in all journals with MSC (2010): 35R30, 35R25, 35R35


Additional Information

V. Bacchelli
Affiliation: Department of Mathematics, Politecnico di Milano, 20100 Milan, Italy
Email: valeria.bacchelli@polimi.it

M. Di Cristo
Affiliation: Department of Mathematics, Politecnico di Milano, 20100 Milan, Italy
Email: michele.dicristo@polimi.it

E. Sincich
Affiliation: Department of Mathematics, Università di Trieste, 34014 Trieste, Italy
Address at time of publication: Laboratory for Multiphase Processes, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
Email: esincich@units.it, eva.sincich@ung.si

S. Vessella
Affiliation: Department of Mathematics, Università di Firenze, 50121 Florence, Italy
Email: sergio.vessella@unifi.it

DOI: https://doi.org/10.1090/S0002-9947-2014-05807-8
Received by editor(s): September 1, 2011
Received by editor(s) in revised form: January 23, 2012
Published electronically: April 7, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society